On problems equivalent to (min,+)-convolution

Marek Cygan, Marcin Mucha, Karol Wȩgrzycki, Michał Włodarczyk

Research output: Contribution to journalArticlepeer-review

43 Scopus citations


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[i])n-1 i=0 , wherec[k] = mini=0, k {a[i] + b[k -i]}, given sequences (a[i])n-1 i=0 and (b[i])n-1 i=0 . This can easily be done in O(n2) time, but no O(n2-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.

Original languageEnglish
Article numberA14
JournalACM Transactions on Algorithms
Issue number1
StatePublished - 1 Jan 2019
Externally publishedYes


  • (min,+)- convolution
  • Fine-grained complexity
  • conditional lower bounds
  • knapsack
  • subquadratic equivalence

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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