Edge separators for graphs of bounded genus with applications

$n$-vertex graph of positive genus $g$ and maximal degree $k$ has an $O(\sqrt{gkn})$ edge separator. This bound is best possible to within a constant factor. The separator can be found in $O(g+n)$ time provided that we start with an imbedding of the graph in its genus surface. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric problem, to efficient embeddings of graphs of genus $g$ into various classes of graphs including trees, meshes and hypercubes and to showing lower bounds on crossing numbers of $K_n,K_{m,n}$ and $Q_n$ drawn on surfaces of genus $g$

Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, such as Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for an improved running time analysis. We illustrate the method with improved algorithms for Max (r,2) -CSP and #Dominating Set. For Max (r,2) -CSP instances with domain size r and m constraints, the running time improves from r m/6 to r m/7.5 for cubic instances and from r 0.19⋅m to r 0.18⋅m for general instances, omitting subexponential factors. For #Dominating Set instances with n vertices, the running time improves from 1.4143 n to 1.2458 n for cubic instances and from 1.5673 n to 1.5183 n for general instances. It is likely that other algorithms relying on local transformations can be improved using our method, which exploits a non-local property of graphs

Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, notably Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for a more precise and improved running time analysi