Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six

11/14/2019
by   Suman K. Bera, et al.
0

We consider the problem of counting all k-vertex subgraphs in an input graph, for any constant k. This problem (denoted sub-cnt_k) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for sub-cnt_k. Towards a better understanding of the limits of these techniques, we ask: for what values of k can sub-cnt_k be solved in linear time? We discover a chasm at k=6. Specifically, we prove that for k < 6, sub-cnt_k can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all k ≥ 6, sub-cnt_k cannot be solved even in near-linear time.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

10/16/2020

Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles

Counting homomorphisms of a constant sized pattern graph H in an input g...
10/12/2020

Counting Subgraphs in Degenerate Graphs

We consider the problem of counting the number of copies of a fixed grap...
08/29/2018

Counting Independent Sets in Cocomparability Graphs

We show that the number of independent sets in cocomparability graphs ca...
11/04/2020

PCP Theorems, SETH and More: Towards Proving Sub-linear Time Inapproximability

In this paper we propose the PCP-like theorem for sub-linear time inappr...
08/25/2019

Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP

Local search is a widely-employed strategy for finding good solutions to...
07/07/2021

On a k-matching algorithm and finding k-factors in random graphs with minimum degree k+1 in linear time

We prove that for k+1≥ 3 and c>(k+1)/2 w.h.p. the random graph on n vert...
07/05/2018

Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

We investigate the problem #IndSub(Φ) of counting all induced subgraphs ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.