Parameterized Counting of Partially Injective Homomorphisms

We study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queri...

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Bibliographic Details
Published in:Algorithmica Vol. 83; no. 6; pp. 1829 - 1860
Main Author: Roth, Marc
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2021
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Classification
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Graphs
Homomorphisms
Inequalities
Mathematics of Computing
Parameterization
Parameters
Theorems
Theory of Computation
Matroid lattices
Graph homomorphisms
Parameterized complexity theory
Counting problems
Möbius function
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:We study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem in fixed-parameter tractable and # W [ 1 ] -complete cases. The proof relies on the framework of linear combinations of homomorphisms as independently discovered by Chen and Mengel (PODS 16) and by Curticapean, Dell and Marx in the recent breakthrough result regarding the exact complexity of the subgraph counting problem (STOC 17). Moreover, we invoke Rota’s NBC-Theorem to obtain an explicit criterion for fixed-parameter tractability based on treewidth. The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs. As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-021-00805-y