1. 1 Property Testing in Sparse and General Graphs Michael Krivelevich Tel Aviv University

2. 2 Graph Property Testing Very general setting: P = graph property to test (k-colorability, planarity, non-existence of a copy of H, etc.) Input: graph G on n vertices, n→∞ Promise: G  P (positive) or: G is ε-far from P (negative) (ε-percentage of description of P should be changed to get H  P) Algorithm A (typically randomized): Queries description of P G  P  Pr[ A accepts G] ≥ 2/3 G is ε-far from P  Pr[ A rejects G] ≥ 2/3 G  P, Pr[ A accepts G] =1 – one-sided error algorithm Should be specified!

3. 3 Property Testing in Dense Graphs -Formally defined in GGR’98 (appeared implicitly in combinatorial papers in 70’s, 80’s) Input graph description: adjacency matrix G=(V,E), V=[n] Algorithm: queries the adjacency matrix of G Query: whether (i,j)  E(G)? (vertex pair query) Distance: G is is ε-far from P if ≥εn 2 entries in A(G) need to be changed to get H  P

4. 4 Property Testing in Dense Graphs – Brief Summary “… It’s all about REGULARITY.” (AFNS’06) Very strong (and fruitful) connection between property testing in dense graphs and the Szemerédi Regularity Lemma and its versions (started in AFKS’99 and culminated in AFNS’06) Have reached very good understanding of this setting (though of course quite a few challenging problems remain)

5. 5 Dense Graph Model - limitations Suitable/tailored for dense graphs only Degenerate for many graph properties Ex. : P = “ G is connected” - Always answer “YES” ( dist(G,P)≤ n-1 << εn 2 ) A typical algorithm: - sample S [n], |S|=O(1) - look inside to check whether G[S]  P - returns a.s. empty set S for |E(G)|=o(n 2 ) useless/irrelevant

6. 6 Property Testing in Bounded Degree Graphs Introduced by GR’97 Assumption: Δ(input graph G) ≤ d=const; ε<< 1/d Graph representation: by incidence lists L(v i )=(v i,1,…,v i,d ) – list of neighbors of v i Query: who is the j-th neighbor of v i ? (neighbor query) Distance: G is ε-far from P if need ≥ εdn modifications in incidence lists to get H  P

7. 7 Bounded Degree Graphs – an Example Th. (GR’97): Connectivity in bounded degree model can be tested in O(1/ε 2 ) queries Proof: Assume: G is ε-far from being connected G has ≥ εn connected components G has ≥ εn/2 con. components of size ≤ 2/ε (= small components) ≥ ε/2 percentage of all vertices in small components

8. 8 Property Testing in Bounded Degree Graphs (cont.) Algorithm: Repeat O(1/ε) times: 1. Sample a random vertex v  R V 2. Explore the connected component C(v) of v till accumulate 2/ε vertices 3. If |C(v)| ≤ 2/ε – reject If never reject – accept One-sided error algorithm with complexity O(1/ε 2 ) More careful analysis (1/ε) queries

9. 9 Testing bounded degree graphs – basic tools Random sampling Local search (exploring the neighborhood/ball of a vertex) Random walks (a random neighbor of a random neighbor of a random neighbor…)

10. 10 Bounded degree – first results Results from GR’97: Can test: -connectivity: connectivity in (1/ε) queries 2-edge connectivity: (1/ε 2 ) 3-edge connectivity: (1/ε 3 ) k-vertex connectivity, k=2,3: (1/ε k ) - one-sided error algorithms - cycle-freeness in O(1/ε 3 ) queries - two-sided error algorithm Proof idea: G is ε-far from a forest  many small components with a cycle, or large components C i with large surplus e(C i )-v(C i ) Uses structural connectivity results (block, cactus, etc.)

11. 11 Testing bipartiteness in bounded degree graphs P = “G is bipartite” Lower bound (GR’97): Ω(√n) queries - in very sharp contrast to the dense case Proof idea: Negative distribution D N = Hamilton cycle + random perfect matching (O(1)-far from being bipartite a.s.) Positive distribution D P =Hamilton cycle + random perfect matching between vertices of different parity = D N = D P Any tester: can’t distinguish between D P, D N before having seen a cycle Takes Ω(√n) queries by birthday paradox

12. 12 Testing bipartiteness in bounded degree graphs (cont.) Th. (GR’99): There is a one-sided error algorithm for testing bipartiteness in the bounded degree model in (√n) queries. Algorithm: Repeat T= O(1/ε) times: 1. Choose a random vertex s  R V 2. Perform K:= (√n) random walks of length L:=polylog(n) starting from s 3. If get to the same endvertex by an odd and an even path – reject If no rejection - accept

13. 13 Testing bipartiteness in bounded degree graphs (cont.) Analysis: very elaborate - relatively easy for rapidly mixing case [ s Pr[a random walk of length L starting from s] = Θ(1/n) )] - for general case: no rapid mixing  small cut (M’89) use them to decompose the graph and the problem

14. 14 Testing k-colorability P = “G is k-colorable”; k≥3 – fixed Obviously can be done in O(n) queries (just get all O(dn) edges of G) Th. (BOT’02): For every fixed k ≥3, testing k-colorability in the bounded degree model requires Ω(n) queries  No room for sophisticated testing algorithms

15. 15 Testing k-colorability (cont.) Proof Idea: For one-sided error: Can use classical result of Erdős’62: Th.: There exists G=(V,E), |V|=n, Δ(G)=O(1), G is ε-far from 3-colorable, but: every δn edges form a 3-colorable graph  tester has to obtain ≥ δn edges to catch G 0 G with χ(G 0 )>3 For two-sided error algorithm: -Two distributions (positive, negative) over instances of systems of linear equations; Any algorithm can’t distinguish between them in o(n) time - Then: gap preserving reductions from linear equations to 3- colorability

16. 16 Testing in non-expanding bounded degree graphs Czumaj, Shapira, Sohler’07 Notion of hereditary non-expanding graphs: Def: G is λ-expanding if for every V 0 V(G), |V 0 | ≤n/2, |N(V 0 )|≥ λ |V_0| Def: Graph family F is non-expanding if there exists n 0 =n 0 ( F ) s.t. for all G  F, |V(G)|≥ n 0, G is not (1/log 2 n)-expanding Ex.: F =planar graphs – non-expanding (exists separator of size O(√|V(G)|) Use: G  non-expanding family F, bounded degree  can repeatedly cut G to decompose it into constant sized pieces H 1,H 2,…, number of edges between pieces ≤ ε n/2

17. 17 Testing in non-expanding graphs (cont.) Th. (CSS): P= hereditary property (closed under taking induced subgraphs, say, 3-colorability) Assume: Input G  non-expanding family F of bounded degree subgraphs  P can be tested over F in constant time f(ε) Proof idea: Decompose G=(H 1,H 2,…) as above G=negative instance  many of H i ’s are witnesses  can be found by random sampling + local search

18. 18 Testing planarity Th. (BSS’08) P = “G is planar” P can be tested in time O ε (1) in bounded degree graphs by a 2-sided error algorithm (proved more: every minor-closed property P is testable in constant time) Proof idea: Local statistics in planar graphs differ substantially from those in graphs ε-far from planar (related to hyper-finite graphs, converging sequences of sparse graphs, etc.)

19. 19 Testing planarity (cont.) Remarks: 1. Get two-sided error algorithm, query complexity exp(exp(exp(1/ε))). Better query complexity? 2. Two-sided vs one-sided Ex: G= bounded degree expander of high girth (Θ(log n)) (say, LPS graph) - Θ(1)-far from planar - every c logn edges form a forest  planar subgraph  LB=Ω(log n) can strengthen to Ω(√ n) of GR’97 Conj: P= “G is H-minor free” P can be tested with a one-sided error algorithm in O(√n) queries

20. 20 Bounded degree graphs –open questions Characterization of testable properties? (testable := testable in O ε (1) queries) or at least: wide classes of testable properties One-sided vs two-sided? Comparative study for various properties Testing in restricted graph classes? (á la CSS) Tolerant testing? Estimating distance to a given property?

21. 21 Bounded degree model - limitations Opposite/similar to the dense model Suitable/tailored only for bounded degree graphs Distance notion is “hardwired” – measured always w.r.t. to dn Degenerates for certain properties (e.g. √ n-colorability – always answer “YES”)

22. 22 Testing in graphs of general density - Introduced in KKR’03 Main principles: 1.Distance in measured w.r.t. to the actual size of the input graph (latter can be approximated first if necessary) G=(V,E) is ε-far from P if ≥ ε|E| edges need to be changed to get H  P (appeared already in PR’02) 2.Queries allowed: a) vertex pair queries: whether (i,j)  E(G)? (like in the dense model) b) neighbor queries: j-th neighbor of i  V(G)? (like in the sparse model) c) degree queries: what is d G (i)? No inherent limitation on input graph density

23. 23 Testing bipartiteness in general graphs Th. (KKR’03): 1.Testing bipartiteness can be done in (min(√n, n/d)) queries, where d=2|E|/|V| is the average degree of G; 2.Lower bound of Ω(min(√n, n/d)) - continuous interpolation between the sparse and the dense cases queries n√n√ d n√n√ n

24. 24 Testing bipartiteness for general graphs - proofs Upper bound: Case d≤√n – same as in the bounded degree model K:= O ε (√n), L:=polylog ε (n) Repeat T= O(1/ε) times: 1. Choose a random vertex s  R V 2. Perform K random walks of length L starting from s 3. A 0 = endpoints of walks corresponding to paths of even length A 1 = endpoints of walks corresponding to paths of odd length 4. If A 0 ∩ A 1 ≠Ø – reject, found an odd cycle Never rejected - accept

25. 25 Testing bipartiteness for general graphs – proofs (cont.) Upper bound: Case d≥√n Now:K:= O ε (√(n/d)), L:=polylog ε (n) A 0, A 1 – as before Check whether A 0 or A 1 spans an edge (here use vertex pair queries) If happens – reject Never happens - accept

26. 26 Testing bipartiteness for general graphs – proofs (cont.) Lower bound: Negative distribution D N = G n,d – random d-regular graph Positive distribution D P =G n/2,n/2,d – random bipartite d-regular graph - choose an equipartition V=(V 1,V 2 ) u.a.r. - construct a random d-regular bipartite graph between V 1, V 2 Proof idea: ALG = arbitrary algorithm o(n/d) vertex pair queries  a.s. do not produce an edge have seen o(√n) vertices  a.s. no neighbor query closes a cycle (birthday paradox) o(min(n/d, √n)) queries – both items apply, can’t distinguish between D P, D N

27. 27 Testing triangle-freeness in general graphs Result of AKKR’06 Property P to test = “G is K 3 -free” Most interesting part – Lower Bound d:=average degree of the input graph d≤ n 1-δ(n), δ(n)→ 0  Ω(n 1/3 ) queries are needed d=Θ(n)  O ε (1) queries are enough (AFKS’99)  Threshold-like behavior for query complexity, abrupt change around d=Θ(n) Proof Idea: Cayley graphs, set of generators – random subset of a dense 3AP-free set (c.f. A’02 for the dense case)

28. 28 Comparative study of strength of different query types - BKKR’08 Test case: k-colorability, k≥3 fixed Models to compare:  vertex pair queries  neighbor queries  combined model (pair+neighbor queries)  new query type – group query Group query: v  V - vertex, S – vertex subset ? Whether there is an edge between v and S in G ? YES/NO (and then can find a random edge between v and S in O(log n) queries if needed) -motivated by Group Testing

29. 29 Comparative study of strength of different query types -results On the qualitative level: vertex pair, neighbor < combined model < group query vertex pair queries are better for dense graphs, neighbor queries are better for sparse graphs for group queries: UB=O(n/d) LB= Ω(n/d) (d := average degree of the input graph) Say, in testing bipartiteness

30. 30 Testing general graphs – open problems  Results for (other) concrete problems? (testing H-freeness, k-colorability, etc.)  Develop technology for proving lower bounds  One-sided vs two-sided error algorithms?  What if given ability to sample a random edge? (to eliminate hiding small dense hard instances)  Further query types, their comparison? Query types driven by practical applications?