the matching problem in general graphs is in quasi-ncjakub.tarnawski.org/slides_focs_2017.pdffenner,...
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![Page 1: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/1.jpg)
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The Matching Problemin General Graphs is in quasi-NC
Jakub Tarnawski
joint work with Ola Svensson
October 16, 2017
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 2: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/2.jpg)
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Perfect matching problem
Given a graph, can we pair upall vertices using edges?
very tough instance:graph is non-bipartite!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 3: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/3.jpg)
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Perfect matching problem
Given a graph, can we pair upall vertices using edges?
very tough instance:graph is non-bipartite!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 4: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/4.jpg)
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Perfect matching problem
Given a graph, can we pair upall vertices using edges?
very tough instance:graph is non-bipartite!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 5: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/5.jpg)
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Perfect matching problem
Benchmark problem in computer science
Algorithms:I bipartite: Jacobi [XIX century, weighted!]
I general: Edmonds [1965]
I since then, tons of researchand still active
I many models of computation:monotone circuits, extended formulations,parallel, distributed, streaming/sublinear, ...
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 6: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/6.jpg)
3/30
Perfect matching problem
Benchmark problem in computer science
Algorithms:I bipartite: Jacobi [XIX century, weighted!]
I general: Edmonds [1965]
I since then, tons of researchand still active
I many models of computation:monotone circuits, extended formulations,parallel, distributed, streaming/sublinear, ...
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 7: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/7.jpg)
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Parallel complexity
Class NC: problems that paralellize completely
poly n processors
poly log n time
it’s in Randomized NC
Main open question: is matching in NC?
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 8: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/8.jpg)
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Parallel complexity
Class NC: problems that paralellize completely
poly n processors
poly log n time
it’s in Randomized NC
Main open question: is matching in NC?Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 9: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/9.jpg)
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Parallel complexity
Class NC: problems that paralellize completely
poly n processors
poly log n time
it’s in Randomized NC
Main open question: is matching in NC?Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 10: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/10.jpg)
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Parallel complexity
I Matching is in Randomized NC [Lovasz 1979]:has randomized algorithm that uses:I polynomially many processorsI polylog time
I Search version is in Randomized NC:I [Karp, Upfal, Wigderson 1986]I [Mulmuley, Vazirani, Vazirani 1987]
Can we derandomize all efficient computation?
Can we derandomize one of these algorithms?
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 11: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/11.jpg)
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Parallel complexity
I Matching is in Randomized NC [Lovasz 1979]:has randomized algorithm that uses:I polynomially many processorsI polylog time
I Search version is in Randomized NC:I [Karp, Upfal, Wigderson 1986]I [Mulmuley, Vazirani, Vazirani 1987]
Can we derandomize all efficient computation?
Can we derandomize one of these algorithms?
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 12: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/12.jpg)
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Parallel complexity
I Matching is in Randomized NC [Lovasz 1979]:has randomized algorithm that uses:I polynomially many processorsI polylog time
I Search version is in Randomized NC:I [Karp, Upfal, Wigderson 1986]I [Mulmuley, Vazirani, Vazirani 1987]
Can we derandomize all efficient computation?
Can we derandomize one of these algorithms?
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 13: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/13.jpg)
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Is matching in NC?Yes, for restricted graph classes:
I bipartite regular [Lev, Pippenger, Valiant 1981]I bipartite convex [Dekel, Sahni 1984]I incomparability graphs [Kozen, Vazirani, Vazirani 1985]I bipartite graphs with small number of perfect matchings [Grigoriev, Karpinski 1987]I claw-free [Chrobak, Naor, Novick 1989]I K3,3-free (decision version) [Vazirani 1989]I planar bipartite [Miller, Naor 1989]I dense [Dahlhaus, Hajnal, Karpinski 1993]I strongly chordal [Dahlhaus, Karpinski 1998]I P4-tidy [Parfenoff 1998]I bipartite small genus [Mahajan, Varadarajan 2000]I graphs with small number of perfect matchings [Agrawal, Hoang, Thierauf 2006]I planar (search version) [Anari, Vazirani 2017]
but not known for:I general
I bipartite
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 14: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/14.jpg)
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Is matching in NC?Yes, for restricted graph classes:
I bipartite regular [Lev, Pippenger, Valiant 1981]I bipartite convex [Dekel, Sahni 1984]I incomparability graphs [Kozen, Vazirani, Vazirani 1985]I bipartite graphs with small number of perfect matchings [Grigoriev, Karpinski 1987]I claw-free [Chrobak, Naor, Novick 1989]I K3,3-free (decision version) [Vazirani 1989]I planar bipartite [Miller, Naor 1989]I dense [Dahlhaus, Hajnal, Karpinski 1993]I strongly chordal [Dahlhaus, Karpinski 1998]I P4-tidy [Parfenoff 1998]I bipartite small genus [Mahajan, Varadarajan 2000]I graphs with small number of perfect matchings [Agrawal, Hoang, Thierauf 2006]I planar (search version) [Anari, Vazirani 2017]
but not known for:I general
I bipartite
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 15: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/15.jpg)
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Is matching in NC?Yes, for restricted graph classes:
I bipartite regular [Lev, Pippenger, Valiant 1981]I bipartite convex [Dekel, Sahni 1984]I incomparability graphs [Kozen, Vazirani, Vazirani 1985]I bipartite graphs with small number of perfect matchings [Grigoriev, Karpinski 1987]I claw-free [Chrobak, Naor, Novick 1989]I K3,3-free (decision version) [Vazirani 1989]I planar bipartite [Miller, Naor 1989]I dense [Dahlhaus, Hajnal, Karpinski 1993]I strongly chordal [Dahlhaus, Karpinski 1998]I P4-tidy [Parfenoff 1998]I bipartite small genus [Mahajan, Varadarajan 2000]I graphs with small number of perfect matchings [Agrawal, Hoang, Thierauf 2006]I planar (search version) [Anari, Vazirani 2017]
but not known for:I generalI bipartite
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 16: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/16.jpg)
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Is matching in NC?
Fenner, Gurjar and Thierauf [2015] showed:I Bipartite matching is in quasi-NC
(npoly log n processors, poly log n time, deterministic)
I Approach fails for non-bipartite graphs
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 17: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/17.jpg)
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Is matching in NC?
Fenner, Gurjar and Thierauf [2015] showed:I Bipartite matching is in quasi-NC
(npoly log n processors, poly log n time, deterministic)
I Approach fails for non-bipartite graphs
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 18: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/18.jpg)
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Our result
We show: general matching is in quasi-NC:I npoly log n processorsI poly log n timeI deterministic
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 19: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/19.jpg)
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Outline
1 Isolating weight functions[Mulmuley, Vazirani, Vazirani 1987]
2 Bipartite case[Fenner, Gurjar, Thierauf 2015]
3 Difficulties of general case& our approach
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 20: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/20.jpg)
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1. Isolating weight functions[Mulmuley, Vazirani, Vazirani 1987]
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 21: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/21.jpg)
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Isolating weight functions
How to solve unweighted problem?
Make it weighted
But we choose the weight function – do it smartly!
Weight function w : E → Z+ is isolatingif there is a unique min-weight perfect matching
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 22: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/22.jpg)
11/30
Isolating weight functions
How to solve unweighted problem?
Make it weighted
But we choose the weight function – do it smartly!
Weight function w : E → Z+ is isolatingif there is a unique min-weight perfect matching
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 23: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/23.jpg)
11/30
Isolating weight functions
How to solve unweighted problem?
Make it weighted
But we choose the weight function – do it smartly!
Weight function w : E → Z+ is isolatingif there is a unique min-weight perfect matching
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 24: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/24.jpg)
11/30
Isolating weight functions
How to solve unweighted problem?
Make it weighted
But we choose the weight function – do it smartly!
Weight function w : E → Z+ is isolatingif there is a unique min-weight perfect matching
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 25: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/25.jpg)
11/30
Isolating weight functions
How to solve unweighted problem?
Make it weighted
But we choose the weight function – do it smartly!
Weight function w : E → Z+ is isolatingif there is a unique min-weight perfect matching
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 26: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/26.jpg)
11/30
Isolating weight functions
How to solve unweighted problem?
Make it weighted
But we choose the weight function – do it smartly!
Weight function w : E → Z+ is isolatingif there is a unique min-weight perfect matching
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 27: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/27.jpg)
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[Mulmuley, Vazirani, Vazirani 1987]
isolating weight function
matching
determinant computationin NC
random sampling
Isolation Lemma
something deterministic?
?
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 28: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/28.jpg)
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[Mulmuley, Vazirani, Vazirani 1987]
isolating weight function
matching
determinant computationin NC
random sampling
Isolation Lemma
something deterministic?
?
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 29: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/29.jpg)
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[Mulmuley, Vazirani, Vazirani 1987]
isolating weight function
matching
determinant computationin NC
random sampling
Isolation Lemma
something deterministic?
?
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 30: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/30.jpg)
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Isolation Lemma
Weight function w : E → Z+ is isolatingif there is a unique min-weight perfect matching
Isolation Lemma [MVV 1987]If each w (e) picked randomly from 1, 2, ..., n3,then P[w isolating] ≥ 1− 1
n
I holds more generally,for any set family in place of matchings!
I many applications in complexity theory
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 31: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/31.jpg)
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Isolation Lemma
Weight function w : E → Z+ is isolatingif there is a unique min-weight perfect matching
Isolation Lemma [MVV 1987]If each w (e) picked randomly from 1, 2, ..., n3,then P[w isolating] ≥ 1− 1
n
I holds more generally,for any set family in place of matchings!
I many applications in complexity theory
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 32: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/32.jpg)
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Derandomize the Isolation Lemma
I Challenge:get an isolating weight functiondeterministically in NC
I We prove:can construct nO(log2 n) weight functions in quasi-NCsuch that one of them is isolating
I We do it without looking at the graph
I Implies: matching is in quasi-NC
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 33: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/33.jpg)
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Derandomize the Isolation Lemma
I Challenge:get an isolating weight functiondeterministically in NC
I We prove:can construct nO(log2 n) weight functions in quasi-NCsuch that one of them is isolating
I We do it without looking at the graph
I Implies: matching is in quasi-NC
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 34: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/34.jpg)
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2. Bipartite case[Fenner, Gurjar, Thierauf 2015]
Goal: how to construct nO(log n) weight functionssuch that one of them is isolating?
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 35: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/35.jpg)
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Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 36: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/36.jpg)
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Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 37: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/37.jpg)
16/30
Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 38: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/38.jpg)
16/30
Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 39: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/39.jpg)
16/30
Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 40: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/40.jpg)
16/30
Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 41: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/41.jpg)
16/30
Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 42: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/42.jpg)
16/30
Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 43: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/43.jpg)
16/30
Isolating matchingsWhat if w is not isolating?
I there are perfect matchings M , M ′with w (M) = w (M ′) minimum
I symmetric difference= alternating cycles
I in each cycle C ,w (GREEN) = w (RED)(otherwise could get lighter matching)
I define discrepancy of a cycle:dw (C ) := w (GREEN)− w (RED)
I dw (C ) = 0
C
If (∀C ) dw (C ) 6= 0, then w isolating!
New objective: assign 6= 0 discrepancy to every cycle
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 44: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/44.jpg)
17/30
Removing cycles
New objective: assign 6= 0 discrepancy to every cycle
LemmaFor any n4 cycles,can find a weight function w thatassigns all of them 6= 0 discrepancy.
If ≤ n4 cycles in the graph: done!
Not so easy, but we can cope with all 4-cycles.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 45: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/45.jpg)
17/30
Removing cycles
New objective: assign 6= 0 discrepancy to every cycle
LemmaFor any n4 cycles,can find a weight function w thatassigns all of them 6= 0 discrepancy.
If ≤ n4 cycles in the graph: done!
Not so easy, but we can cope with all 4-cycles.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 46: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/46.jpg)
17/30
Removing cycles
New objective: assign 6= 0 discrepancy to every cycle
LemmaFor any n4 cycles,can find a weight function w thatassigns all of them 6= 0 discrepancy.
If ≤ n4 cycles in the graph: done!
Not so easy, but we can cope with all 4-cycles.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 47: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/47.jpg)
17/30
Removing cycles
New objective: assign 6= 0 discrepancy to every cycle
LemmaFor any n4 cycles,can find a weight function w thatassigns all of them 6= 0 discrepancy.
If ≤ n4 cycles in the graph: done!
Not so easy, but we can cope with all 4-cycles.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 48: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/48.jpg)
17/30
Removing cycles
New objective: assign 6= 0 discrepancy to every cycle
LemmaFor any n4 cycles,can find a weight function w thatassigns all of them 6= 0 discrepancy.
If ≤ n4 cycles in the graph: done!
Not so easy, but we can cope with all 4-cycles.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 49: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/49.jpg)
18/30
Removing cycles
Active subgraph:those edges that are in a min-weight perfect matching
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
0
3
0
1 1
1 1dw (C1) = 1 6= 0dw (C2) = 1 6= 0
C2
C1
=⇒
By assigning 6= 0 discrepancy to 4-cycles, we can remove them.Then continue restricted to the smaller active subgraph!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 50: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/50.jpg)
18/30
Removing cycles
Active subgraph:those edges that are in a min-weight perfect matching
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
0
3
0
1 1
1 1
dw (C1) = 1 6= 0dw (C2) = 1 6= 0
C2
C1
=⇒
By assigning 6= 0 discrepancy to 4-cycles, we can remove them.Then continue restricted to the smaller active subgraph!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 51: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/51.jpg)
18/30
Removing cycles
Active subgraph:those edges that are in a min-weight perfect matching
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
0
3
0
1 1
1 1dw (C1) = 1 6= 0dw (C2) = 1 6= 0
C2
C1
=⇒
By assigning 6= 0 discrepancy to 4-cycles, we can remove them.Then continue restricted to the smaller active subgraph!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 52: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/52.jpg)
18/30
Removing cycles
Active subgraph:those edges that are in a min-weight perfect matching
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
0
3
0
1 1
1 1dw (C1) = 1 6= 0dw (C2) = 1 6= 0
C2
C1
=⇒
By assigning 6= 0 discrepancy to 4-cycles, we can remove them.Then continue restricted to the smaller active subgraph!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 53: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/53.jpg)
18/30
Removing cycles
Active subgraph:those edges that are in a min-weight perfect matching
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
0
3
0
1 1
1 1dw (C1) = 1 6= 0dw (C2) = 1 6= 0
C2
C1
=⇒
By assigning 6= 0 discrepancy to 4-cycles, we can remove them.Then continue restricted to the smaller active subgraph!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 54: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/54.jpg)
18/30
Removing cycles
Active subgraph:those edges that are in a min-weight perfect matching
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
0
3
0
1 1
1 1dw (C1) = 1 6= 0dw (C2) = 1 6= 0
C2
C1
=⇒
By assigning 6= 0 discrepancy to 4-cycles, we can remove them.Then continue restricted to the smaller active subgraph!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 55: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/55.jpg)
18/30
Removing cycles
Active subgraph:those edges that are in a min-weight perfect matching
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
0
3
0
1 1
1 1dw (C1) = 1 6= 0dw (C2) = 1 6= 0
C2
C1
=⇒
By assigning 6= 0 discrepancy to 4-cycles, we can remove them.Then continue restricted to the smaller active subgraph!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 56: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/56.jpg)
18/30
Removing cycles
Active subgraph:those edges that are in a min-weight perfect matching
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
0
3
0
1 1
1 1dw (C1) = 1 6= 0dw (C2) = 1 6= 0
C2
C1
=⇒
By assigning 6= 0 discrepancy to 4-cycles, we can remove them.Then continue restricted to the smaller active subgraph!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 57: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/57.jpg)
19/30
Isolating in stages
Crucial idea:I Can find w1 such that 4-cycles
are assigned 6= 0 discrepancy
I Can find w2 such that (≤ 8)-cyclesare removed from active subgraph
I Can find w3 such that (≤ 16)-cyclesare removed from active subgraph
I ...I Can find wlog n such that all cycles
are removed from active subgraph =⇒ done!
Actually, not sure how to find in NC some wi that is good...But always some wi of a special form is good.Try all combinations (w1,w2, ...,wlog n) obliviously!There are nO(log n) many.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 58: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/58.jpg)
19/30
Isolating in stages
Crucial idea:I Can find w1 such that 4-cycles
are removed from active subgraph
I Can find w2 such that (≤ 8)-cyclesare removed from active subgraph
I Can find w3 such that (≤ 16)-cyclesare removed from active subgraph
I ...I Can find wlog n such that all cycles
are removed from active subgraph =⇒ done!
Actually, not sure how to find in NC some wi that is good...But always some wi of a special form is good.Try all combinations (w1,w2, ...,wlog n) obliviously!There are nO(log n) many.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 59: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/59.jpg)
19/30
Isolating in stages
Crucial idea:I Can find w1 such that 4-cycles
are removed from active subgraphI Can find w2 such that (≤ 8)-cycles
are removed from active subgraphI Can find w3 such that (≤ 16)-cycles
are removed from active subgraphI ...I Can find wlog n such that all cycles
are removed from active subgraph =⇒ done!
Actually, not sure how to find in NC some wi that is good...But always some wi of a special form is good.Try all combinations (w1,w2, ...,wlog n) obliviously!There are nO(log n) many.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 60: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/60.jpg)
19/30
Isolating in stages
Crucial idea:I Can find w1 such that 4-cycles
are removed from active subgraphI Can find w2 such that (≤ 8)-cycles
are removed from active subgraphI Can find w3 such that (≤ 16)-cycles
are removed from active subgraphI ...I Can find wlog n such that all cycles
are removed from active subgraph =⇒ done!
Actually, not sure how to find in NC some wi that is good...But always some wi of a special form is good.Try all combinations (w1,w2, ...,wlog n) obliviously!There are nO(log n) many.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 61: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/61.jpg)
19/30
Isolating in stages
Crucial idea:I Can find w1 such that 4-cycles
are removed from active subgraphI Can find w2 such that (≤ 8)-cycles
are removed from active subgraphI Can find w3 such that (≤ 16)-cycles
are removed from active subgraphI ...I Can find wlog n such that all cycles
are removed from active subgraph =⇒ done!
Actually, not sure how to find in NC some wi that is good...But always some wi of a special form is good.Try all combinations (w1,w2, ...,wlog n) obliviously!There are nO(log n) many.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 62: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/62.jpg)
19/30
Isolating in stages
Crucial idea:I Can find w1 such that 4-cycles
are removed from active subgraphI Can find w2 such that (≤ 8)-cycles
are removed from active subgraphI Can find w3 such that (≤ 16)-cycles
are removed from active subgraphI ...I Can find wlog n such that all cycles
are removed from active subgraph =⇒ done!
Actually, not sure how to find in NC some wi that is good...But always some wi of a special form is good.Try all combinations (w1,w2, ...,wlog n) obliviously!There are nO(log n) many.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 63: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/63.jpg)
19/30
Isolating in stages
Crucial idea:I Can find w1 such that 4-cycles
are removed from active subgraphI Can find w2 such that (≤ 8)-cycles
are removed from active subgraphI Can find w3 such that (≤ 16)-cycles
are removed from active subgraphI ...I Can find wlog n such that all cycles
are removed from active subgraph =⇒ done!
Actually, not sure how to find in NC some wi that is good...But always some wi of a special form is good.Try all combinations (w1,w2, ...,wlog n) obliviously!There are nO(log n) many.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 64: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/64.jpg)
20/30
3. Difficulties of general case& our approach
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 65: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/65.jpg)
21/30
Bipartite key property fails
Bipartite key propertyOnce we assign a cycle 6= 0 discrepancy,it will disappear from the active subgraph.
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 66: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/66.jpg)
22/30
Polyhedral perspective
I PM: perfect matching polytope(convex hull of all perfect matchings)
I F: set of points in PM that minimize w
I F is a face of PMI w isolating ⇐⇒ |F| = 1 (F is a vertex)
PM
F
w
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 67: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/67.jpg)
22/30
Polyhedral perspective
I PM: perfect matching polytope(convex hull of all perfect matchings)
I F: set of points in PM that minimize w
I F is a face of PMI w isolating ⇐⇒ |F| = 1 (F is a vertex)
PM
F
w
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 68: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/68.jpg)
22/30
Polyhedral perspective
I PM: perfect matching polytope(convex hull of all perfect matchings)
I F: set of points in PM that minimize w
I F is a face of PM
I w isolating ⇐⇒ |F| = 1 (F is a vertex)
F
PM
F
w
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 69: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/69.jpg)
22/30
Polyhedral perspective
I PM: perfect matching polytope(convex hull of all perfect matchings)
I F: set of points in PM that minimize w
I F is a face of PMI w isolating ⇐⇒ |F| = 1 (F is a vertex)
F
PM
F
w
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 70: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/70.jpg)
22/30
Polyhedral perspective
I PM: perfect matching polytope(convex hull of all perfect matchings)
I F: set of points in PM that minimize w
I F is a face of PMI w isolating ⇐⇒ |F| = 1 (F is a vertex)
F
PM
F
w
w not isolating
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 71: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/71.jpg)
22/30
Polyhedral perspective
I PM: perfect matching polytope(convex hull of all perfect matchings)
I F: set of points in PM that minimize w
I F is a face of PMI w isolating ⇐⇒ |F| = 1 (F is a vertex)
PM
F
ww isolating
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 72: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/72.jpg)
23/30
LP formulation
Edmonds [1965]PM described as set of x ∈ RE such that:I xe ≥ 0 for every edge e
I x(δ(v )) = 1 for every vertex v
I x(δ(S)) ≥ 1 for every odd set S of vertices
So every face F is given as:
F = x ∈ PM : xe = 0 for some edges e,x(δ(S)) = 1 for some odd sets S
(δ(S) = edges crossing S)
Bipartite key property fails!
I In bipartite case:F = x ∈ PM : xe = 0 for some edges e(F given by the active subgraph)
I Now, faces are exponentially harderI Need 2Ω(n) inequalities [Rothvoss 2013]
F
PM
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 73: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/73.jpg)
23/30
LP formulation
Edmonds [1965]PM described as set of x ∈ RE such that:I xe ≥ 0 for every edge e
I x(δ(v )) = 1 for every vertex v
I x(δ(S)) ≥ 1 for every odd set S of vertices
So every face F is given as:
F = x ∈ PM : xe = 0 for some edges e,x(δ(S)) = 1 for some odd sets S
(δ(S) = edges crossing S)
Bipartite key property fails!
I In bipartite case:F = x ∈ PM : xe = 0 for some edges e(F given by the active subgraph)
I Now, faces are exponentially harderI Need 2Ω(n) inequalities [Rothvoss 2013]
F
PM
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 74: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/74.jpg)
23/30
LP formulation
Edmonds [1965]PM described as set of x ∈ RE such that:I xe ≥ 0 for every edge e
I x(δ(v )) = 1 for every vertex v
I x(δ(S)) ≥ 1 for every odd set S of vertices
So every face F is given as:
F = x ∈ PM : xe = 0 for some edges e,x(δ(S)) = 1 for some odd sets S
(δ(S) = edges crossing S)
Bipartite key property fails!
I In bipartite case:F = x ∈ PM : xe = 0 for some edges e(F given by the active subgraph)
I Now, faces are exponentially harderI Need 2Ω(n) inequalities [Rothvoss 2013]
F
PM
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 75: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/75.jpg)
23/30
LP formulation
Edmonds [1965]PM described as set of x ∈ RE such that:I xe ≥ 0 for every edge e
I x(δ(v )) = 1 for every vertex v
I x(δ(S)) ≥ 1 for every odd set S of vertices
So every face F is given as:
F = x ∈ PM : xe = 0 for some edges e,x(δ(S)) = 1 for some odd sets S
(δ(S) = edges crossing S)
Bipartite key property fails!
I In bipartite case:F = x ∈ PM : xe = 0 for some edges e(F given by the active subgraph)
I Now, faces are exponentially harderI Need 2Ω(n) inequalities [Rothvoss 2013]
F
PM
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 76: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/76.jpg)
24/30
How bipartite key property fails
S
1
1
1
00
000
0
C
want:dw (C ) 6= 0dw (C ) = 2 6= 0
PM: convex hull of all four matchings:
F: convex hull of matchings of weight 1:
F ( PM but still has all edges...F ( PM but still has all edges...F = x ∈ PM : x(δ(S)) = 1
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 77: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/77.jpg)
24/30
How bipartite key property fails
S
1
1
1
00
000
0
C
want:dw (C ) 6= 0dw (C ) = 2 6= 0
PM: convex hull of all four matchings:
F: convex hull of matchings of weight 1:
F ( PM but still has all edges...F ( PM but still has all edges...F = x ∈ PM : x(δ(S)) = 1
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 78: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/78.jpg)
24/30
How bipartite key property fails
S
1
1
1
00
000
0
C
want:dw (C ) 6= 0
dw (C ) = 2 6= 0
PM: convex hull of all four matchings:
F: convex hull of matchings of weight 1:
F ( PM but still has all edges...F ( PM but still has all edges...F = x ∈ PM : x(δ(S)) = 1
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 79: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/79.jpg)
24/30
How bipartite key property fails
S
1
1
1
00
000
0
C
want:dw (C ) 6= 0
dw (C ) = 2 6= 0
PM: convex hull of all four matchings:
F: convex hull of matchings of weight 1:
F ( PM but still has all edges...F ( PM but still has all edges...F = x ∈ PM : x(δ(S)) = 1
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 80: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/80.jpg)
24/30
How bipartite key property fails
S
1
1
1
00
000
0
C
want:dw (C ) 6= 0
dw (C ) = 2 6= 0
PM: convex hull of all four matchings:
F: convex hull of matchings of weight 1:
F ( PM but still has all edges...F ( PM but still has all edges...F = x ∈ PM : x(δ(S)) = 1
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 81: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/81.jpg)
24/30
How bipartite key property fails
S
1
1
1
00
000
0
C
want:dw (C ) 6= 0
dw (C ) = 2 6= 0
PM: convex hull of all four matchings:
F: convex hull of matchings of weight 1:
F ( PM but still has all edges...
F ( PM but still has all edges...F = x ∈ PM : x(δ(S)) = 1
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 82: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/82.jpg)
24/30
How bipartite key property fails
S
1
1
1
00
000
0
C
want:dw (C ) 6= 0
dw (C ) = 2 6= 0
PM: convex hull of all four matchings:
F: convex hull of matchings of weight 1:
F ( PM but still has all edges...
F ( PM but still has all edges...F = x ∈ PM : x(δ(S)) = 1
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 83: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/83.jpg)
24/30
How bipartite key property fails
S
1
1
1
00
000
0
C
want:dw (C ) 6= 0
dw (C ) = 2 6= 0
PM: convex hull of all four matchings:
F: convex hull of matchings of weight 1:
F ( PM but still has all edges...
F ( PM but still has all edges...F = x ∈ PM : x(δ(S)) = 1
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 84: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/84.jpg)
25/30
How we cope
technical path
Main ingredients:I Laminar family of tight cut constraintsI Tight cut constraints decompose the instance
⇒ divide-and-conquer approach
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 85: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/85.jpg)
25/30
How we cope
technical path
Main ingredients:I Laminar family of tight cut constraintsI Tight cut constraints decompose the instance
⇒ divide-and-conquer approach
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 86: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/86.jpg)
25/30
How we cope
technical path
Main ingredients:I Laminar family of tight cut constraintsI Tight cut constraints decompose the instance
⇒ divide-and-conquer approach
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 87: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/87.jpg)
26/30
LaminarityEvery face F is given as:
F = x ∈ PM : xe = 0 for some edges e,x(δ(S)) = 1 for some odd sets S
Great news: “some” can be chosen to be a laminar family!
(at most n/2 constraints instead of exponentially many to describe a face)
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 88: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/88.jpg)
26/30
LaminarityEvery face F is given as:
F = x ∈ PM : xe = 0 for some edges e,x(δ(S)) = 1 for some odd sets S
Great news: “some” can be chosen to be a laminar family!
(at most n/2 constraints instead of exponentially many to describe a face)
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 89: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/89.jpg)
27/30
Tight odd cuts are not all bad
exactly one edge crossing
I once we fix a boundary edge...
I ... the instance decomposes into two independent ones
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 90: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/90.jpg)
27/30
Tight odd cuts are not all bad
exactly one edge crossing
I once we fix a boundary edge...
I ... the instance decomposes into two independent ones
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 91: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/91.jpg)
27/30
Tight odd cuts are not all bad
exactly one edge crossing
I once we fix a boundary edge...
I ... the instance decomposes into two independent ones
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 92: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/92.jpg)
27/30
Tight odd cuts are not all bad
exactly one edge crossing
I once we fix a boundary edge...
I ... the instance decomposes into two independent ones
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 93: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/93.jpg)
27/30
Tight odd cuts are not all bad
exactly one edge crossing
I once we fix a boundary edge...
I ... the instance decomposes into two independent ones
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 94: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/94.jpg)
27/30
Tight odd cuts are not all bad
exactly one edge crossing
I once we fix a boundary edge...
I ... the instance decomposes into two independent ones
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 95: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/95.jpg)
27/30
Tight odd cuts are not all bad
exactly one edge crossing
I once we fix a boundary edge...I ... the instance decomposes into two independent ones
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 96: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/96.jpg)
27/30
Tight odd cuts are not all bad
exactly one edge crossing
I once we fix a boundary edge...I ... the instance decomposes into two independent ones
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 97: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/97.jpg)
28/30
Divide & conquer
Simplest case of laminar family: only one tight odd set
Between friends: cycles that do not cross tight odd setsbehave like in the bipartite case and can thus be removed
I then every boundary edge determines entire matching
I so: at most n2 perfect matchingsI easy to isolate
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 98: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/98.jpg)
28/30
Divide & conquer
Simplest case of laminar family: only one tight odd set
Between friends: cycles that do not cross tight odd setsbehave like in the bipartite case and can thus be removed
I then every boundary edge determines entire matching
I so: at most n2 perfect matchingsI easy to isolate
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 99: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/99.jpg)
28/30
Divide & conquer
Simplest case of laminar family: only one tight odd set
Between friends: cycles that do not cross tight odd setsbehave like in the bipartite case and can thus be removed
I then every boundary edge determines entire matching
I so: at most n2 perfect matchingsI easy to isolate
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 100: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/100.jpg)
28/30
Divide & conquer
Simplest case of laminar family: only one tight odd set
Between friends: cycles that do not cross tight odd setsbehave like in the bipartite case and can thus be removed
I then every boundary edge determines entire matching
I so: at most n2 perfect matchingsI easy to isolate
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 101: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/101.jpg)
28/30
Divide & conquer
Simplest case of laminar family: only one tight odd set
Between friends: cycles that do not cross tight odd setsbehave like in the bipartite case and can thus be removed
I then every boundary edge determines entire matchingI so: at most n2 perfect matchings
I easy to isolate
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 102: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/102.jpg)
28/30
Divide & conquer
Simplest case of laminar family: only one tight odd set
Between friends: cycles that do not cross tight odd setsbehave like in the bipartite case and can thus be removed
I then every boundary edge determines entire matchingI so: at most n2 perfect matchingsI easy to isolate
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 103: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/103.jpg)
29/30
Our dichotomy
Dichotomy:
I remove cycles not crossing tight odd-sets
I use tight odd-sets to decompose problem(divide & conquer)
Details: see paper or talk to me :)
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 104: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/104.jpg)
29/30
Our dichotomy
Dichotomy:
I remove cycles not crossing tight odd-sets
I use tight odd-sets to decompose problem(divide & conquer)
Details: see paper or talk to me :)
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 105: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/105.jpg)
30/30
Future work
I go down to NCI even for bipartite graphsX for planar graphs: [Anari, Vazirani 2017]
I derandomize Isolation Lemma in other casesX matroid intersection: [Gurjar, Thierauf 2017]X totally unimodular polytopes: [Gurjar, Thierauf, Vishnoi 2017]I any efficiently solvable 0/1-polytope?
Exact MatchingGiven: graph with some edges red, number k .Is there a perfect matching with exactly k red edges?
I randomized complexity: even Randomized NCI deterministic complexity: is it in P?
Thank you!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 106: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/106.jpg)
30/30
Future work
I go down to NCI even for bipartite graphsX for planar graphs: [Anari, Vazirani 2017]
I derandomize Isolation Lemma in other casesX matroid intersection: [Gurjar, Thierauf 2017]X totally unimodular polytopes: [Gurjar, Thierauf, Vishnoi 2017]I any efficiently solvable 0/1-polytope?
Exact MatchingGiven: graph with some edges red, number k .Is there a perfect matching with exactly k red edges?
I randomized complexity: even Randomized NCI deterministic complexity: is it in P?
Thank you!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 107: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/107.jpg)
30/30
Future work
I go down to NCI even for bipartite graphsX for planar graphs: [Anari, Vazirani 2017]
I derandomize Isolation Lemma in other casesX matroid intersection: [Gurjar, Thierauf 2017]X totally unimodular polytopes: [Gurjar, Thierauf, Vishnoi 2017]I any efficiently solvable 0/1-polytope?
Exact MatchingGiven: graph with some edges red, number k .Is there a perfect matching with exactly k red edges?
I randomized complexity: even Randomized NCI deterministic complexity: is it in P?
Thank you!
Ola Svensson, Jakub Tarnawski Matching is in quasi-NC
![Page 108: The Matching Problem in General Graphs is in quasi-NCjakub.tarnawski.org/slides_focs_2017.pdfFenner, Gurjar and Thierauf [2015] showed: I Bipartitematching is in quasi-NC (npolylogn](https://reader035.vdocuments.us/reader035/viewer/2022071413/610ab91ea7712e4ff73c316d/html5/thumbnails/108.jpg)
30/30
Future work
I go down to NCI even for bipartite graphsX for planar graphs: [Anari, Vazirani 2017]
I derandomize Isolation Lemma in other casesX matroid intersection: [Gurjar, Thierauf 2017]X totally unimodular polytopes: [Gurjar, Thierauf, Vishnoi 2017]I any efficiently solvable 0/1-polytope?
Exact MatchingGiven: graph with some edges red, number k .Is there a perfect matching with exactly k red edges?
I randomized complexity: even Randomized NCI deterministic complexity: is it in P?
Thank you!Ola Svensson, Jakub Tarnawski Matching is in quasi-NC