Min-Max Graph Partitioning and Small Set Expansion

Abstract

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal-size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O( √ log n log k)approximation algorithm. This improves over an O(log 2 n) approximation for the second version due to Svitkina and Tardos [ST04], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)-approximation algorithm for graphs that exclude any fixed minor.Our algorithm uses a new procedure for solving the Small-Set Expansion problem. In this problem, we are given a graph G and the goal is to find a non-empty set S ⊆ V of size |S| ≤ ρn with minimum edge-expansion. We give an O( log n log (1/ρ)) bicriteria approximation algorithm for the general case of Small-Set Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.

Minimum nonuniform graph partitioning with unrelated weights

“…This statement was proved in Bansal et al (2011) implicitly, so for completeness we prove it in the Appendix -see Theorem A.1. For graphs with excluded minors and bounded genus graphs,…”

“…The notion of orthogonal separators was introduced in Chlamtac, Makarychev, and Makarychev (2006), where it was used in an algorithm for Unique Games. Later, Bansal et al (2011) showed that the following holds. If the SDP solution satisfies constraints (3), (4), (6), and (7), then for every ε ∈ (0, 1), δ ∈ (0, 1), and i ∈ [k], there exist a distortion D i = O ε ( log n log(1/(δρ i ))), and a probability distribution over subsets of V such that for a random set S i ⊂ V ("orthogonal separator") distributed according to this distribution, we have for α = 1/n,…”

“…Our relaxation for the problem is based on the SDP relaxation for the Small Set Expansion (SSE) problem of Bansal et al (2011). We write the SSE relaxation for every cluster P i and then add consistency constraints similar to constraints used in Unique Games.…”

“…We add constraint (2) saying that µ i (P i ) ≤ ρ i . We further add spreading constraints (4) from Bansal et al (2011) (see also Louis and Makarychev (2014)). The spreading constraints above are satisfied in the integral solution: If u / ∈ P i , thenū i = 0 and both sides of the inequality equal 0.…”

“…Let us examine a somewhat naïve algorithm for the problem inspired by the algorithm of Bansal et al (2011) for Small Set Expansion. We shall maintain the set of active (yet unassigned) vertices A(t).…”