Feasible Combinatorial Matrix Theory

Ariel Fernandez & Michael Soltys

August 29, 2013

Comb Matrix - Soltys MFCS’13 IST Austria Title - 1/21

Statistical Archeology: sequence dating (Flinders Petrie, 1899)900 pre-dynastic Egyptian graves containing 800 representatives ofpottery.

The “graves-versus-varieties” matrix contains vast amount ofinformation, such as sequential ordering.

[Kendall 1969]

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Paleogenomics: DNA sequence organization of ancient livingorganisms using similarities and differences between chromosomesof extant organisms.

[Chauve et al 2008]

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Consecutive-ones Property: C1P

Consider a slight relaxation, (k , δ)-C1P: each row has at most kblocks of 1s and the gap between any two blocks is at most δ.

So (1, 0)-C1P is C1P, and deciding if an A has (k , δ)-C1P is:

I polytime for (1, 0)

I NP-hard for every k ≥ 2, δ ≥ 1 except (2, 1)

What about (2, 1)? [Patterson 2012]

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Konig’s Min-Max

0

1 0 1 1

1 0

1

00

0

1

1 0

00

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Problems

Cover(A, α):

∀i , j ≤ r(A)(A(i , j) = 1→ α(1, i) = 1 ∨ α(2, j) = 1)

MinCover(A, α):

Cover(A, α) ∧ ∀α′ ≤ c(α)(Cover(A, α′)→ Σα′ ≥ Σα)

KMM(A, α, β):

MinCover(A, α) ∧MaxSelect(A, β)→ Σα = Σβ

a ΣB1 formula.

But classical proof is ΠB2 — is there a ΣB

1 proof?

Comb Matrix - Soltys MFCS’13 IST Austria Problem - 7/21

Related Theorems:

I Menger’s: size of min cut equals max nr of disjoint s, t-paths(“Min-Cut Max-Flow”)

I Hall’s: ∀k ∈ [n] |Si1 ∪ . . . ∪ Sik | ≥ k , then there exists aSystem of Distinct Representatives.

I Dilworth’s: Min nr of chains needed to partition a posetequals size of max anti-chain of that poset.

Can they all be shown equivalent to KMM with ΣB0 proofs?

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ΣB1 Proof of KMM

Let lA be the min nr of lines necessary to cover A

Let oA be the max selection of ones in A

Comb Matrix - Soltys MFCS’13 IST Austria Main Proof - 9/21

0

1 0 1 1

1 0

1

00

0

1

1 0

00

lA=oA= 3

We want to show with ΣB1 induction that oA = lA.

Comb Matrix - Soltys MFCS’13 IST Austria Main Proof - 10/21

LA uses ΣB0 induction

LA ` oA ≤ lA

LA over Z is equivalent to VTC0

And oA ≤ lA follows more or less from the Pigeonhole Principle:

if we can select oA 1s, no two on the same line, then we shallrequire at least lA lines to cover those 1s.

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∃LA ` oA ≥ lA

Showing oA ≥ lA is more difficult; we use ΣB1 induction.

Lots of cases, but the interesting case is:

0

where we reduce the general case to the case of the blue matrixwhose cover requires as many lines as rows.

Comb Matrix - Soltys MFCS’13 IST Austria Main Proof - 12/21

00 00 00 0 0 0 0 0 0

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10 00 00 0 0 0 0 0 0

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0

0 00 00 0 0 0 0 0 01

0 0 0 1 0 0 0 0 0

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1

0 00 00 0 0 0 0 0 01

0 0 0 1 0 0 0 0 0

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1

0 00 00 0 0 0 0 0 01

0 0 0 1 0 0 0 0 0

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H

1 1 1

1

1

1

1

1

1

1

1

1

1

1

1

1

A

B

C

D

E

FG

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J

1 1 1

1

1

1

1

1

1

1

1

1

1

1

1

1

H

1

1

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ΣB0 Proof of Equivalence

For example, ΣB0 proof of Menger→ KMM

yx

I Left graph has a matching of size k ⇐⇒Right graph has k disjoint {x , y}-paths

I Left graph has a cover of size k ⇐⇒Right graph has an {x , y}-cut of size k

KMM → Mengcomplicated,[Aharoni 1983]

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Some open problems

I Frankl’s Theorem: t a positive integer, and m ≤ n (2t−1)t ; if A

is m × n, and its rows are distinct, then there exists a columnthat when deleted, the resulting matrix has at most 2t−1 − 1pairs of equal rows. (Bondy’s Theorem when t = 1.)

I Complexity of decompositions: A = P1 + P2 + · · ·+ Pn + X ;n boys and n girls, each boy introduced to exactly k girls andvice versa. Compute a pairing where each boy & girl has beenpreviously introduced.

I Projective Geometry

1 1 1 0 0 0 01 0 0 1 1 0 01 0 0 0 0 1 10 1 0 1 0 1 00 1 0 0 1 0 10 0 1 1 0 0 10 0 1 0 1 1 0

Desargues Thm

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