2016/10/28 by Hans L. Bodlaender, Bodlaender, Hans L., Tom C. van der Zanden +1
Computer Science · #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #Interconnection Networks and Systems
paper · pdf · doi:10.48550/arxiv.1610.09130
openalex publication_date 2016/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 2o(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≥ 16. 2Θ(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.