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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).
The Maximum Likelihood Decoding Problem (MLD) and the Multivariate Quadratic System Problem (MQ) are known to be NP-hard. In this paper we present a polynomial-time reduction from any instance of MLD to an instance of MQ, and viceversa.
We assess the computational complexity of several decision problems concerning (Murray-von Neumann) equivalence classes of projections of AF-algebras whose Elliott classifier is lattice-ordered. We construct polytime reductions among many of these problems.
In this paper we consider the problems of testing isomorphism of tensors, $p$-groups, cubic forms, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. These problems can all be cast as
Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem with four colors is NP-complete, and they showed that the infinite variant of this problem is undecidable. In this paper, we st
We study the NP-hard textsc{$k$-Sparsest Cut} problem ($k$SC) in which, given an undirected graph $G = (V, E)$ and a parameter $k$, the objective is to partition vertex set into $k$ subsets whose maximum edge expansion is minimized. Herein, the edge