lp-based parameterized algorithms for separation problems
DESCRIPTION
LP-Based Parameterized Algorithms for Separation Problems. D. Lokshtanov , N.S. Narayanaswamy V. Raman , M.S. Ramanujan S. Saurabh. Message of this talk. It was open for quite a while whether Odd Cycle Transversal and Almost 2-Sat are FPT . - PowerPoint PPT PresentationTRANSCRIPT
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LP-Based Parameterized Algorithms for Separation Problems
D. Lokshtanov, N.S. NarayanaswamyV. Raman, M.S. Ramanujan S. Saurabh
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Message of this talk
It was open for quite a while whether Odd Cycle Transversal and Almost 2-Sat are FPT.
A simple branching algorithm for Vertex Cover known since the mid 90’s solves both problems in time .
Some more work gives .
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Results
(Above LP) Multiway Cut
(Above LP) Vertex Cover
Almost 2-SAT
Odd Cycle Transversal
≤≤
≤
[CPPW11]
2.32𝑘−𝐿𝑃
2.32𝑘
2.32𝑘
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How does one get a 4k-LP algorithm?
Branching: on both sides k-LP decreases by at least ½.
How to improve? Decrease k-LP more.
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Multiway Cut
In: Graph G, set T of vertices, integer k.Question: such that no component of G\S has at least two vertices of T?
FPT by Marx, 04Faster FPT by Chen et al, 07Fastest FPT and FPT/k-LP by Cygan et al, 11
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Vertex Cover
Long story...Here: .
In: G, tQuestion: such that every edge in G has an endpoint in S?
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Almost 2-SAT
FPT by Razgon and O’Sullivan, 08Here: .
In: 2-SAT formula, integer kQuestion: Can we remove k variables from and make it satisfiable?
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Odd Cycle Transversal
FPT: by Reed et al.Here: .
In: G, kQuestion: such that G\S is bipartite?
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Vertex CoverIn: G, tQuestion: such that every edge in G has an endpoint in S?
Minimize ∀𝑢𝑣∈𝐸 (𝐺 ) :𝑥𝑢+𝑥𝑣≥1𝑥𝑖≥0
Z
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Vertex Cover Above LP
In: G, tQuestion: such that every edge in G has an endpoint in S?Running Time: , where LP is the value of the optimum LP solution.
𝜇=𝑡− 𝐿𝑃
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Odd Cycle Transversal Almost 2-Sat
x y
z
x y
z
𝑥∨¬ 𝑦¬𝑥∨ 𝑦
𝑥∨¬ 𝑧
¬𝑥∨ 𝑧
¬ 𝑦∨ 𝑧𝑦∨¬ 𝑧
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Almost 2-SAT Vertex Cover/t-LP
𝑥
¬𝑥
𝑦
¬ 𝑦
𝑧
¬𝑧
𝑥∨ 𝑦𝑦∨¬ 𝑧
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Nemhauser Trotter Theorem
(a) There is always an optimal solution to Vertex Cover LP that sets variables to .
(b) For any –solution there is a matching from the 1-vertices to the 0-vertices, saturating the 1-vertices.
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Nemhauser Trotter Proof
¿𝟏𝟐
¿𝟏𝟐
𝟏𝟐
𝟎
𝟏
+ +
- - -
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Reduction Rule
If exists optimal LP solution that sets xv to 1, then exists optimal vertex cover that selects v.
Remove v from G and decrease t by 1.
Correctness follows from Nemhauser TrotterPolynomial time by LP solving.
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Branching
Pick an edge uv. Solve (G\u, t-1) and (G\v, t-1).
since otherwise there is an optimal LP solution for G that sets u to 1.
Then
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Branching - Analysis
LP – t drops by ½ ... in both branches!
Total time:
Caveat: The reduction does not increase the measure!
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Moral
Nemhauser Trotter reduction + classic «branch on an edge» gives time algorithm for Vertex Cover and time algorithm for Odd Cycle Transversal and Almost 2-Sat.
Can we do better?
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Surplus
The surplus of a set I is |N(I)| – |I|. The surplus can be negative!
In any -LP solution, the total weight is n/2 + surplus(V0)/2.
Solving the Vertex Cover LP is equivalent to finding an independent set I of minimum surplus.
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Surplus and Reductions
If «all ½» is the unique LP optimum then surplus(I) > 0 for all independent sets.
Can we say anything meaningful for independent sets of surplus 1? 2? k?
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Surplus Branching Lemma
Let I be an independent set in G with minimum surplus. There exists an optimal vertex cover C that either contains I or avoids I.
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Surplus Branching Lemma Proof
I N(I) R
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Branching Rule
Find an independent set I of minimum surplus. Solve (G\I, t-|I|) and (G\N(I), t-|N(I)|).
LP(G\I) > LP(G) - |I|, since otherwise LP(G) has an optimal solution that sets I to 1.
So
t-LP drops by at least ½.
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Branching Rule Analysis Cont’d
Analyzing the (G\N(I), t-N(I)) side:
So
t-LP drops by at least
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Branching Summary
The measure k-LP drops by (½, surplus(I)/2).
Will see that independent sets of surplus 1 can be reduced in polynomial time!
Measure drops by (½,1) giving a time algorithm for Vertex Cover
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Reducing Surplus 1 sets.Lemma: If surplus(I) = 1, I has minimum surplus and N(I) is not independent then there exists an optimum vertex cover containing N(I).
I N(I)R
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Reducing Surplus 1 sets.Reduction Rule: If surplus(I) = 1, I has minimum surplus and N(I) is independent then solve (G’,t-|I|) where G’ is G with N[I] contracted to a single vertex v.
I N(I)R
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Summary
Nemhauser Trotter lets us assume surplus > 0
More rules let us assume surplus > 1 ()*
If surplus then branching yields time for Vertex Cover
The correctness of these rules were also proved by NT!
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Can we do better?
Can get down to by more clever branching rules. Yields for Almost 2-SAT and Odd Cycle Transversal.
Should not be the end of the story.
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Better OCT?
Can we get down to for Odd Cycle Transversal?
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LP Branching in other cases
I believe many more problems should have FPT algorithms by LP-guided branching.
What about ... (Directed) Feedback Vertex Set, parameterized by solution size k?
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Thank You!