Testing SubdivisionFreeness Property Testing Meets Structural Graph Theory
Testing Subdivision-Freeness – Property Testing Meets Structural Graph Theory – Yuichi Yoshida (NII & PFI) Joint with Ken-ichi Kawarabayashi (NII & JST ERATO Kawarabayashi Project)
Property testing Graph G is ε-far from property P : We need to modify ε -fraction of edges to make G satisfy P. An ε-tester for property P decides whether G satisfies P or ε-far from P w. p. 2/3. All graphs P Yes w. p. 2/3 ε-far No w. p. 2/3
Property testing We want to design constant-time testers, which runs in time independent of n. We need to define a model on how to access G. • Adjacency matrix model • Bounded-degree model
Adjacency matrix model Graph G is given as query access OG: V 2 → {0, 1} OG(u, v) = 1 iff u and v are adjacent. ε-far: we need to add / remove εn 2 edges. Many properties are constant-time testable! • Triangle-freeness, 3 -colorability, etc. A property is constant-time testable iff it can be well-approximated using Szemeredi-type regular partitions. [AFNS 09]
Adjacency matrix model This model is not suitable to handle properties sensitive to O(n) edge modifications. Ex. Connectivity • Every graph is 1/n-close. • We can always accept. Nothing ε-far All graphs Connected
Bounded-degree model d: degree bound. Query access OG: V × [d] → V ∪ {⊥}. OG(v, i): i-th neighbor of v. ε-far: we need to add / remove εdn edges. Again many properties are testable. • Triangle-freeness, connectivity, planarity, etc. Can we characterize constant-time testable properties? Not yet
Three reasons of testability in boundeddegree model Properties Why is it testable? △-freeness, H-freeness [GR 02] Locally determined k-edge-connectivity [GR 02] k edge-disjoint spanning trees [ITY’ 12] Min-max theorems in edge-augmentation / matroid theory. Planarity, H-minor-freeness Existence of separators [BSS 08, HKNO 09] Is there any other kind of testable properties?
Main contribution A subdivision H’ of H : H H’ G is H-subdivision-free (H-sdf) if G contains no Hsubdivision as its subgraph. [Main Theorem] Kt-subdivision-freeness is constant-time testable for any t.
Subdivision-freeness Is H-sdf locally determined? No. A 3 -regular random graph contains a K 4 subdivision but no constant-size K 4 -subdivision exists.
Subdivision-freeness Related to edge augmentation / matroid theory? Seems not. • Edge augmentation handles properties closed under edge additions. • H-sd is apparently nothing to do with matroids.
Subdivision-freeness Does H-sdf imply separators? A separator is a cut of size o(n) that divides G into subgraphs of size at most 2 n/3. No. A 3 -regular random graph is K 5 -sdf, but no separator exists. ≤ 2 n/3 Ω(n) ≤ 2 n/3
What makes H-subdivision-freeness testable behind the scenes?
Testing Kt-subdivision-freeness Polynomial-time tester for Kt-sdf • Decompose G into components • of constant size, or • with large clique minor and no small cut • Design a tester that works locally given the decomposition. Constant-time tester for Kt-sdf • Use “partitioning oracle” to obtain query access to the decomposition. • Run the poly-time tester on the decomposition.
Minor H is a minor of G if H is obtained by removing vertices, removing edges and contracting edges. G H H-minor-free: does not contain H as its minor H-minor-free ⇒ H-subdivision-free
Separation (A, B) • A, B ⊆ V with A ∪ B = V • No edge between A B and B A A B v The order of (A, B) = |A ∩ B| l-global separation: |A – B|, |A – B| ≥ l|A ∩ B|.
Dangerous A vertex v is dangerous w. r. t. S ⊆ V if we cannot exclude the possibility that v is a branch of a Ktsubdivision just by looking at S. Ex. v is not dangerous when t = 4. S v
A sufficient condition that Kt-subdivision exists ∀t and l, ∃t’, c, and r such that • Kt’-minor • no l-global separation of order ≤ t – 2. • ≥ c dangerous vertices w. r. t. r-radius balls ⇒ Kt-subdivision ≥t– 1 O(1) Kt-subdivision dangerous Kt’-minor
Decomposition lemma ∀t’ and l, ∃s, we can decompose G by removing n / l edges into components 1) of size ≤ s, and 2) with Kt’-minor and no l-local separation. ≤s ≤s
Polynomial-time tester for Kt-sdf ∀t’ and l, ∃s, we can decompose G by removing n / l edges into components 1) of size ≤ s, and 2) with Kt’-minor and no l-local separation. 1) easy to test 2) Count dangerous vertices: • Kt-sdf: few dangerous vertices. (say, εn/1000) (From the sufficient condition) • ε-far: many dangerous vertices. (say, εn/2) (Otherwise, we can remove them)
Partitioning oracle [HKNO 09] (ε, k)-partitioning oracle O for G provides query access to a partition P of V such that • P is a function of G and random bits in O. • |P[v]| ≤ k for every v. • #{(u, v) ∈ E : P[u] ≠ P[v], P[u] or P[v] is not a singleton} ≤ εn holds whp. • P[v] can be computed in constant time. ≤k # of edges crossing is ≤ εn
Partitioning oracle for H-minor-free graphs G: H-minor-free. ∀ε, ∃k such that G can be decomposed by removing εn edges into components of size ≤ k. Proof. Recursively delete separators. (ε, k)-partitioning oracle for H-minor-free graphs with P having < εn singletons. [HKNO 09]
Partitioning oracle for H-sdf graphs. ∀t and l, ∃s, we can decompose G by removing n / l edges into components ① of size ≤ s, and ② with Kt’-minor and no l-local separation. (ε, k)-partitioning oracle for H-sdf graphs with P having < εn singletons from ①. Essentially, we can assume query access to ②.
Conclusions and future work Kt-subdivision-freeness is constant-time testable. Open problems: • Query complexity: 2^(d^poly(ε/2 poly(t))). • Every graph property is testable in H-minor-free graphs [NS 11]. What can we test in H-sdf graphs, or simply expander graphs? • Can we test Kt-subdivision-freeness in adjacency list model?
Constant-time tester for Kt-sdf If we have query access to ②, we can test Kt-sdf by counting dangerous vertices in ②. Since the partitioning oracle have error, we may regard non-dangerous vertices in ② as dangerous in the partition. ⇒ we can bound.
Graph decomposition [Lemma] For l, t > 0, there is s such that any graph G has a vertex set X of order at most n/l such that G − X consists of connected components {Ci} with the following property: either ≤s • a ≤s ≤t-2 ≤ �� t ≥t-1
Partitioning Oracle Let G = (V, E) be a (ε, ρ(ε))-hyperfinite graph w. r. t. U ⊆ V. There is an (εd, ρ(O(ε 3)))-partitioning oracle such that w. p. 99/100, the following holds. The number of singleton vertices in U in the partition P is at most εn Let q be the number of (non-adaptive) queries to the oracle. Then, the query complexity is q*2^(d^O(ρ(O(ε 3)))).
Partitioning Oracle Let ρ be a function from ℝ+ to ℝ+. A graph G = (V, E) is ρ-hyperfinite w. r. t U ⊆ V, if for every ε > 0, we can remove εn edges so that every vertex in U is contained in a connected component of at most ρ(ε) vertices. Every H-minor-free graph is ρ-hyperfinite w. r. t. V for some ρ = ρ(ε, d, H). [HKNO 09]
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