1. 1 Algorithmic Aspects in Property Testing of Dense Graphs Oded Goldreich – Weizmann Institute Dana Ron - Tel-Aviv University

2. 2 Let G = (V,E) be a graph.  If G has property P: accept w.h.p.  If G is “  -far” from P: reject w.h.p. “  -far”: “  -fraction of the graph” should be modified to obtain P. A Testing Algorithm for graph property P can query the graph G on the neighborhood relations of vertices in G. Graph Property Testing

3. 3 Models for Testing Graph Properties 1 2 … d 1 n Bounded-Degree Graphs Model [Goldreich, R] : (graph is represented by n incidence lists of size d)  queries: who is i’th neighbor of v?   -far:  d  n edges should be modified.  suitable: (almost)-regular sparse graphs (in particular, constant-degree graphs) Dense Graphs Model [Goldreich, Goldwasser, R] : (graph is represented by n x n adjacency matrix)  queries: is (u,v)  E ?   -far:  n 2 edges should be modified.  suitable: dense graphs v 1 u

4. 4 Algorithmic Aspects of Testing Graph Properties Intuitively, while in the bounded-degree model there is place for algorithmic design (e.g., various forms of local search, random walks), in the dense-graphs model it seems that all one can do is take random induced subgraphs, so that it boils down to combinatorics. Formally, [Alon, Fischer, Krivelevich, Szegedy] [Goldreich & Trevisan] showed that if one ignores a quadratic blow-up in the complexity, then indeed it can be assumed w.l.o.g that the algorithm simply takes a uniform sample of vertices and queries all pairs in the sample (thus obtaining the induced subgraph). We refer to this algorithm as the canonical algorithm

5. 5 Algorithmic Aspects in the Dense-Graphs Model But what if one does care about a quadratic blow up? Are there cases where we can reduce the complexity by doing something more sophisticated than taking a random induces subgraph ? In particular, can adaptivity help? (A tester is adaptive if its queries depend on answers to previously asked queries – the canonical tester is clearly non-adaptive.) Note: In the bounded degree model testers are “adaptive by nature” [Raskhodnikova & Smith].

6. 6 Adaptivity in the Dense-Graphs Model This question (can adaptivity help in the dense-graphs model) was first posed for testing bipartiteness. [Alon&Krivelevich] strengthened [GGR] and proved that it suffices for the canonical algorithm to take a sample of Õ(1/  ) vertices (so that its query complexity is Õ(1/  2 ) ) [Bogdanov&Trevisan] proved that  (1/  2 ) queries are necessary for any non-adaptive algorithm, while for an adaptive algorithm the lower bound is  (1/  3/2 ) [Gonen&R] gave an adaptive algorithm whose complexity is Õ(1/  3/2 ) for the case that the degree of the graph is O(  n) (the lower bounds of [BT] hold for this case).

7. 7 Adaptivity in the Dense-Graphs Model [GonR]: For every  testing bipartiteness of graphs with max- degree O(  n) can be done adaptively with Õ( 1/   ) queries, while by [BT]  (1/  2 ) queries are necessary for non-adaptive algorithms. This result leaves open the question of whether there is a single property for which such a gap exists for every  In this work we show that such a gap (4/3 in the exponent) exists for Clique-Collection, a larger gap (3/2 ) exists for BiClique-Collection, and we conjecture that a certain property gives an almost maximum gap (2-  for any constant  ). We also show that there are cases in which non- adaptive (but carefully designed) algorithms can be optimal. (Won’t discuss in this talk.)

8. 8 Main Two Properties Studied Clique Collection (denoted CC): Graphs that consist of a union of cliques (of any number and size) Bi-Clique Collection (denoted BCC): Graphs that consist of a union of bi-cliques (each bi-clique is complete bipartite subgraph with each side an ind. set) Property corresponds to a “perfect clustering” Property corresponds to a “perfect bi-clustering” (e.g., when have a relation over two types of entities (

9. 9 Main Results Thm 1: - There exists an adaptive tester for CC with query complexity and running time Õ (1/  ). - Every non-adaptive algorithm for CC must have query complexity  (1/  4/3 ). ( Furthermore, result is tight.) Thm 2: - There exists an adaptive tester for BCC with query complexity and running time Õ (1/  ). - Every non-adaptive algorithm for BCC must have query complexity  (1/  3/2 ).

10. 10 Conjecture: almost quadratic gap A Super-Cycle of length t consists of t ind. sets that are arranged on a cycle such that each pair of adjacent ind. sets have complete bipartite graph between them. A t -Super-Cycle Collection ( t -SCC) is a union of super- cycles of length at most t. Can show that every non-adaptive algorithm for t -SCC must have query complexity  (1/  2-2/t ). Conjecture that exists an adaptive tester for t -SCC with query complexity and running time Õ(1/  ) (holds under a promise).

11. 11 Adaptive Algorithm for CC Basic observation: If G ( = (V,E) )  CC then the following holds for every v  V: for every two neighbors u and w of v, (u,w)  E, and for every neighbor u and non-neighbor z we have: (u,z)  E. v z w u High-level idea of Algorithm: select sample of (“start”) vertices. For each start vertex v take sample of neighbors and of non-neighbors of v and verify that above holds. (To sample neighbors and non-neighbors, take sample S and query all pairs (v,x) for x  S ) Remember: total number of queries should be Õ (1/  ) so should be thrifty… v S N S (v)

12. 12 Adaptive Algorithm for CC Let s 1 =  (1), s 2 =  (log 3 (1/  )), m =  (log(1/  )). For j = 1,…,m select s 1 2 j start vertices and for each start vertex v do: 1.Select random subset S of s 2 /(2 j  ) vertices. 2.Determine N S (v) (neighbors of v in S). 3.Select sample of s 2 /(2 j  ) pairs in N S (v) x N S (v) and check that each is an edge (if too few neighbors then consider all pairs). 4.Select sample of s 2 /(2 j  ) pairs in N S (v) x (S\N S (v)) and check that each is not an edge If all checks pass then accept, otherwise, reject v S N S (v) Query complexity and running time:

13. 13 Illustrative example for Alg for CC Consider following case of graph that should be rejected: The graph consists of subsets of size  (  n) where in each there are a constant fraction of missing edges. In first iteration select constant num of vertices. For each v selected, select subset S of size Õ(1/  ). W.h.p, |N S (v)| = log c (1/  ) and among them have pair with no edge between them. Since alg is adaptive it can “focus” on the neighbors. If it is not adaptive it “can’t know” where to ask the queries.

14. 14 Analysis of Alg for CC Accepts every graph in CC w.p. 1. To prove that rejects w.p. at least 2/3 graphs that are  -far from CC, prove contrapositive statement: if alg accepts w.p. > 1/3 then graph is  -close to CC. This is done by iteratively constructing a collection of (almost) cliques (with few edges between them) using the neighbor sets of “good” vertices: Vertex v is good if N(v) is close to being a clique and few edges going out of N(v).

15. 15 Lower bound for non-adaptive testing of CC Define Two families that are hard to distinguish non-adaptively, in less than 1/  4/3 queries. First family: graphs consist of 1/(3  ) cliques, each of size 3  n; Second family: graphs consist of 1/(6  ) bi-cliques, each of size 2  3  n;

16. 16 Summary Main Conjecture:  t ≥ 3 have property for which adaptive complexity is while non-adaptive is For t = 3 : CC – adaptive is non-adaptive is For t = 4: BCC - adaptive is non-adaptive is u.b (for non-adaptive) is open. For general t proved l.b. and showed that holds for promise problem.

17. 17 Conclusions and Open problems  Adaptivity can be beneficial in the dense-graphs model.  The dense-graphs model is not all about combinatorics: algorithmic aspects play a role.  What is the complexity of adaptive algorithms for testing bipartiteness of general graphs? (Comment: if all degrees at least  1/2 n then complexity is Õ(1/  3/2 ))  Is our conjecture correct? (I.e., there exists a quadratic gap for the property we suggest (or another property))?  For what constants c  [1,2] is there a gap (in the exponent) of a factor of c?  Characterize the class of graph properties for which c=1.

18. 18 Thanks