Counting Patterns in Degenerate Graphs in Constant Space

TL;DR

This paper introduces DAG treedepth as a new graph parameter enabling constant-space algorithms for counting subgraphs and homomorphisms in degenerate graphs, improving efficiency over previous methods.

Contribution

The paper presents a novel DAG treedepth parameter and demonstrates its use in designing space-efficient algorithms for subgraph counting in degenerate graphs, with improved time complexity.

Findings

Constant space algorithms for counting subgraphs in sparse pattern graphs.

Characterization of graphs with DAG treedepth up to two.

Quadratic time counting for pattern graphs up to 11 vertices.

Abstract

For an arbitrary, fixed graph (pattern graph), we study the algorithmic complexity of counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms from the pattern graph to $n$-vertex, $d$-degenerate graphs as input. Recent work by Bressan (Algorithmica, 2021) has shown that this problem has efficient dynamic programming algorithms using a graph parameter called DAG treewidth. Bressan used DAG treewidth to design a fast algorithm for counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms that use polynomial space. Bera, Gishboliner, Levanzov, Seshadhri, and Shapira (SODA, 2021) provided a characterization of graphs with DAG treewidth one. In this paper, we introduce a new graph parameter called DAG treedepth and show that it yields efficient divide and conquer algorithms that use only constant space (in the unit-cost RAM model). Specifically, we show: An algorithm for counting subgraphs isomorphic to sparse pattern graphs using only constant space. We derive an induced minor-based characterization for graphs of DAG treedepth up to two. For pattern graphs upto nine vertices, the induced subgraphs can be counted in $O(n^3)$ time using constant space. An algorithm for counting induced subgraphs that matches the running time given by Bressan but only uses constant space. Apart from the DAG treedepth result, we also focus on DAG treewidth. For DAG treewidth, we show that we can count homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms faster than Bressan's algorithm (2021). We further show that for all pattern graphs up to 11 vertices, we can count induced subgraphs in quadratic time.