On Problems Equivalent to (min,+)-Convolution

Abstract

In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower bounds -reductions from a problem assumed to be hard. The hard problems include 3SUM, All-Pairs Shortest Path, SAT, Orthogonal Vectors, and others.In the (min, +)-convolution problem, the goal is to compute a sequence (c. This can easily be done in O(n 2 ) time, but no O(n 2−ε ) algorithm is known for ε > 0. In this paper, we undertake a systematic study of the (min, +)-convolution problem as a hardness assumption.First, we establish the equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The (min, +)-convolution problem has been used as a building block in algorithms for many problems, notably problems in stringology. It has also appeared as an ad hoc hardness assumption. Second, we investigate some of these connections and provide new reductions and other results. We also explain why replacing this assumption with the SETH might not be possible for some problems.

Fast algorithms for knapsack via convolution and prediction

“…Based on this algorithm, we compute an uncertain solution for the knapsack convolution in almost linear time and via Theorem 3.4 compute a ⋆ b in time O(v max n). Finally, we use the recent technique of [11] to reduce the 0/1 knapsack problem to the knapsack convolution. This yields an O(v max t + n) time algorithm for solving the 0/1 knapsack problem when the item values are bounded by v max .…”

“…Recent evidence suggests that a classic O(nt) dynamic-programming solution for the knapsack problem [2] might be the fastest in the worst case. In fact, solving the knapsack problem was shown to be equivalent to the (min, +) convolution problem [11], which is thought to be facing a quadratic-time barrier. The two-dimensional extension, called the (min, +) matrix product problem, appears in several conditional hardness results.…”

“…The past few years have seen increased attention towards several variants of convolution problems (e.g., [1,3,4,8,11,14,15]). Most importantly, many problems, such as tree sparsity, subset sum, and 3-sum, have been shown to have conditional lower bounds on their running time via their intimate connection with (min, +) convolution.…”

“…In particular, previous studies have shown that (max,+) convolution, knapsack, and tree sparsity are computationally (almost) equivalent [11]. However, these hardness results are obtained by constructing instances with arbitrarily high item values (in the case of knapsack) or vertex weights (in the case of tree sparsity).…”

“…If there is no size or value constraint, it has been shown that knapsack and (max, +) convolution are computationally equivalent with respect to subquadratic algorithms [11]. In other words, any subquadratic solution for knapsack yields a subquadratic solution for (max, +) convolution and vice versa.…”