25 research outputs found
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 time
and polynomial space, and an algorithm that uses time
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
Learning-Based Heuristic for Combinatorial Optimization of the Minimum Dominating Set Problem using Graph Convolutional Networks
A dominating set of a graph is a subset of vertices
such that every vertex
outside the dominating set is adjacent to a vertex within the set. The
minimum dominating set problem seeks to find a dominating set of minimum
cardinality and is a well-established NP-hard combinatorial optimization
problem. We propose a novel learning-based heuristic approach to compute
solutions for the minimum dominating set problem using graph convolutional
networks. We conduct an extensive experimental evaluation of the proposed
method on a combination of randomly generated graphs and real-world graph
datasets. Our results indicate that the proposed learning-based approach can
outperform a classical greedy approximation algorithm. Furthermore, we
demonstrate the generalization capability of the graph convolutional network
across datasets and its ability to scale to graphs of higher order than those
on which it was trained. Finally, we utilize the proposed learning-based
heuristic in an iterative greedy algorithm, achieving state-of-the-art
performance in the computation of dominating sets
Solving Edge Clique Cover Exactly via Synergistic Data Reduction
The edge clique cover (ECC) problem - where the goal is to find a minimum cardinality set of cliques that cover all the edges of a graph - is a classic NP-hard problem that has received much attention from both the theoretical and experimental algorithms communities. While small sparse graphs can be solved exactly via the branch-and-reduce algorithm of Gramm et al. [JEA 2009], larger instances can currently only be solved inexactly using heuristics with unknown overall solution quality. We revisit computing minimum ECCs exactly in practice by combining data reduction for both the ECC and vertex clique cover (VCC) problems. We do so by modifying the polynomial-time reduction of Kou et al. [Commun. ACM 1978] to transform a reduced ECC instance to a VCC instance; alternatively, we show it is possible to "lift" some VCC reductions to the ECC problem. Our experiments show that combining data reduction for both problems (which we call synergistic data reduction) enables finding exact minimum ECCs orders of magnitude faster than the technique of Gramm et al., and allows solving large sparse graphs on up to millions of vertices and edges that have never before been solved. With these new exact solutions, we evaluate the quality of recent heuristic algorithms on large instances for the first time. The most recent of these, EO-ECC by Abdullah et al. [ICCS 2022], solves 8 of the 27 instances for which we have exact solutions. It is our hope that our strategy rallies researchers to seek improved algorithms for the ECC problem