Improved Algorithms for Sparse MAX-SAT and MAX-k-CSP

Abstract

We give improved deterministic algorithms solving sparse instances of MAX-SAT and MAX-k-CSP. For instances with n variables and cn clauses (constraints), we give algorithms running in time ${{\mathrm{poly}}}(n)\cdot 2^{n(1-\mu )}$ for

• $\mu = \Omega (\frac{1}{c} )$ and polynomial space solving MAX-SAT and MAX-k-SAT,

• $\mu = \Omega (\frac{1}{\sqrt{c}} )$ and exponential space solving MAX-SAT and MAX-k-SAT,

• $\mu = \Omega (\frac{1}{ck^2} )$ and polynomial space solving MAX-k-CSP,

• $\mu = \Omega (\frac{1}{\sqrt{ck^3}} )$ and exponential space solving MAX-k-CSP.

The previous MAX-SAT algorithms have savings $\mu =\Omega (\frac{1}{c^2 \log ^2 c})$ for running in polynomial space [15] and $\mu =\Omega (\frac{1}{c \log c})$ for exponential space [5]. We also give an algorithm with improved savings for satisfiability of depth-2 threshold circuits with cn wires.

R. Chen—Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement no. 615075.

R. Santhanam—Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement no. 615075.