Improved parameterized algorithms for above average constraint satisfaction

• Authors: Eun Jung Kim;Ryan Williams
• Venue: IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
• Year: 2011

For many constraint satisfaction problems, the algorithm which chooses a random assignment achieves the best possible approximation ratio. For instance, a simple random assignment for Max-E3-Sat allows 7/8-approximation and for every ε0 there is no polynomial-time (7/8+ε)-approximation unless P=NP. Another example is the Permutation CSP of bounded arity. Given the expected fraction ρ of the constraints satisfied by a random assignment (i.e. permutation), there is no (ρ+ε)-approximation algorithm for every ε0, assuming the Unique Games Conjecture (UGC). In this work, we consider the following parameterization of constraint satisfaction problems. Given a set of m constraints of constant arity, can we satisfy at least ρm+k constraint, where ρ is the expected fraction of constraints satisfied by a random assignment? Constraint Satisfaction Problems above Average have been posed in different forms in the literature [18,17]. We present a faster parameterized algorithm for deciding whether m/2+k/2 equations can be simultaneously satisfied over F2. As a consequence, we obtain O(k)-variable bikernels for boolean CSPs of arity c for every fixed c, and for permutation CSPs of arity 3. This implies linear bikernels for many problems under the "above average" parameterization, such as Max-c-Sat, Set-Splitting, Betweenness and Max Acyclic Subgraph. As a result, all the parameterized problems we consider in this paper admit 2O(k)-time algorithms. We also obtain non-trivial hybrid algorithms for every Max c-CSP: for every instance I, we can either approximate I beyond the random assignment threshold in polynomial time, or we can find an optimal solution to I in subexponential time.