@inproceedings{59fed29694134d159d1b4290ee130127,

title = "On r-simple k-path",

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.",

author = "Hasan Abasi and Bshouty, {Nader H.} and Ariel Gabizon and Elad Haramaty",

year = "2014",

doi = "https://doi.org/10.1007/978-3-662-44465-8_1",

language = "الإنجليزيّة",

isbn = "9783662444641",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

number = "PART 2",

pages = "1--12",

booktitle = "Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings",

edition = "PART 2",

note = "39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014 ; Conference date: 25-08-2014 Through 29-08-2014",

}