Separate, Measure and Conquer

Abstract

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 analysis. We illustrate the method with improved algorithms for M ax ( r ,2)-C sp and #D ominating S et . An instance of the problem M ax ( r ,2)-C SP , or simply M ax 2-CSP, is parameterized by the domain size r (often 2), the number of variables n (vertices in the constraint graph G ), and the number of constraints m (edges in G ). When G is cubic, and omitting sub-exponential terms here for clarity, we give an algorithm running in time r (1/5) n = r (2/15) m the previous best was r (1/4) n = r (1/6) m . By known results, this improvement for the cubic case results in an algorithm running in time r (9/50) m for general instances; the previous best was r (19/100) m . We show that the analysis of the earlier algorithm was tight: our improvement is in the algorithm, not just the analysis. The same running time improvements hold for M ax C ut , an important special case of M ax 2-CSP, and for Polynomial and Ring CSP, generalizations encompassing graph bisection, the Ising model, and counting. We also give faster algorithms for #D ominating S et , counting the dominating sets of every cardinality 0, … , n for a graph G of order n . For cubic graphs, our algorithm runs in time 3 (1/5) n the previous best was 2 (1/2) n . For general graphs, we give an unrelated algorithm running in time 1.5183 n the previous best was 1.5673 n . The previous best algorithms for these problems all used local transformations and were analyzed by the Measure and Conquer method. Our new algorithms capitalize on the existence of small balanced separators for cubic graphs—a non-local property—and the ability to tailor the local algorithms always to “pivot” on a vertex in the separator. The new algorithms perform much as the old ones until the separator is empty, at which point they gain because the remaining vertices are split into two independent problem instances that can be solved recursively. It is likely that such algorithms can be effective for other problems too, and we present their design and analysis in a general framework.