New Notions and Constructions of Sparsification for Graphs and Hypergraphs

Abstract

A sparsi\xef\xac\x81er of a graph G (Benczu\xc2\xb4r and Karger; Spielman and Teng) is a sparse weighted subgraph \xcb\x9c G that approximately retains the same cut structure of G. For general graphs, non-trivial sparsi\xef\xac\x81cation is possible only by using weighted graphs in which di\xef\xac\x80erent edges have di\xef\xac\x80erent weights. Even for graphs that admit unweighted sparsi\xef\xac\x81ers (that is, sparsi\xef\xac\x81ers in which all the edge weights are equal to the same scaling factor), there are no known polynomial time algorithms that \xef\xac\x81nd such unweighted sparsi\xef\xac\x81ers. We study a weaker notion of sparsi\xef\xac\x81cation suggested by Oveis Gharan, in which the number of cut edges in each cut (S, \xc2\xaf S) is not approximated within a multiplicative factor (1 + \xc7\xab), but is, instead, approximated up to an additive term bounded by \xc7\xab times d \xc2\xb7 |S| + vol(S), where d is the average degree of the graph and vol(S) is the sum of the degrees of the vertices in S. We provide a probabilistic polynomial time construction of such sparsi\xef\xac\x81ers for every graph, and our sparsi\xef\xac\x81ers have a near-optimal number of edges O(\xc7\xab\xe2\x88\x922npolylog(1/\xc7\xab)). We also provide a deterministic polynomial time construction that constructs sparsi\xef\xac\x81ers with a weaker property having the optimal number of edges O(\xc7\xab\xe2\x88\x922n). Our constructions also satisfy a spectral version of the \xe2\x80\x9cadditive sparsi\xef\xac\x81cation\xe2\x80\x9d property. Notions of sparsi\xef\xac\x81cation have also been studied for hypergraphs. Our construction of \xe2\x80\x9cadditive sparsi\xef\xac\x81ers\xe2\x80\x9d with O\xc7\xab(n) edges also works for hypergraphs, and provides the \xef\xac\x81rst non-trivial notion of sparsi\xef\xac\x81cation for hypergraphs achievable with O(n) hyperedges when \xc7\xab and the rank r of the hyperedges are constant. Finally, we provide a new construction of spectral hypergraph sparsi\xef\xac\x81ers, according to the standard de\xef\xac\x81nition, with poly(\xc7\xab\xe2\x88\x921,r)\xc2\xb7nlogn hyperedges, improving over the previous spectral construction (Soma and Yoshida) that used \xcb\x9c O(n3) hyperedges even for constant r and \xc7\xab