Locally Expanding Hypergraphs and the Unique Games Conjecture

We examine the hardness of approximating constraint satisfaction problems with k-variable constraints, known as k-CSP's. We are specif- ically interested in k-CSP's whose constraints are unique, which means that for any assignment to any k 1 of the variables, there is a unique assignment to the last variable satisfying the constraint. One fundamental example of these CSP's is Ek-Lin-p, the problem of satisfying as many equations as possible from an over-determined system of linear equations modulo a prime p, where each equation contains exactly k variables. The central question in much of the recent work on inapproximability has been the Unique Games Conjecture, which posits a very strong hard- ness of approximation for 2-CSP's with unique constraints, such as E2- Lin-p. Many strong inapproximability results have been proven assuming that it is true, including a recent result of Raghavendra ("Optimal algo- rithms and inapproximability results for every CSP?" in STOC. ACM, 2008.) giving an approximation algorithm for every CSP, whose perfor- mance is essentially optimal if the Unique Games Conjecture is true. To date, however, not much progress has been made on resolving the conjec- ture. In this paper, we give a reduction from unique 3-CSP's to unique 2- CSP's which is sometimes approximation-preserving, depending on the combinatorial structure of the underlying hypergraph of the 3-CSP. The underlying hypergraph of a 3-CSP is the hypergraph in which each vertex represents a variable, and every hyperedge represents a constraint. Every constraint c yields a hyperedge between the three variables involved in c. The reduction only works when the underlying hypergraph of the 3-CSP satisfies a (hypergraph) expansion property, which we call local expansion. We prove that the Unique Games Conjecture is equivalent to a hard- ness result for unique 3-CSP's whose underlying hypergraphs are local expanders. We give a precise formulation of the desired hardness result as a conjecture, which we call the Expanding Unique 3-CSP's Conjecture. We also give a restricted, but still equivalent, conjecture that E3-Lin-p is