Dominating Clique Problem Research Articles

A Graph-Neural-Network-Powered Solver Framework for Graph Optimization Problems.

Backtracking combined with branching heuristics is a prevalent approach for tackling constraint satisfaction problems (CSPs) and combinatorial optimization problems (COPs). While branching heuristics specifically designed for certain problems can be theoretically efficient, they are often complex and difficult to implement in practice. On the other hand, general branching heuristics can be applied across various problems, but at the risk of suboptimality. We introduce a solver framework that leverages the Shannon entropy in branching heuristics to bridge the gap between generality and specificity in branching heuristics. This enables backtracking to follow the path of least uncertainty, based on probability distributions that conform to problem constraints. We employ graph neural network (GNN) models with loss functions derived from the probabilistic method to learn these probability distributions. We have evaluated our approach by its applications to two NP-hard problems: the (minimum) dominating-clique problem and the edge-clique-cover problem. Compared with the state-of-the-art solvers for both problems, our solver framework outputs competitive results. Specifically, for the (minimum) dominating-clique problem, our approach generates fewer branches than the solver presented by Culberson et al. (2005). For the edge-clique-cover problem, our approach produces smaller-sized edge clique covers (ECCs) than the solvers referenced by Conte et al. (2020) and Kellerman (1973).

Algorithms for dominating clique problems

We handle in this paper three dominating clique problems, namely, the decision problem to detect whether a graph has a dominating clique and two optimization versions asking to compute a maximum- and a minimum-size dominating clique of a graph G, if G has a dominating clique. For the three problems we propose exact moderately exponential algorithms with worst-case running time upper bounds improving those by Kratsch and Liedloff [D. Kratsch, M. Liedloff, An exact algorithm for the minimum dominating clique problem, Theoret. Comput. Sci. 385 (1–3) (2007) 226–240]. We then study the three problems in sparse and dense graphs also providing improved running time upper bounds. Finally, we propose some exponential time approximation algorithms for the optimization versions.

Dominating set is fixed parameter tractable in claw-free graphs

We show that the Dominating Set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw ( K 1 , 3 , the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2 O ( k 2 ) n O ( 1 ) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k . Note that this parameterization of Dominating Set is W [ 2 ] -hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general. The most general class of graphs for which an FPT algorithm was previously known for this parameterization of Dominating Set is the class of K i , j -free graphs, which exclude, for some fixed i , j ∈ N , the complete bipartite graph K i , j as a subgraph. For i , j ≥ 2 , the class of claw-free graphs and any class of K i , j -free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of Dominating Set is known to be fixed-parameter tractable. We also show that, in some sense, it is the presence of the claw that makes this parameterization of the Dominating Set problem hard. More precisely, we show that for any t ≥ 4 , the Dominating Set problem parameterized by the solution size is W [ 2 ] -hard in graphs that exclude the t -claw K 1 , t as an induced subgraph. Our arguments also imply that the related Connected Dominating Set and Dominating Clique problems are W [ 2 ] -hard in these graph classes. Finally, we show that for any t ∈ N , the Clique problem parameterized by solution size, which is W [ 1 ] -hard on general graphs, is FPT in t -claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.

An exact algorithm for the minimum dominating clique problem

A subset of vertices D ⊆ V of a graph G = ( V , E ) is a dominating clique if D is a dominating set and a clique of G . The existence problem ‘Given a graph G , is there a dominating clique in G ?’ is NP-complete, and thus both the Minimum and the Maximum Dominating Clique problems are NP-hard. We present an O ( 1.338 7 n ) time and polynomial space algorithm that for an input graph on n vertices either computes a minimum dominating clique or reports that the graph has no dominating clique. The algorithm uses the Branch & Reduce paradigm and its time analysis is based on the Measure & Conquer approach. We also establish a lower bound of Ω ( 1.259 9 n ) for the worst case running time of the algorithm. Finally using memorization we obtain an O ( 1.323 4 n ) time and exponential space algorithm for the same problem.

Connected domination and dominating clique in trapezoid graphs

The class of trapezoid graphs is a subclass of cocomparability graphs and contains both the interval graphs and the permutation graphs. In this paper we present an O( n) time algorithm for the minimum cardinality connected dominating set problem and an O( n+ m) time algorithm for the minimum cardinality dominating clique problem in trapezoid graphs. Both algorithms require the trapezoid diagram to be given as input.

Weighted domination of cocomparability graphs

It is shown in this paper that the weighted domination problem and its three variants, the weighted connected domination, total domination, and dominating clique problems are NP-complete on cobipartite graphs when arbitrary integer vertex weights are allowed and all of them can be solved in polynomial time on cocomparability graphs if vertex weights are integers and less than or equal to a constant c. The results are interesting because cocomparability graphs properly contain cobipartite graphs and the cardinality cases of the above problems are trivial on cobipartite graphs. On the other hand, an O(¦V¦ 2) algorithm is given for the weighted independent perfect domination problem of a cocomparability graph G = ( V. E).

The parallel solution of domination problems on chordal and strongly chordal graphs

We present efficient parallel algorithms for the domination problem on strongly chordal graphs and related problems, such as the set cover problem for α-acyclic hypergraphs and the dominating clique problem for strongly chordal graphs. Moreover, we present an efficient parallel algorithm which checks, for any chordal graph, whether it has a dominating clique.

Incorporating negative-weight vertices in certain vertex-search graph algorithms

It is shown in this paper that the weighted domination problem and its three variants, the weighted connected domination, total domination, and dominating clique problems are NP-complete on cobipartite graphs when arbitrary integer vertex weights are allowed and all of them can be solved in polynomial time on cocomparability graphs if vertex weights are integers and less than or equal to a constant c. The results are interesting because cocomparability graphs properly contain cobipartite graphs and the cardinality cases of the above problems are trivial on cobipartite graphs. On the other hand, an O(¦V¦2) algorithm is given for the weighted independent perfect domination problem of a cocomparability graph G = (V.E).

Efficient algorithms for the minimum weighted dominating clique problem on permutation graphs

Given a graph G=( V, E) with real weights assigned to its vertices, a clique of G that also dominates its vertex set V, is called a dominating clique (DC) of G. Given a permutation graph G with all its vertices having nonnegative weights, and its permutation representation, the problem addressed in this paper is that of finding any minimum weight DC of G. We improve the existing O(| V| 2) algorithm for this problem to O(| V|log| V|). The space complexity of our algorithm is O(| V|). We also present a | V| processor, O(log| V|) time, O(| V|log| V|) space parallel EREW PRAM algorithm for this problem.

Finding dominating cliques efficiently, in strongly chordal graphs and undirected path graphs

We study a new version of the domination problem in which the dominating set is required to be a clique. The minimum dominating clique problem is NP-complete for split graphs and, hence, for chordal graphs. We show that for two other important subclasses of chordal graphs the problem is solvable efficiently. We present an O( m · log n) algorithm for strongly chordal graphs and an O( n 4) algorithm for undirected path graphs.

On domination problems for permutation and other graphs

There is an increasing interest in results on the influence of restricting NP-complete graph problems to special classes of perfect graphs as, e.g., permutation graphs. It was shown that several problems restricted to permutation graphs are solvable in polynomial time [2, 3, 4, 6, 7, 14, 16]. In this paper we give 1. (i) an algorithm with time bound O( n 2) for the weighted independent domination problem on permutation graphs (which is an improvement of the O( n 3) solution given in [7]); 2. (ii) a polynomial time solution for the weighted feedback vertex set problem on permutation graphs; 3. (iii) an investigation of (weighted) dominating clique problems for several graph classes including an NP-completeness result for weakly triangulated graphs as well as polynomial time bounds.