Finding Small Satisfying Assignments Faster Than Brute Force: A Fine-grained Perspective into Boolean Constraint Satisfaction
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To study the question under which circumstances small solutions can be found faster than by exhaustive search (and by how much), we study the fine-grained complexity of Boolean constraint satisfaction with size constraint exactly k. More precisely, we aim to determine, for any finite constraint family, the optimal running time f(k)n^g(k) required to find satisfying assignments that set precisely k of the n variables to 1. Under central hardness assumptions on detecting cliques in graphs and 3-uniform hypergraphs, we give an almost tight characterization of g(k) into four regimes: (1) Brute force is essentially best-possible, i.e., g(k) = (1± o(1))k, (2) the best algorithms are as fast as current k-clique algorithms, i.e., g(k)=(ω/3± o(1))k, (3) the exponent has sublinear dependence on k with g(k) ∈ [Ω(√(k)), O(√(k))], or (4) the problem is fixed-parameter tractable, i.e., g(k) = O(1). This yields a more fine-grained perspective than a previous FPT/W[1]-hardness dichotomy (Marx, Computational Complexity 2005). Our most interesting technical contribution is a f(k)n^4√(k)-time algorithm for SubsetSum with precedence constraints parameterized by the target k – particularly the approach, based on generalizing a bound on the Frobenius coin problem to a setting with precedence constraints, might be of independent interest.
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