Abstract 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.
The problem of finding MDS remains NP-hard also for several very restricted graph classes. For example, NP-hardness of finding MDS for grids is known, the proof is attributed to Leighton @cite_25 . For unit disk graphs, MDS problem is also NP-hard @cite_48 . Additionally, Chleb ' i k and Chleb ' i kov ' a have shown that for bipartite graphs with maximum degree @math and general graphs with maximum degree @math , it is NP-hard to approximate MDS within ratios @math and @math , respectively @cite_29 .
@cite_48 @cite_29 @cite_25
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This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree. The MAXIMUM INDEPENDENT SET problem in d-box graphs, i.e., in the intersection graphs of axis-parallel rectangles in Rd, is a challenge open problem. For any fixed d ≥ 2 the problem is NP-hard and no approximation algorithm with ratio o(logd-1 n) is known. In some restricted cases, e.g., for d-boxes with bounded aspect ratio, a PTAS exists [17]. In this paper we prove APX-hardness (and hence non-existence of a PTAS, unless P = NP), of the MAXIMUM INDEPENDENT SET problem in d-box graphs for any fixed d ≥ 3. We state also first explicit lower bound 443 442 on efficient approximability in such case. Additionally, we provide a generic method how to prove APX-hardness for many NP-hard graph optimization problems in d-box graphs for any fixed d ≥ 3. In 2-dimensional case we give a generic approach to NP-hardness results for these problems in highly restricted intersection graphs of axis-parallel unit squares (alternatively, in unit disk graphs). The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets @math and @math , where @math is an independent set and @math induces a forest. The set @math in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.