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]
Comb Matrix - Soltys MFCS’13 IST Austria Introduction - 2/21
Paleogenomics: DNA sequence organization of ancient livingorganisms using similarities and differences between chromosomesof extant organisms.
[Chauve et al 2008]
Comb Matrix - Soltys MFCS’13 IST Austria Introduction - 3/21
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]
Comb Matrix - Soltys MFCS’13 IST Austria Introduction - 4/21
Konig’s Min-Max
0
1 0 1 1
1 0
1
00
0
1
1 0
00
Comb Matrix - Soltys MFCS’13 IST Austria Introduction - 5/21
Comb Matrix - Soltys MFCS’13 IST Austria Introduction - 6/21
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?
Comb Matrix - Soltys MFCS’13 IST Austria Problem - 8/21
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.
Comb Matrix - Soltys MFCS’13 IST Austria Main Proof - 11/21
∃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
Comb Matrix - Soltys MFCS’13 IST Austria Interesting Case - 13/21
10 00 00 0 0 0 0 0 0
Comb Matrix - Soltys MFCS’13 IST Austria Interesting Case - 14/21
0
0 00 00 0 0 0 0 0 01
0 0 0 1 0 0 0 0 0
Comb Matrix - Soltys MFCS’13 IST Austria Interesting Case - 15/21
1
0 00 00 0 0 0 0 0 01
0 0 0 1 0 0 0 0 0
Comb Matrix - Soltys MFCS’13 IST Austria Interesting Case - 16/21
1
0 00 00 0 0 0 0 0 01
0 0 0 1 0 0 0 0 0
Comb Matrix - Soltys MFCS’13 IST Austria Interesting Case - 17/21
H
1 1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
A
B
C
D
E
FG
Comb Matrix - Soltys MFCS’13 IST Austria Interesting Case - 18/21
J
1 1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
H
1
1
Comb Matrix - Soltys MFCS’13 IST Austria Interesting Case - 19/21
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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