Matching is in quasi-NC Jakub Tarnawski joint work with Ola - - PowerPoint PPT Presentation

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Dec 17, 2023 92 likes •332 views

Matching is in quasi-NC Jakub Tarnawski joint work with Ola Svensson June 10, 2017 1/10 Jakub Tarnawski Matching is in quasi-NC Perfect matching problem Basic question in computer science Decision problem: Does given graph contain a

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Matching is in quasi-NC

Jakub Tarnawski joint work with Ola Svensson June 10, 2017

Jakub Tarnawski Matching is in quasi-NC

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Perfect matching problem

◮ Basic question in computer science ◮ Decision problem: Does given graph contain a perfect matching? 5 1 4 2 6 3

Jakub Tarnawski Matching is in quasi-NC

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Perfect matching problem

◮ Basic question in computer science ◮ Decision problem: Does given graph contain a perfect matching? 5 1 4 2 6 3

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Perfect matching problem

◮ Matching is in P (Edmonds 1965): has deterministic polytime algorithm ◮ Parallel algorithm?

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Perfect matching problem

◮ Matching is in P (Edmonds 1965): has deterministic polytime algorithm ◮ It is also in Randomized NC (Lovász 1979): has randomized algorithm that uses:

◮ polynomially many processors ◮ polylog time

Jakub Tarnawski Matching is in quasi-NC

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Perfect matching problem

◮ Matching is in P (Edmonds 1965): has deterministic polytime algorithm ◮ It is also in Randomized NC (Lovász 1979): has randomized algorithm that uses:

◮ polynomially many processors ◮ polylog time

◮ Deterministic parallel complexity still not resolved: is matching in NC?

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Is matching in NC?

Yes, for restricted graph classes: ◮ strongly chordal ◮ graphs with small number of perfect matchings ◮ dense ◮ P✹-tidy ◮ claw-free ◮ incomparability graphs ◮ bipartite planar ◮ bipartite regular ◮ bipartite convex but not known for: ◮ bipartite ◮ planar (finding PM)

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Is matching in NC?

◮ Fenner, Gurjar and Thierauf (2015) showed: bipartite matching is in quasi-NC (npoly log n processors, polylog time)

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Is matching in NC?

◮ Fenner, Gurjar and Thierauf (2015) showed: bipartite matching is in quasi-NC (npoly log n processors, polylog time) ◮ We show: matching is in quasi-NC (for general graphs)

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Isolating weight functions

Difficulty in parallelization: how to coordinate machines to search for the same matching?

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Isolating weight functions

Difficulty in parallelization: how to coordinate machines to search for the same matching? Answer: look for a min-weight perfect matching

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Isolating weight functions

Weight function w : E → Z+ is isolating if there is a unique perfect matching M with minimum w(M) Mulmuley, Vazirani and Vazirani (1987) Given isolating w, can find perfect matching in NC ✵ ✵ ✵ ✵ ✵ ✵ ✷ ✵

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Isolating weight functions

Weight function w : E → Z+ is isolating if there is a unique perfect matching M with minimum w(M) Mulmuley, Vazirani and Vazirani (1987) Given isolating w, can find perfect matching in NC 3 1 4 2 T(G) =     ✵ X✶✷ X✶✸ X✶✹ −X✶✷ ✵ ✵ X✷✹ −X✶✸ ✵ ✵ X✸✹ −X✶✹ −X✷✹ −X✸✹ ✵     ◮ build Tutte’s matrix with entries Xuv ✷ ◮ det T(G) = ✵ (as polynomial) ⇐ ⇒ graph has perfect matching

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Isolating weight functions

Weight function w : E → Z+ is isolating if there is a unique perfect matching M with minimum w(M) Mulmuley, Vazirani and Vazirani (1987) Given isolating w, can find perfect matching in NC 3 1 4 2 T w(G) =     ✵ ✷w(✶,✷) ✷w(✶,✸) ✷w(✶,✹) −✷w(✶,✷) ✵ ✵ ✷w(✷,✹) −✷w(✶,✸) ✵ ✵ ✷w(✸,✹) −✷w(✶,✹) −✷w(✷,✹) −✷w(✸,✹) ✵     ◮ build Tutte’s matrix with entries Xuv := ✷w(u,v) ◮ det T w(G) = ✵ (as scalar) ⇐ ⇒ graph has perfect matching

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Isolating weight functions

Weight function w : E → Z+ is isolating if there is a unique perfect matching M with minimum w(M) Mulmuley, Vazirani and Vazirani (1987) Given isolating w, can find perfect matching in NC 3 1 4 2 T w(G) =     ✵ ✷w(✶,✷) ✷w(✶,✸) ✷w(✶,✹) −✷w(✶,✷) ✵ ✵ ✷w(✷,✹) −✷w(✶,✸) ✵ ✵ ✷w(✸,✹) −✷w(✶,✹) −✷w(✷,✹) −✷w(✸,✹) ✵     ◮ build Tutte’s matrix with entries Xuv := ✷w(u,v) ◮ det T w(G) = ✵ (as scalar) ⇐ ⇒ graph has perfect matching ◮ we can compute determinant in NC

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Isolating weight functions

Isolation Lemma [MVV 1987] If each w(e) picked randomly from {✶, ✷, ..., n✷}, then P[w isolating] ≥ ✶

✷.

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Isolating weight functions

Isolation Lemma [MVV 1987] If each w(e) picked randomly from {✶, ✷, ..., n✷}, then P[w isolating] ≥ ✶

✷.

Randomized NC algorithm [MVV 1987] ◮ Sample w (the only random component) ◮ Compute determinant (possible in NC) ◮ Answer YES iff it is nonzero

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Derandomize the Isolation Lemma

◮ Challenge: deterministically get small set

f weight functions (to be checked in parallel)

◮ We prove: can construct nO(log✷ n) weight functions such that one of them is isolating ◮ Can even do it without looking at the graph ◮ Implies: matching is in quasi-NC First step to derandomizing Polynomial Identity Testing? (for polynomial being det T(G))

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Derandomize the Isolation Lemma

◮ Challenge: deterministically get small set

f weight functions (to be checked in parallel)

◮ We prove: can construct nO(log✷ n) weight functions such that one of them is isolating ◮ Can even do it without looking at the graph ◮ Implies: matching is in quasi-NC First step to derandomizing Polynomial Identity Testing? (for polynomial being det T(G))

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Derandomize the Isolation Lemma

◮ Challenge: deterministically get small set

f weight functions (to be checked in parallel)

◮ We prove: can construct nO(log✷ n) weight functions such that one of them is isolating ◮ Can even do it without looking at the graph ◮ Implies: matching is in quasi-NC First step to derandomizing Polynomial Identity Testing? (for polynomial being det T(G))

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Future work

◮ go down to NC (even for bipartite case) ◮ derandomize Isolation Lemma in other cases (totally unimodular polytopes?) ◮ derandomize Exact Matching (is in Randomized NC; is it in P?)

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Future work

◮ go down to NC (even for bipartite case) ◮ derandomize Isolation Lemma in other cases (totally unimodular polytopes?) ◮ derandomize Exact Matching (is in Randomized NC; is it in P?)

Thank you!

Jakub Tarnawski Matching is in quasi-NC