Johan M. M. van Rooij

Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution

In this paper, we show that algorithms on tree decompositions can be made faster with the use of generalisations of fast subset convolution. Amongst others, this gives algorithms that, for a graph, given with a tree decomposition of width k, solve the dominated set problem in O(n k 2 3 k ) time and the problem to count the number of perfect matchings in O ∗ (2 k ) time. Using a generalisation of fast subset convolution, we obtain faster algorithms for all [ρ,σ]-domination problems with finite or cofinite ρ and σ on tree decompositions. These include many well known graph problems. We give additional results on many more graph covering and partitioning problems.

Design by Measure and Conquer, A Faster Exact Algorithm for Dominating Set

The measure and conquer approach has proven to be a powerful tool to analyse exact algorithms for combinatorial problems, like Dominating Set and Independent Set. In this paper, we propose to use measure and conquer also as a tool in the design of algorithms. In an iterative process, we can obtain a series of branch and reduce algorithms. A mathematical analysis of an algorithm in the series with measure and conquer results in a quasiconvex programming problem. The solution by computer to this problem not only gives a bound on the running time, but also can give a new reduction rule, thus giving a new, possibly faster algorithm. This makes design by measure and conquer a form of computer aided algorithm design. When we apply the methodology to a Set Cover modelling of the Dominating Set problem, we obtain the currently fastest known exact algorithms for Dominating Set: an algorithm that uses

Inclusion/Exclusion Meets Measure and Conquer

O(1.5134^n)

Exact algorithms for dominating set

time and polynomial space, and an algorithm that uses

A bottom-up method and fast algorithms for MAX INDEPENDENT SET

O(1.5063^n)

Partition Into Triangles on Bounded Degree Graphs

time.

Maximum independent set in graphs of average degree at most three in O (1.08537 n )

Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times.In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number.This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set, and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k-Set Splitting and its generalisation k-Not-All-Equal Satisfiability.

Inclusion/Exclusion Branching for Partial Dominating Set and Set Splitting

The measure and conquer approach has proven to be a powerful tool to analyse exact algorithms for combinatorial problems like Dominating Set and Independent Set. This approach is used in this paper to obtain a faster exact algorithm for Dominating Set. We obtain this algorithm by considering a series of branch and reduce algorithms. This series is the result of an iterative process in which a mathematical analysis of an algorithm in the series with measure and conquer results in a convex or quasiconvex programming problem. The solution, by means of a computer, to this problem not only gives a bound on the running time of the algorithm, but can also give an indication on where to look for a new reduction rule, often giving a new, possibly faster algorithm. As a result, we obtain an O(1.4969^n) time and polynomial space algorithm.

Faster algorithms on branch and clique decompositions

We first propose a new method, called “bottom-up method”, that, informally, “propagates” improvement of the worst-case complexity for “sparse” instances to “denser” ones and we show an easy though non-trivial application of it to the min set cover problem. We then tackle max independent set. Following the bottom-up method we propagate improvements of worst-case complexity from graphs of average degree d to graphs of average degree greater than d. Indeed, using algorithms for max independent set in graphs of average degree 3, we tackle max independent set in graphs of average degree 4, 5 and 6. Then, we combine the bottom-up technique with measure and conquer techniques to get improved running times for graphs of maximum degree 4, 5 and 6 but also for general graphs. The best computation bounds obtained for max independent set are O*(1.1571n), O*(1.1918n) and O*(1.2071n), for graphs of maximum (or more generally average) degree 4, 5 and 6 respectively, and O*(1.2127n) for general graphs. These results improve upon the best known polynomial space results for these cases.

Polynomial space algorithms for counting dominating sets and the domatic number

We consider the Partition Into Triangles problem on bounded degree graphs. We show that this problem is polynomial-time solvable on graphs of maximum degree three by giving a linear-time algorithm. We also show that this problem becomes