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 article, 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 Strong Exponential Time Hypothesis might not be possible for some problems.
In REF , the authors present an equivalence class for (min, +)-convolution, including some variants of the classical knapsack problem and problems related to subadditive sequences.
On Problems Equivalent to (min,+)-Convolution
TALG
Computer Science Mathematics