How random walks led to advances in testing minor-freeness C. - PowerPoint PPT Presentation

How random walks led to advances in testing minor-freeness C. Seshadhri (UC Santa Cruz) WOLA 2019 1

My coauthors Akash Kumar, Purdue Andrew Stolman, UCSC WOLA 2019 2

Classics • [Kuratowski 30, Wagner 37] G is not planar, iff it contains a K 5 or K 3,3 minor – From geometry to topology WOLA 2019 3 https://en.wikipedia.org/wiki/Planar_graph#/media/File:Goldner-Harary_graph.svg

Minors x Vertex x disjoint connected subgraphs Vertex disjoint paths • H is minor of G, if H obtained by deletions and edge contractions in G • Forbidden minor characterization: G is planar iff it does not contain K 5 and K 3,3 minors – G is forest, iff it doesn ’ t have K 3 minor WOLA 2019 4

Robertson-Seymour I - XX x Vertex x disjoint connected subgraphs Vertex disjoint paths • If property P is closed on taking minors, P has finite forbidden minor characterization • Planarity, outerplanarity, bounded genus embeddable, treewidth < k,… – Each P has a finite list F of forbidden minors WOLA 2019 5

Algorithmic classics • Given non-planar G, find forbidden minor in it • [Hopcroft-Tarjan 74] O(n) time algorithm to decide planarity WOLA 2019 6

Robertson-Seymour: algorithms Disjoint connected subgraphs Is contained? Disjoint paths • There is O(n 3 ) algorithm to decide if G contains H- minor – Thus, O(n 3 ) for any minor-closed property [Kawarabayashi-Kobayashi-Reed12] O(n 2 ) algorithm • • Grand generalization of Hopcroft-Tarjan, worse running time WOLA 2019 7

What if you can’t read all of G? o(n) algorithms for planarity WOLA 2019 8

[Goldreich-Ron 02] The query model v v v • G is bounded degree, stored as adjacency list – n vertices, d degree bound • You can pick random vertices/seeds • You can crawl from these seeds – BFS, Random walks WOLA 2019 9

Distance to planarity Arbitrary set of εnd edges Still not planar! • G is ε-far from planar if > εnd edges need to be removed to make G planar • G is ε-far from H-minor freeness if > εnd edges need to be removed to make H-minor free WOLA 2019 10

The testing problem Certificate of non-planarity Graphs “far” from P P • If G is ε-far from planar, reject w.p. > 2/3 • (Two-sided) If G is planar, accept w.p. > 2/3 • (One-sided) If G is planar, accept w.p. 1 • (One-sided) If G is ε-far from planar, find forbidden minor w.p. > 2/3 WOLA 2019 11

[Benjamini-Schramm-Shapira 08] Graphs “far” from P P • Two-sided tester for all [Goldreich-Ron 02, Czumaj-Goldreich- • Ron-S-Shapira-Sohler 14] minor-closed properties One-sided ! lower in exp(exp(exp(d/ε)) bound queries – Forbidden minor is poly(log n) sized WOLA 2019 12

Post BSS08 Graphs “far” from P P Two-sided One-sided [Hassidim-Kelner-Nguyen-Onak 09] [Czumaj-Goldreich-Ron-S-Shapira- • • Sohler 14] exp(d/ε) ! for cycle-freeness [Levi-Ron 15] (d/ε) log(1/ε) • [Fichtenburger-Levi-Vasudev-Wotzel17] • [Yoshida-Ito 11, Edelman-Hassidim- • ! "/$ for K 2,r -minor Nguyen-Onak 11] freeness poly(d/ε) for bounded treewidth classes WOLA 2019 13

Post BSS08 Graphs “far” from P P Two-sided One-sided [Hassidim-Kelner-Nguyen-Onak 09] [Czumaj-Goldreich-Ron-S-Shapira- • • ! Sohler 14] exp(d/ε) poly(d/ ε ) one-sided ! for cycle-freeness [Levi-Ron 15] (d/ε) log(1/ε) tester for • tester for [Fichtenburger-Levi-Vasudev-Wotzel17] • planarity? [Yoshida-Ito 11, Edelman-Hassidim- • planarity? ! "/$ for K 2,r -minor Nguyen-Onak 11] freeness poly(d/ε) for bounded treewidth classes WOLA 2019 14

Sorry, this is a marketing slide BSS08 Goldreich17 WOLA 2019 15

And now… Graphs “far” from P P One-sided Two-sided [Kumar-S-Stolman 18] [Kumar-S-Stolman 19] ! " # " $(&) for all minor- poly(d/ε) for all minor- closed properties closed properties Based on (new?) toolkit using spectral graph theory for minor-freeness WOLA 2019 16

One-sided tester Planarity, outerplanarity, series-parallel, bounded genus embeddable, treewidth < k [Kumar-S-Stolman 18] Fix minor-closed property P. (By [RS], there is finite list of forbidden minors.) There is ! ∗ ($ %) - time randomized algorithm: If G is ε-far from P, algorithm produces a forbidden minor in G – O*() hides poly(1/ε).n o(1) – Doubly exponential dependence on r, size of largest minor in G WOLA 2019 17

Two-sided tester [Kumar-S-Stolman 19] Fix minor-closed property P. There is O "# $%&& time two-sided tester for P – Previously, poly(1/ε) not known for planarity WOLA 2019 18

Cute corollary Consider d bounded degree G with at least (3+ε)n edges. There is O*(dn 1/2 )-time algorithm that finds K 5 or K 3,3 minor in G – Analogous theorem for any minor-closed property WOLA 2019 19

Less graph minors, more random walks • No Robertson-Seymour machinery – No brambles, treewidth, etc. – In searching for H-minor, H does not play major role • It’s all spectral graph theory – Finding minors through random walks WOLA 2019 20

How did it all start? Let’s try to find K 5 minors WOLA 2019 21

[Goldreich-Ron 99] • If G is ɛ-far from bipartite, ! algorithm to find odd cycle – The inspiration for our result – Finding cycles through random walks 22

The rapid mixing case: G is expander v 1 s v 2 • G is disjoint collection of expanders – ℓ = log n • Pick random starting vertex s • Perform 5 ℓ -length rws from s to reach v 1 , v 2 ,…, v 5 – Perform " random walks from v 1 …v 5 to form K 5 minor 23

Connecting the dots • Perform ! ℓ -length random walks from v i – Birthday paradox: guaranteed to have two walks end at the same vertex • Guaranteed to connect all (v i , v j ) pairs – Union bound 24

Paths don’t imply minors GOOD BAD • Paths unlikely to be (internally) vertex disjoint • In expander, intersections are “localized” – We can contract away intersections to get K 5 25

Just run this algorithm on any graph? WOLA 2019 26

[GR99] The general case v 1 s v 2 • Every graph can be decomposed into “expander-like” pieces – Remove εdn edges, get disjoint pieces with mixing time poly(log n) • [Trevisan 05, Arora-Barak-Steurer 15] Deep connection with UGC/approx algorithms WOLA 2019 27

The sublinear constraint v 1 s v 2 • G can be decomposed into G’, disjoint collection of “expander-like” pieces • Yes, but o(n) algorithm cannot know G’ • Algorithm performs random walks on G, and hopes to simulate expander algorithm on G’…? WOLA 2019 28

The [GR99] decomposition P i s i • (There is k st) Pick s 1 , s 2 , …, s k uar • We can remove εdn edges and get pieces P 1 , P 2 ,…P k where: • ℓ -rws from s i (in G) reach all vertices in P i with roughly the same probability (> 1/n 1/2 ) WOLA 2019 29

The [GR99] decomposition P i s i • ℓ -rws from s i (in G) reach all vertices in P i with roughly the same probability • The expander analysis goes through – If G is far from bipartite, then constant fraction (by total size) of P i are far from bipartite WOLA 2019 30

Problem #1 for minor finding s i • ℓ -rws from s i (in G) reach all vertices in P i with roughly the same probability – Only have guarantee from one vertex in P i – Enough for finding cycle • K 5 needs walks from 5 “starting” vertices WOLA 2019 31

Problem #2 for minor finding • ℓ -rws from s i (in G) reach all vertices in P i with roughly the same probability • These walks leave P i , and we have no control on intersection – No problem for odd-cycle • How to argue about minors? WOLA 2019 32

Fixed source and destination • [Czumaj-Goldreich-Ron-S-Shapira-Sohler 14] ! tester H-minor freees, when H is cycle • [Fichtenburger-Levi-Vasudev-Wotzel17] n 2/3 algorithm if H is K 2,r or cactus graph • All about finding multiple paths between the same two vertices WOLA 2019 33

Fundamental problem • For any decomposition… • Need to walk ℓ > (log n) steps to reach most vertices in each piece – There could be εn cut edges • So walks will leave piece whp, and we don’t know how to control the behavior outside WOLA 2019 34

The [GR99] decomposition Somehow strengthen this decomposition? P i More starting vertices s i within in each piece? • (There is k st) Pick s 1 , s 2 , …, s k uar • We can remove εdn edges and get pieces P 1 , P 2 ,…P k where: • ℓ -rws from s i (in G) reach all vertices in P i with roughly the same probability (> 1/n 1/2 ) WOLA 2019 35

The [GR99] decomposition P i s i Stuck here for years • (There is k st) Pick s 1 , s 2 , …, s k uar • We can remove εdn edges and get pieces P 1 , P 2 ,…P k where: • ℓ -rws from s i (in G) reach all vertices in P i with roughly the same probability (> 1/n 1/2 ) WOLA 2019 36

Revisit the expander case: When can random walks find minors? WOLA 2019 37

Leaking random walks At least ℓ 10 • ℓ = n δ (think little more than poly(log n)) p s, ` = Prob. vector of ℓ rw from s • s is “leaky” if: k p s,` k 2 2  ` − 10 • It means: ℓ -rws from s reach at least poly( ℓ ) vertices WOLA 2019 38