Approximating unique games

ABSTRACT

The UNIQUE GAMES problem is the following: we are given a graph G = (V, E), with each edge e = (u, v) having a weight we and a permutation πuv on [k]. The objective is to find a labeling of each vertex u with a label fu ∈ [k] to minimize the weight of unsatisfied edges---where an edge (u, v) is satisfied if fv = πuv(fu).The Unique Games Conjecture of Khot [8] essentially says that for each ε > 0, there is a k such that it is NP-hard to distinguish instances of Unique games with (1-ε) satisfiable edges from those with only ε satisfiable edges. Several hardness results have recently been proved based on this assumption, including optimal ones for Max-Cut, Vertex-Cover and other problems, making it an important challenge to prove or refute the conjecture.In this paper, we give an O(log n)-approximation algorithm for the problem of minimizing the number of unsatisfied edges in any Unique game. Previous results of Khot [8] and Trevisan [12] imply that if the optimal solution has OPT = εm unsatisfied edges, semidefinite relaxations of the problem could give labelings with min {k²ε^1/5, (ε log n)^1/2}m unsatisfied edges. In this paper we show how to round a LP relaxation to get an O(log n)-approximation to the problem; i.e., to find a labeling with only O(εm log n) = O(OPT log n) unsatisfied edges.

REFERENCES

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory
          Subjects: Graph algorithms

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Linear approximation
  G.2 DISCRETE MATHEMATICS
      G.2.1 Combinatorics
          Subjects: Permutations and combinations

General Terms:
Algorithms, Theory