Into the Square: On the Complexity of Some Quadratic-time Solvable Problems

Michele Borassi (1) , Pierluigi Crescenzi (2, 3) , Michel Habib (4)

1 IMT Institute of Advanced Studies
2 ERABLE - Equipe de recherche européenne en algorithmique et biologie formelle et expérimentale
3 DSI - Dipartimento di Sistemi e Informatica
4 LNC - Laboratoire de Neurosciences Cognitives [Marseille]

We analyze several quadratic-time solvable problems, and we show that these problems are not solvable in truly subquadratic time (that is, in time O(n2−ϵ) for some ϵ>0), unless the well known Strong Exponential Time Hypothesis (in short, SETH) is false. In particular, we start from an artificial quadratic-time solvable variation of the k-Sat problem (already introduced and used in the literature) and we will construct a web of Karp reductions, proving that a truly subquadratic-time algorithm for any of the problems in the web falsifies SETH. Some of these results were already known, while others are, as far as we know, new. The new problems considered are: computing the betweenness centrality of a vertex (the same result was proved independently by Abboud et al.), computing the minimum closeness centrality in a graph, computing the hyperbolicity of a graph, and computing the subset graph of a collection of sets. On the other hand, we will show that testing if a directed graph is transitive and testing if a graph is a comparability graph are subquadratic-time solvable (our algorithm is practical, since it is not based on intricate matrix multiplication algorithms).

Mots clés: Strong Exponential Time Hypothesis; Quadratic-time algorithms; reductions; transitive closure

Domaines: Algorithme et structure de données [cs.DS]

Michele Borassi, Pierluigi Crescenzi, Michel Habib. Into the Square: On the Complexity of Some Quadratic-time Solvable Problems. Electronic Notes in Theoretical Computer Science, 2016, 322, pp.51-67. ⟨10.1016/j.entcs.2016.03.005⟩. ⟨hal-01390131⟩