An Efficient Sparse Regularity Concept

Amin Coja-Oghlan,Colin Cooper,Alan Frieze

doi:10.1137/080730160

Abstract

Let ${\bf A}$ be a $0/1$ matrix of size $m\times n$, and let p be the density of ${\bf A}$ (i.e., the number of ones divided by $m\cdot n$). We show that ${\bf A}$ can be approximated in the cut norm within $\varepsilon\cdot mnp$ by a sum of cut matrices (of rank 1), where the number of summands is independent of the size $m\cdot n$ of ${\bf A}$, provided that ${\bf A}$ satisfies a certain boundedness condition. This decomposition can be computed in polynomial time. This result extends the work of Frieze and Kannan [Combinatorica, 19 (1999), pp. 175–220] to sparse matrices. As an application, we obtain efficient $1-\varepsilon$ approximation algorithms for “bounded” instances of MAX CSP problems.

Highlights

Introduction and resultsFor many fundamental optimization problems there are known NP-hardness of approximation results, showing that is it NP-hard to compute the optimum exactly but even to approximate the optimum within a factor bounded away from 1
We show that A can be approximated in the cut norm within ε · mnp by a sum of cut matrices, where the number of summands is independent of the size m · n of A, provided that A satisfies a certain boundedness condition
If G = (V, E) is a graph on n vertices of density p = 2n−2|E|, its MAX CUT can be approximated within a factor of 1 − ε in time poly(exp((εp)−2) · n)

Summary

Suppose that

Instead of scaling as a tower function T ((C/ε)9), the running time of the algorithm WeakPartition from Corollary 3 is bounded by exp(O(C/ε)2) in terms of C, ε. This may still seem impractical, this is just a worst-case upper bound, and it is quite conceivable that it is practically much easier to find a good approximation in the cut norm than a good regular partition. As Theorem 1 shows, one can approximate a (C, γ)-bounded adjacency matrix by a sum of O(C/ε) cut matrices (if the actual partition of the vertex set is not needed), avoiding the exponential dependence on C/ε.

ApxMatrix for Theorem

But for

Di satisfies

Di and

Dψi with

Findings

The number of edges within the set