Separate, Measure and Conquer: Faster Polynomial-Space Algorithms for Max 2-CSP and Counting Dominating Sets

Abstract

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, 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.

Faster Graph Coloring in Polynomial Space

“…Since we may exhaust the separators a logarithmic number of times, and computing a new separator might introduce a penalty term each time, the measure also includes a logarithmic term that counteracts these artificial increases in measure, and will in the end only contribute a polynomial factor to the running time. For an in-depth treatment of the method we refer to [20]. Since we use the Separate, Measure and Conquer method when the average degree drops to at most 8 /3, we slightly generalize the separation computation from [20], where the bound on the size of the separator depended only on the maximum degree.…”

“…For an in-depth treatment of the method we refer to [20]. Since we use the Separate, Measure and Conquer method when the average degree drops to at most 8 /3, we slightly generalize the separation computation from [20], where the bound on the size of the separator depended only on the maximum degree. A separation (L, S, R) of a graph G is a partition of the vertex set of G such that every path from a vertex in L to a vertex in R contains a vertex from S. We will use the lemma for graphs with maximum degree 3 and graphs with maximum degree 3 and average degree at most 8 /3, for which path decompositions of width at most n /6 + o(n) and n /9 + o(n) can be computed in polynomial time, respectively [13,15].…”

“…A separation (L, S, R) of G is a partition of its vertex set into the three sets L, S, R such that no vertex in L is adjacent to any vertex in R. The sets L, S, R are also known as the left set, separator, and right set. Using a similar notion to [20], a separation (L, S, R) of G is balanced with respect to some measure µ, and a branching constant…”

“…However, we note that the 2 n3/5+o(n) running time for counting independent sets can easily be obtained from previous results. Namely, the problem of counting independent sets is a polynomial PCSP with domain size 2, as shown in [27], and the algorithm of [20] for polynomial PCSPs preprocesses all degree-2 vertices, leaving a cubic graph on n 3 vertices that is solved in 2 n/5+o(n) time. Improving on this bound is challenging, and degree-3 vertices with all neighbors of degree 2 need special attention since branching on them affects the degree-3 vertices of the graph exactly the same way as for the much more general polynomial PCSP problem, whereas for other degree-3 vertices one can take advantage of the asymmetric nature of the typical independent set branching (i.e., we can delete the neighbors when counting the independent sets containing the vertex we branch on).…”

An Improved Fixed-Parameter Algorithm for Max-Cut Parameterized by Crossing Number

“…This is one of the 21 problems shown to be NP-hard by Karp's seminal work [12]. To overcome this intractability, numerous researches have been done from the viewpoints of approximation algorithms [8,13,11,25], exponential-time exact algorithms [7,26], and fixed-parameter algorithms [3,16,17,22]. There are several graph classes for which the Max-Cut problem admits polynomial time algorithms [1,9].…”