Reductions between Expansion Problems

Abstract

The Small-Set Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge expansion of small sets in graphs. This hardness assumption is closely connected to the Unique Games Conjecture (Khot, STOC 2002). In particular, the Small-Set Expansion Hypothesis implies the Unique Games Conjecture (Raghavendra, Steurer, STOC 2010).Our main result is that the Small-Set Expansion Hypothesis is in fact equivalent to a variant of the Unique Games Conjecture. More precisely, the hypothesis is equivalent to the Unique Games Conjecture restricted to instance with a fairly mild condition on the expansion of small sets. Alongside, we obtain the first strong hardness of approximation results for the Balanced Separator and Minimum Linear Arrangement problems. Before, no such hardness was known for these problems even assuming the Unique Games Conjecture.These results not only establish the Small-Set Expansion Hypothesis as a natural unifying hypothesis that implies the Unique Games Conjecture, all its consequences and, in addition, hardness results for other problems like Balanced Separator and Minimum Linear Arrangement, but our results also show that the Small-Set Expansion Hypothesis problem lies at the combinatorial heart of the Unique Games Conjecture.The key technical ingredient is a new way of exploiting the structure of the Unique Games instances obtained from the Small-Set Expansion Hypothesis via (Raghavendra, Steurer, 2010). This additional structure allows us to modify standard reductions in a way that essentially destroys their local-gadget nature. Using this modification, we can argue about the expansion in the graphs produced by the reduction without relying on expansion properties of the underlying Unique Games instance (which would be impossible for a local-gadget reduction).

Inapproximability of Treewidth and Related Problems

“…To the best of our knowledge, there is no hardness of approximation for MCLA in the literature. Its cousin MLA was recently proved SSE-hard to approximate within any constant factor (Raghavendra et al, 2012), and we observe that the same hardness applies to the MCLA problem.…”

“…On the hardness side, our work builds upon the work by Raghavendra et al (2012), which showed that the SSE Conjecture implies superconstant hardness of approximation for MLA (and for c-balanced separator). The only other hardness of relative approximation that we are aware of for these problems is a result of the work of Ambühl et al (2007), showing that MLA does not have a PTAS unless NP has randomized subexponential time algorithms.…”

“…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. In this light, the Small Set Expansion conjecture serves as a natural unified conjecture that yields all of the implications of UGC and also hardness for expansion-like problems that could not be resolved with the UGC.…”

“…An improved algorithm for c-balanced separator will also improve the approximation algorithms for the other problems. On the other hand, hardness of approximating c-balanced separator (Raghavendra et al, 2012) does not necessarily imply hardness of approximating layout problems.…”

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

“…For a constant ∈ (−1, 1), let N k, denote the infinite graph over R k where the weight of an edge (x, y) is the probability that two standard Gaussian random vectors X, Y with correlation 3 equal x and y respectively. The first k eigenvalues of N k, are at least 1 − (see [RST10b]). The following lemma bounds the expansion of small sets in N k, .…”