Fast Witness Counting
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We study the witness-counting problem: given a set of vectors V in the d-dimensional vector space over F_2, a target vector t, and an integer k, count all ways to sum-up exactly k different vectors from V to reach t. The problem is well-known in coding theory and received considerable attention in complexity theory. Recently, it appeared in the context of hardware monitoring. Our contribution is an algorithm for witness counting that is optimal in the sense of fine-grained complexity. It runs in time O^*(2^d) with only a logarithmic dependence on m=|V|. The algorithm makes use of the Walsh-Hadamard transform to compute convolutions over F_2^d. The transform, however, overcounts the solutions. Inspired by the inclusion-exclusion principle, we introduce correction terms. The correction leads to a recurrence that we show how to solve efficiently. The correction terms are obtained from equivalence relations over F_2^d. We complement our upper bound with two lower bounds on the problem. The first relies on # ETH and prohibits an 2^o(d)-time algorithm. The second bound states the non-existence of a polynomial kernel for the decision version of the problem.
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