Some Probabilistic Results on Width Measures of Graphs

TL;DR

This paper establishes that for graphs generated by a simple random model, key width parameters such as treewidth and cliquewidth are asymptotically almost surely linear in size, challenging the universality of FPT algorithms based on these parameters.

Contribution

It provides asymptotic lower bounds on several width measures of random graphs, questioning the broad applicability of FPT algorithms relying on these parameters.

Findings

Omega(n) lower bounds on width parameters for random graphs

Asymptotic almost sure linear growth of these parameters

Implications for the effectiveness of FPT algorithms

Abstract

Fixed parameter tractable (FPT) algorithms run in time f(p(x)) poly(|x|), where f is an arbitrary function of some parameter p of the input x and poly is some polynomial function. Treewidth, branchwidth, cliquewidth, NLC-width, rankwidth, and booleanwidth are parameters often used in the design and analysis of such algorithms for problems on graphs. We show asymptotically almost surely (aas), there are Omega(n) lower bounds on the treewidth, branchwidth, cliquewidth, NLC-width, and rankwidth of graphs drawn from a simple random model. This raises important questions about the generality of FPT algorithms using the corresponding decompositions.