Space complexity of reachability testing in labelled graphs

Vidhya Ramaswamy, Jayalal Sarma, K. S. Sunil

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Fix an algebraic structure (A, ∗). Given a graph G = (V, E) and the labelling function φ (φ: E → A) for the edges, two nodes s, t ∈ V, and a subset F ⊆ A, the A-Reach problem asks if there is a path p (need not be simple) from s to t whose yield (result of the operation in the ordered set of the labels of the edges constituting the path) is in F. On the complexity frontier of this problem, we show the following results. – When A is a group whose size is polynomially bounded in the size of the graph (hence equivalently presented as a multiplication table at the input), and the graph is undirected, the A-Reach problem is in L. Building on this, using a decomposition in [4], we show that, when A is a fixed quasi-group, and the graph is undirected, the A-Reach problem is in L. In contrast, we show NL-hardness of the problem over bidirected graphs, when A is a matrix group over ℚ. When A is a fixed aperiodic monoid, we show that the problem is NL-complete. – As our main theorem, we prove a dichotomy for graphs labelled with fixed aperiodic monoids by showing that for every fixed aperiodic monoid A, A-Reach problem is either in L or is NL-complete. – We show that there exists a monoid M, such that the reachability problem in general DAGs can be reduced to A-Reach problem for planar non-bipartite DAGs labelled with M. In contrast, we show that if the planar DAGs that we obtain above are bipartite, the problem can be further reduced to reachability testing in planar DAGs and hence is in UL.

Original languageEnglish (US)
Title of host publicationLanguage and Automata Theory and Applications - 11th International Conference, LATA 2017, Proceedings
EditorsFrank Drewes, Carlos Martín-Vide, Bianca Truthe
PublisherSpringer Verlag
Pages351-363
Number of pages13
ISBN (Print)9783319537320
DOIs
StatePublished - 2017
Externally publishedYes
Event11th International Conference on Language and Automata Theory and Applications, LATA 2017 - Umea, Sweden
Duration: Mar 6 2017Mar 9 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10168 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference11th International Conference on Language and Automata Theory and Applications, LATA 2017
Country/TerritorySweden
City Umea
Period3/6/173/9/17

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science

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