Some probabilistic results on width measures of graphs

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), booleanwidth βw(G) is O(rw(G) log rw(G)), where rw is rankwidth. More importantly, we show aas Ω(n) lower bounds on the treewidth, branchwidth, cliquewidth, NLCwidth, and rankwidth of graphs drawn from a simple random model. This raises important questions about the generality of FPT algorithms using the corresponding decompositions. 1 The Introduction 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. Notice that as long as it is safe to assume that p is O(1), the run time is polynomial in the length of the input, even with f exponential or worse. For problems on graphs, parameter p is usually a measure of complexity of some tree decomposition of a graph, which is referred to as the graph’s width. There has been much progress in the development of FPT graph algorithms recently. The attention seems to have shifted from treewidth (tw) [21, 17] to newer width measures [14]: branchwidth (bw) [20], cliquewidth (cwd) [9], NLCwidth (nlcw) [23], rankwidth (rw) [19], and booleanwidth (βw) [6]. It is known [8, 18, 16] graph G has nlcw(G) ≥ cwd(G) ≥ rw(G) (1) tw(G) + 2 ≥ bw(G) + 1 ≥ rw(G) (2)