No abstract

We study the problem of graph summarization. Given a large graph we aim at producing a concise lossy representation that can be stored in main memory and used to approximately answer queries about the original graph much faster than by using the exact representation. In this paper we study a very natural type of summary: the original set of vertices is partitioned into a small number of supernodes connected by superedges to form a complete weighted graph. The superedge weights are the edge densities between vertices in the corresponding supernodes. The goal is to produce a summary that minimizes the reconstruction error w.r.t. the original graph. By exposing a connection between graph summarization and geometric clustering problems (i.e., k-means and k-median), we develop the first polynomial-time approximation algorithm to compute the best possible summary of a given size.

We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any disjoint source-sink pairs on the directed hypercube can be connected with edge-disjoint paths, then monotonicity of functions f : {0, 1}n → R can be tested with O(n) queries, for any totally ordered range R. More generally, if at least an α(n) fraction of the pairs can always be connected with edge-disjoint paths then the query-complexity is O(n/α(n)).We construct a family of instances of = Ω(2 n ) pairs in n-dimensional hypercubes such that no more than roughly a 1 √ n fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [LR01], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≈ n 3/2 . Additionally, our construction can also be used to obtain a strong counterexample to Szymanski's conjecture on routing in the hypercube. In particular, we show that for any δ > 0, the n-dimensional hypercube is not n 1 2 −δ -realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths.We also prove a lower bound of Ω(n/ ) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.

We develop a query-efficient sample extractor for juntas, that is,

We study the problem of testing isomorphism (equivalence up to relabelling of the variables)