New Bounds for the CLIQUE-GAP Problem Using Graph Decomposition Theory

Abstract

Halldórsson et al (ICALP proceedings of the 39th international colloquium conference on automata, languages, and programming, vol part I, Springer, pp 449-460, 2012) investigated the space complexity of the following problem CLIQUE-GAP(r, s):In particular, they give matching upper and lower bounds for CLIQUE-GAP(r, s) for any r and s = c log(n), for some constant c. The space complexity of the CLIQUE-GAP problem for smaller values of s is left as an open question. In this paper, we answer this open question. Specifically, for any r and for s =Õ(log(n)), we prove that the space complexity of CLIQUE-GAP problem is˜ ( ms 2 r 2 ). Our lower bound is based on a new connection between graph decomposition theory (Chung et al in Studies in pure mathematics, Birkhäuser, Basel, pp 95-101, 1983; Chung in SIAM J Algebr Discrete Methods 2(1):1-12, 1981) and the multi-party set disjointness problem in communication complexity.

Independent Sets in Vertex-Arrival Streams

“…We can 3-approximate MIS for a stream of unit squares in the plane using Õ(α(G)) space, and achieving better than a 5 2 -approximation to α(G) requires Ω(n) space. Unit interval intersection graphs given as an explicit vertex stream require Ω(n) space to get a better than 5 3 approximation to α(G), making it harder than the implicit stream equivalent. Figures 2 and 1 illustrate these results and put them in context to previously known bounds (see also Section 1.3).…”

“…A corresponding Õ n 2 c 2 space random sampling algorithm shows this is tight up to logarithmic factors. Braverman et al [5] showed that space Ω( m c 2 ) is needed, even if c = o(log n), where m is the number of edges of the input graph. This bound though only holds for small values of m.…”

“…Braverman et al [5] showed that in a vertex arrival model, where every vertex arrives together with all its incident edges (as opposed to the explicit vertex stream model considered here where every vertex arrives together with its incident edges connecting to vertices that have previously arrived), space Ω( m c 3 ) is required for computing a c-approximate MIS. In their construction the input graph has Θ(nc) edges, which thus yields a lower bound of Ω( n c 2 ).…”

“…Otherwise, α(G) = 5. Hence, any algorithm giving a better than 5 3 (one-sided) approximation factor could distinguish them and solve INDEX.…”

One Sketch to Rule Them All

“…These well-known limitations of sampling motivated an alternative class of techniques based on sketching or streaming algorithms. Here, custom online algorithms and data structures are designed for specific metrics of interest that can yield provable resource-accuracy tradeoffs (e.g., [17,18,20,31,36,38,43]).…”