Improved Parameterized Algorithms for Constraint Satisfaction

Results from inapproximability provide several sharp thresholds on the approximability of important optimization problems. We give several improved parameterized algorithms for solving constraint satisfaction problems above a tight threshold. Our results include the following: • Improved algorithms for any Constraint Satisfaction Problem. Take any boolean Max-CSP with at most c variables per constraint such that a random assignment satisfies a constraint with probability p. There is an algorithm such that for every instance of the problem with m constraints, the algorithm decides whether at least pm + k constraints can be satisfied in O(2m) time. This improves on results of [Alon et al., SODA 2010] who gave a 2 ) +m algorithm, and [Crowston et al., SWAT 2010] who gave a 2 log k) + m algorithm. We observe that an O(2) time algorithm for every ε > 0 would imply that 3SAT is in subexponential time, so it seems unlikely that our runtime dependence on k can be significantly improved. Our proof also shows that every Max-c-CSP has a linear kernel (of at most c(c+ 1)k/2 variables) under this parameterization, and that every Max-c-CSP admits a nontrivial hybrid algorithm. • Better algorithms for problems approximable by SDP. There is an algorithm for Max Cut such that for every graph with m edges, the algorithm outputs a cut with value at least αOPT+k in 2m time (provided such a cut exists), where α = .878 . . . is the Goemans-Williamson constant. Also there is an algorithm for Max-2-Sat such that for 2CNF with m clauses, the algorithm outputs an assignment with value at least αOPT + k in 2m time (provided such an assignment exists), where α = .940 . . . is the Lewin et al. constant. We present these results based on a more general claim for Max-2-CSP. Supported by an EPSRC grant. Email: [email protected] Supported by the Josef Raviv Memorial Fellowship. Email: [email protected]