Graph expansion and the unique games conjecture

Abstract

The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a d-regular graph, the edge expansion/conductance Φ(S) of a subset S ⊆ V is defined as Φ(S) = |E(S,V \S)| d|S| . Approximating the conductance of small linear sized sets (size δn) is a natural optimization question that is a variant of the well-studied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1 − ε), and close to 0 expansion.In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following:-We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs.This is the first non-trivial "reverse" reduction from a natural optimization problem to Unique Games.-Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UG-hard to approximate small set expansion.-On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].

Inapproximability of Treewidth and Related Problems

“…In the work of Raghavendra and Steurer (2010), the Small Set Expansion (SSE) Conjecture was introduced, and it was shown that it implies the UGC, and that the SSE Conjecture follows if one assumes that the UGC is true for somewhat expanding graphs. In follow-up work by Raghavendra et al (2012), it was shown that the SSE Conjecture is in fact equivalent to the UGC on somewhat expanding graphs, and that the SSE Conjecture implies NP-hardness of approximation for balanced separator and MLA.…”

“…Similarly, the approximability of many related graph layout problems is also unresolved, including Minimum Cut Linear Arrangement and Interval Graph Completion. In this paper, we make an important step to resolve this problem by showing that Treewidth, Pathwidth, and a host of related graph layout problems are hard to approximate to within any constant factor, under the Small Set Expansion (SSE) conjecture (Raghavendra & Steurer, 2010).…”

Algorithmic Extensions of Cheeger’s Inequality to Higher Eigenvalues and Partitions

“…The sparsest small set problem has been shown to be closely related to the Unique Games problem (see [RS10,ABS10] ). Recently, Arora et.…”

On the Unique Games Conjecture

“…We don't describe it here, but do point out that it is hypercube based and the fact that the (integral) optimum is low is related to the fact that small subsets of a boolean hypercube are almost fully expanding. This partly motivates Theorem 5.2 giving a connection between the UGC and the Small Set Expansion problem [94].…”

“…Raghavendra and Steurer [94] give a reduction from the Small Set Expansion problem to the Unique Game problem. Whereas we have several reductions from the Unique Game problem to other optimization problems, this is the first reduction in the other direction.…”