On Problems Equivalent to (min, +)-Convolution

In the recent years, significant progress has been made in explaining apparent hardness ofimproving over naive solutions for many fundamental polynomially solvable problems. Thiscame in the form of conditional lower bounds – reductions to one of problems assumed to behard. These include 3SUM, All-Pairs Shortest Paths, SAT and Orthogonal Vectors, and others.In the (min,+)-convolution problem, the goal is to compute a sequence (c[i])i=0 , wherec[k] =mini=0,...,k{a[i] + b[k − i]}, given sequences (a[i])i=0 and (b[i])i=0 . This can easily bedone in O(n2) time, but no O(n2−ε) algorithm is known for ε > 0. In this paper we undertakea systematic study of the (min,+)-convolution problem as a hardness assumption.As the first step, we establish equivalence of this problem to a group of other problems, in-cluding variants of the classic knapsack problem and problems related to subadditive sequences.The (min,+)-convolution has been used as a building block in algorithms for many problems,notably problems in stringology. It has also already appeared as an ad hoc hardness assumption.We investigate some of these connections and provide new reductions and other results. This work is part of a project TOTAL that has received funding from the European Research Council (ERC)under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 677651).Institute of Informatics, University of Warsaw, Poland, {cygan, mucha, k.wegrzycki,m.wlodarczyk}@mimuw.edu.pl