Improved Lower Bounds for Graph Embedding Problems

TL;DR

This paper establishes tight subexponential lower bounds for various graph embedding problems by introducing new combinatorial problems, showing these bounds are likely optimal under the Exponential Time Hypothesis.

Contribution

It introduces String Crafting and Orthogonal Vector Crafting problems to derive simplified hardness results for graph embedding problems on restricted classes.

Findings

No algorithms run in $2^{o(n/\log n)}$ time for several graph problems assuming ETH.

Provides tight lower bounds matching the $2^{ heta(n/\log n)}$ complexity.

Framework aids in establishing lower bounds for restricted graph classes.

Abstract

In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these cannot be solved in time $2^{o(|s|/\log{|s|})}$, unless the Exponential Time Hypothesis fails. These results are used to obtain simplified hardness results for several graph embedding problems, on more restricted graph classes than previously known: assuming the Exponential Time Hypothesis, there do not exist algorithms that run in $2^{o(n/\log n)}$ time for Subgraph Isomorphism on graphs of pathwidth 1, Induced Subgraph Isomorphism on graphs of pathwidth 1, Graph Minor on graphs of pathwidth 1, Induced Graph Minor on graphs of pathwidth 1, Intervalizing 5-Colored Graphs on trees, and finding a tree or path decomposition with width at most $c$ with a minimum number of bags, for any fixed $c\geq 16$. $2^{\Theta(n/\log n)}$ appears to be the "correct" running time for many packing and embedding problems on restricted graph classes, and we think String Crafting and Orthogonal Vector Crafting form a useful framework for establishing lower bounds of this form.