On r-simple k-path

Hasan Abasi, Nader H. Bshouty, Ariel Gabizon, Elad Haramaty

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length greater than 4logk/logr. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex. We then give a randomized algorithm that runs in time that solves the r-SIMPLE k-PATH on a graph with n vertices with one-sided error. We also show that a randomized algorithm with running time poly(n) ·2(c/2)k/ r with c<1 gives a randomized algorithm with running time poly(n)·2 cn for the Hamiltonian path problem in a directed graph - an outstanding open problem. So in a sense our algorithm is optimal up to an O(logr) factor in the exponent. The crux of our method is to use low degree testing to efficiently test whether a polynomial contains a monomial where all individual degrees are bounded by a given r.

Original languageEnglish
Title of host publicationMathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings
Pages1-12
Number of pages12
EditionPART 2
DOIs
StatePublished - 2014
Event39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014 - Budapest, Hungary
Duration: 25 Aug 201429 Aug 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 2
Volume8635 LNCS

Conference

Conference39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014
Country/TerritoryHungary
CityBudapest
Period25/08/1429/08/14

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science

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