On the complete width and edge clique cover problems

A complete graph is the graph in which every two vertices are adjacent. For a graph \(G=(V,E)\), the complete width of G is the minimum k such that there exist k independent sets \(\mathtt {N}_i\subseteq V\), \(1\le i\le k\), such that the graph \(G'\) obtained from G by adding some new edges between certain vertices inside the sets \(\mathtt {N}_i\), \(1\le i\le k\), is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on \(3K_2\)-free bipartite graphs and polynomially solvable on \(2K_2\)-free bipartite graphs and on \((2K_2,C_4)\)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on \(\overline{3K_2}\)-free co-bipartite graphs and polynomially solvable on \(C_4\)-free co-bipartite graphs and on \((2K_2, C_4)\)-free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most \(2^k\) vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most \(2^k\) vertices. Finally we determine all graphs of small complete width \(k\le 3\).     

The Travelling Salesman Problem on Permuted Monge Matrices

We consider traveling salesman problems (TSPs) with a permuted Monge matrix as cost matrix where the associated patching graph has a specially simple structure: a multistar, a multitree or a planar graph. In the case of multistars, we give a complete, concise and simplified presentation of Gaikov's theory. These results are then used for designing an O(m3 + mn) algorithm in the case of multitrees, where n is the number of cities and m is the number of subtours in an optimal assignment. Moreover we show that for planar patching graphs, the problem of finding an optimal subtour patching remains NP-complete.     

Lower bounds for positive semidefinite zero forcing and their applications

The positive semidefinite zero forcing number of a graph is a parameter that is important in the study of minimum rank problems. In this paper, we focus on the algorithmic aspects of computing this parameter. We prove that it is NP-complete to find the positive semidefinite zero forcing number of a given graph, and this problem remains NP-complete even for graphs with maximum vertex degree 7. We present a linear time algorithm for computing the positive semidefinite zero forcing number of generalized series–parallel graphs. We introduce the constrained tree cover number and apply it to improve lower bounds for positive semidefinite zero forcing. We also give formulas for the constrained tree cover number and the tree cover number on graphs with special structures.     

The max quasi-independent set problem

In this paper, we deal with the problem of finding quasi-independent sets in graphs. This problem is formally defined in three versions, which are shown to be polynomially equivalent. The one that looks most general, namely, f-max quasi-independent set, consists of, given a graph and a non-decreasing function f, finding a maximum size subset Q of the vertices of the graph, such that the number of edges in the induced subgraph is less than or equal to f(|Q|). For this problem, we show an exact solution method that runs within time O^*(2^{\fracd-27/23d+1n})O^{*}(2^{\frac{d-27/23}{d+1}n}) on graphs of average degree bounded by d. For the most specifically defined γ-max quasi-independent set and k-max quasi-independent set problems, several results on complexity and approximation are shown, and greedy algorithms are proposed, analyzed and tested.     

Standard directed search strategies and their applications

Given a digraph that contains an intruder hiding on vertices or along edges, a directed searching problem is to find the minimum number of searchers required to capture the intruder. In this paper, we propose the standard directed search strategies, which can simplify arguments in proving search numbers, lower bounds, and relationships between search models. Using these strategies, we characterize k-directed-searchable graphs when k≤3. We also use standard directed search strategies to investigate the searching problems on oriented grids. We give tight lower and upper bounds for the directed search number of general oriented grids. Finally, we show how to compute the directed search number of a Manhattan grid. Yang’s research was supported in part by NSERC and MITACS.     

Fibonacci index and stability number of graphs: a polyhedral study

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a connected variant of them are also extremal for these particular problems. We also make a polyhedral study by establishing all the optimal linear inequalities for the stability number and the Fibonacci index, inside the classes of general and connected graphs of order n.     

Coloring vertices of claw-free graphs in three colors

We study the computational complexity of the vertex 3-colorability problem in the class of claw-free graphs. Both the problem and the class received much attention in the literature, separately of each other. However, very little is known about the 3-colorability problem restricted to the class of claw-free graphs beyond the fact the problem is NP-complete under this restriction. In this paper we first strengthen this negative fact by revealing various further restrictions under which the problem remains NP-complete. Then we derive a number of positive results that deal with polynomially solvable cases of the problem in the class of claw-free graphs.     

The complexity of VLSI power-delay optimization by?interconnect resizing

The lithography used for 32 nanometers and smaller VLSI process technologies restricts the interconnect widths and spaces to a very small set of admissible values. Until recently the sizes of interconnects were allowed to change continuously and the implied power-delay optimal tradeoff could be formulated as a convex programming problem, for which classical search algorithms are applicable. Once the admissible geometries become discrete, continuous search techniques are inappropriate and new combinatorial optimization solutions are in order. A first step towards such solutions is to study the complexity of the problem, which this paper is aiming at. Though dynamic programming has been shown lately to solve the problem, we show that it is NP-complete. Two typical VLSI design scenarios are considered. The first trades off power and sum of delays (L ₁), and is shown to be NP-complete by reduction of PARTITION. The second considers power and max delays (L _∞), and is shown to be NP-complete by reduction of SUBSET_SUM.     

Maximum cyclic 4-cycle packings of the complete multipartite graph

A graph G is said to be m-sufficient if m is not exceeding the order of G, each vertex of G is of even degree, and the number of edges in G is a multiple of m. A complete multipartite graph is balanced if each of its partite sets has the same size. In this paper it is proved that the complete multipartite graph G can be decomposed into 4-cycles cyclically if and only if G is balanced and 4-sufficient. Moreover, the problem of finding a maximum cyclic packing of the complete multipartite graph with 4-cycles are also presented. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday.     

Packing <Emphasis Type="Bold">[1, Δ]</Emphasis>-factors in graphs of small degree

Given an undirected, connected graph G with maximum degree Δ, we introduce the concept of a [1, Δ]-factor k-packing in G, defined as a set of k edge-disjoint subgraphs of G such that every vertex of G has an incident edge in at least one subgraph. The problem of deciding whether a graph admits a [1,Δ]-factor k-packing is shown to be solvable in linear time for k = 2, but NP-complete for all k≥ 3. For k = 2, the optimisation problem of minimising the total number of edges of the subgraphs of the packing is NP-hard even when restricted to subcubic planar graphs, but can in general be approximated within a factor of by reduction to the Maximum 2-Edge-Colorable Subgraph problem. Finally, we discuss implications of the obtained results for the problem of fault-tolerant guarding of a grid, which provides the main motivation for research.     

A characterization of linearizable instances of the quadratic minimum spanning tree problem

We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph \(G=(V,E)\) that can be solved as a linear minimum spanning tree problem. We give a characterization of such problems when G is a complete graph, which is the standard case in the QMSTP literature. We extend our characterization to a larger class of graphs that include complete bipartite graphs and cactuses, among others. Our characterization can be verified in \(O(|E|^2)\) time. In the case of complete graphs and when the cost matrix is given in factored form, we show that our characterization can be verified in O(|E|) time. Related open problems are also indicated.     

The minimum vulnerability problem on specific graph classes

Suppose that each edge e of an undirected graph G is associated with three nonnegative integers \(\mathsf{cost}(e)\), \(\mathsf{vul}(e)\) and \(\mathsf{cap}(e)\), called the cost, vulnerability and capacity of e, respectively. Then, we consider the problem of finding \(k\) paths in G between two prescribed vertices with the minimum total cost; each edge e can be shared without any cost by at most \(\mathsf{vul}(e)\) paths, and can be shared by more than \(\mathsf{vul}(e)\) paths if we pay \(\mathsf{cost}(e)\), but cannot be shared by more than \(\mathsf{cap}(e)\) paths even if we pay the cost for e. This problem generalizes the disjoint path problem, the minimum shared edges problem and the minimum edge cost flow problem for undirected graphs, and it is known to be NP-hard. In this paper, we study the problem from the viewpoint of specific graph classes, and give three results. We first show that the problem is NP-hard even for bipartite outerplanar graphs, 2-trees, graphs with pathwidth two, complete bipartite graphs, and complete graphs. We then give a pseudo-polynomial-time algorithm for bounded treewidth graphs. Finally, we give a fixed-parameter algorithm for chordal graphs when parameterized by the number \(k\) of required paths.     

A rearrangement of adjacency matrix based approach for solving the crossing minimization problem

In this paper, we first present a binary linear programming formulation for the crossing minimization problem (CMP) in bipartite graphs. Then we use the models of a modified minimum cost flow problem (MMCF) and a travelling salesman problem (TSP) to approximatively solve the CMP by rearranging the adjacency matrix of the bipartite graph. Our approaches are useful for problems defined on dense bipartite graphs. In addition, we compute the exact crossing numbers for some general dense graphs.     

Parameterized complexity and inapproximability of dominating set problem in chordal and near chordal graphs

In this paper, we study the parameterized complexity of Dominating Set problem in chordal graphs and near chordal graphs. We show the problem is W[2]-hard and cannot be solved in time n ^o(k) in chordal and s-chordal (s>3) graphs unless W[1]=FPT. In addition, we obtain inapproximability results for computing a minimum dominating set in chordal and near chordal graphs. Our results prove that unless NP=P, the minimum dominating set in a chordal or s-chordal (s>3) graph cannot be approximated within a ratio of \fracc3lnn\frac{c}{3}\ln{n} in polynomial time, where n is the number of vertices in the graph and 0<c<1 is the constant from the inapproximability of the minimum dominating set in general graphs. In other words, our results suggest that restricting to chordal or s-chordal graphs can improve the approximation ratio by no more than a factor of 3. We then extend our techniques to find similar results for the Independent Dominating Set problem and the Connected Dominating Set problem in chordal or near chordal graphs.     

Approximation for vertex cover in $$\beta $$-conflict graphs

Conflict graph is a union of finite given sets of disjoint complete multipartite graphs. Vertex cover on this kind of graph is used first to model data inconsistency problems in database application. It is NP-complete if the number of given sets r is fixed, and can be approximated within \(2-\frac{1}{2^r}\) (Miao et al. in Proceedings of the 9th international conference on combinatorial optimization and applications, vol 9486. COCOA 2015, New York. Springer, New York, pp 395–408, 2015). This paper shows a better algorithm to improve the approximation for dense cases. If the ratio of vertex not belongs to any wheel complete multipartite graph is no more than \(\beta <1\), then our algorithm will provide a \((1+\beta +\frac{1-\beta }{k})\)-approximation, where k is a parameter related to degree distribution of wheel complete multipartite graph.     

Half integer extreme points in the linear relaxation of the 2-edge-connected subgraph polyhedron

This paper studies the graphs for which the linear relaxation of the 2-connected spanning subgraph polyhedron has integer or half-integer extreme points. These graphs are called quasi-integer. For these graphs, the linear relaxation of the k-edge connected spanning subgraph polyhedron is integer for all k=4r, r≥1. The class of quasi-integer graphs is closed under minors and contains for instance the class of series-parallel graphs. We discuss some structural properties of graphs which are minimally non quasi-integer graphs, then we examine some basic operations which preserve the quasi-integer property. Using this, we show that the subdivisions of wheels are quasi-integer.     

Maximum cardinality neighbourly sets in quadrilateral free graphs

Neighbourly set of a graph is a subset of edges which either share an end point or are joined by an edge of that graph. The maximum cardinality neighbourly set problem is known to be NP-complete for general graphs. Mahdian (Discret Appl Math 118:239–248, 2002) proved that it is in polynomial time for quadrilateral-free graphs and proposed an \(O(n^{11})\) algorithm for the same, here n is the number of vertices in the graph, (along with a note that by a straightforward but lengthy argument it can be proved to be solvable in \(O(n^5)\) running time). In this paper we propose an \(O(n^2)\) time algorithm for finding a maximum cardinality neighbourly set in a quadrilateral-free graph.     

On the Robust Single Machine Scheduling Problem

The single machine scheduling problem with sum of completion times criterion (SS) can be solved easily by the Shortest Processing Time (SPT) rule. In the case of significant uncertainty of the processing times, a robustness approach is appropriate. In this paper, we show that the robust version of the (SS) problem is NP-complete even for very restricted cases. We present an algorithm for finding optimal solutions for the robust (SS) problem using dynamic programming. We also provide two polynomial time heuristics and demonstrate their effectiveness.     

The QAP-polytope and the graph isomorphism problem

In this paper we propose a geometric approach to solve the Graph Isomorphism (GI in short) problem. Given two graphs \(G_1, G_2\), the GI problem is to decide if the given graphs are isomorphic i.e., there exists an edge preserving bijection between the vertices of the two graphs. We propose an Integer Linear Program (ILP) that has a non-empty solution if and only if the given graphs are isomorphic. The convex hull of all possible solutions of the ILP has been studied in literature as the Quadratic Assignment Problem (QAP) polytope. We study the feasible region of the linear programming relaxation of the ILP and show that the given graphs are isomorphic if and only if this region intersects with the QAP-polytope. As a consequence, if the graphs are not isomorphic, the feasible region must lie entirely outside the QAP-polytope. We study the facial structure of the QAP-polytope with the intention of using the facet defining inequalities to eliminate the feasible region outside the polytope. We determine two new families of facet defining inequalities of the QAP-polytope and show that all the known facet defining inequalities are special instances of a general inequality. Further we define a partial ordering on each exponential sized family of facet defining inequalities and show that if there exists a common minimal violated inequality for all points in the feasible region outside the QAP-polytope, then we can solve the GI problem in polynomial time. We also study the general case when there are k such inequalities and give an algorithm for the GI problem that runs in time exponential in k.     

Discovering classes in microarray data using island counts

We present a new biclustering algorithm to simultaneously discover tissue classes and identify a set of genes that well-characterize these classes from DNA microarray data sets. We employ a combinatorial optimization approach where the object is to simultaneously identify an interesting set of genes and a partition of the array samples that optimizes a certain score based on a novel color island statistic. While this optimization problem is NP-complete in general, we are effectively able to solve problems of interest to optimality using a branch-and-bound algorithm. We have tested the algorithm on a 30 sample Cutaneous T-cell Lymphoma data set; it was able to almost perfectly discriminate short-term survivors from long-term survivors and normal controls. Another useful feature of our method is that can easily handle missing expression data.