Exact Algorithms via Monotone Local Search

Abstract

We give a new general approach for designing exact exponential-time algorithms for subset problems . In a subset problem the input implicitly describes a family of sets over a universe of size n and the task is to determine whether the family contains at least one set. A typical example of a subset problem is W EIGHTED d -SAT. Here, the input is a CNF-formula with clauses of size at most d , and an integer W . The universe is the set of variables and the variables have integer weights. The family contains all the subsets S of variables such that the total weight of the variables in S does not exceed W and setting the variables in S to 1 and the remaining variables to 0 satisfies the formula. Our approach is based on “monotone local search,” where the goal is to extend a partial solution to a solution by adding as few elements as possible. More formally, in the extension problem, we are also given as input a subset X of the universe and an integer k . The task is to determine whether one can add at most k elements to X to obtain a set in the (implicitly defined) family. Our main result is that a c k n O(1) time algorithm for the extension problem immediately yields a randomized algorithm for finding a solution of any size with running time O ((2−1/ c ) n ). In many cases, the extension problem can be reduced to simply finding a solution of size at most k . Furthermore, efficient algorithms for finding small solutions have been extensively studied in the field of parameterized algorithms. Directly applying these algorithms, our theorem yields in one stroke significant improvements over the best known exponential-time algorithms for several well-studied problems, including d -H ITTING S ET , F EEDBACK V ERTEX S ET , N ODE U NIQUE L ABEL C OVER , and W EIGHTED d -SAT. Our results demonstrate an interesting and very concrete connection between parameterized algorithms and exact exponential-time algorithms. We also show how to derandomize our algorithms at the cost of a subexponential multiplicative factor in the running time. Our derandomization is based on an efficient construction of a new pseudo-random object that might be of independent interest. Finally, we extend our methods to establish new combinatorial upper bounds and develop enumeration algorithms.