Monotone Arithmetic Complexity of Graph Homomorphism Polynomials

We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n -vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for o...

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Bibliographic Details
Published in:Algorithmica Vol. 85; no. 9; pp. 2554 - 2579
Main Authors: Komarath, Balagopal, Pandey, Anurag, Rahul, C. S.
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2023
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Circuits
Combinatorial analysis
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Homomorphisms
Mathematics of Computing
Polynomials
Theory of Computation
Homomorphism polynomials
Monotone complexity
Graph homomorphisms
Treewidth
Fixed-parameter algorithms and complexity
Algebraic circuits
Fine-grained complexity
Algebraic branching programs
Algebraic complexity
Algebraic formulas
Graph algorithms
Pathwidth
Treedepth
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n -vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes VBP , V P , and VNP . We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively. Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-023-01108-0