An Order-based Algorithm for Minimum Dominating Set with Application in Graph Mining

Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLS$_o$) algorithm to solve minimum dominating set problem in large graphs. Experimental evaluation is presented for multiple types of problem instances. These instances include unit disk graphs, which represent a model of wireless networks, random scale-free networks, as well as samples from two social networks and real-world graphs studied in network science. Our experiments indicate that RLS$_o$ performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search. The order-based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices. In addition, we propose a multi-start variant of RLS$_o$ that is suitable for solving the minimum weight dominating set problem. The application of RLS$_o$ in graph mining is also briefly demonstrated

Approximating Dominating Set on Intersection Graphs of Rectangles and L-frames

We consider the Minimum Dominating Set (MDS) problem on the intersection graphs of geometric objects. Even for simple and widely-used geometric objects such as rectangles, no sub-logarithmic approximation is known for the problem and (perhaps surprisingly) the problem is NP-hard even when all the rectangles are "anchored" at a diagonal line with slope -1 (Pandit, CCCG 2017). In this paper, we first show that for any epsilon>0, there exists a (2+epsilon)-approximation algorithm for the MDS problem on "diagonal-anchored" rectangles, providing the first O(1)-approximation for the problem on a non-trivial subclass of rectangles. It is not hard to see that the MDS problem on "diagonal-anchored" rectangles is the same as the MDS problem on "diagonal-anchored" L-frames: the union of a vertical and a horizontal line segment that share an endpoint. As such, we also obtain a (2+epsilon)-approximation for the problem with "diagonal-anchored" L-frames. On the other hand, we show that the problem is APX-hard in case the input L-frames intersect the diagonal, or the horizontal segments of the L-frames intersect a vertical line. However, as we show, the problem is linear-time solvable in case the L-frames intersect a vertical as well as a horizontal line. Finally, we consider the MDS problem in the so-called "edge intersection model" and obtain a number of results, answering two questions posed by Mehrabi (WAOA 2017)

Campaign Management under Approval-Driven Voting Rules

Approval-like voting rules, such as Sincere-Strategy Preference-Based Approval voting (SP-AV), the Bucklin rule (an adaptive variant of $k$-Approval voting), and the Fallback rule (an adaptive variant of SP-AV) have many desirable properties: for example, they are easy to understand and encourage the candidates to choose electoral platforms that have a broad appeal. In this paper, we investigate both classic and parameterized computational complexity of electoral campaign management under such rules. We focus on two methods that can be used to promote a given candidate: asking voters to move this candidate upwards in their preference order or asking them to change the number of candidates they approve of. We show that finding an optimal campaign management strategy of the first type is easy for both Bucklin and Fallback. In contrast, the second method is computationally hard even if the degree to which we need to affect the votes is small. Nevertheless, we identify a large class of scenarios that admit fixed-parameter tractable algorithms.Comment: 34 pages, 1 figur

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

Data Reductions and Combinatorial Bounds for Improved Approximation Algorithms

Kernelization algorithms in the context of Parameterized Complexity are often based on a combination of reduction rules and combinatorial insights. We will expose in this paper a similar strategy for obtaining polynomial-time approximation algorithms. Our method features the use of approximation-preserving reductions, akin to the notion of parameterized reductions. We exemplify this method to obtain the currently best approximation algorithms for \textsc{Harmless Set}, \textsc{Differential} and \textsc{Multiple Nonblocker}, all of them can be considered in the context of securing networks or information propagation

The Homogeneous Broadcast Problem in Narrow and Wide Strips

Let $P$ be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let $s\in P$ be a given source node. Each node $p$ can transmit information to all other nodes within unit distance, provided $p$ is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source $s$ can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem (in the latter, $s$ must be able to reach every node within a specified number of hops), with the restriction that all points lie inside a strip of width $w$. We almost completely characterize the complexity of both the regular and the hop-bounded versions as a function of the strip width $w$.Comment: 50 pages, WADS 2017 submissio