On the complexity of the multicut problem in bounded tree-width graphs and digraphs

Given an edge- or vertex-weighted graph or digraph and a list of source-sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NP-hard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source-sink pairs is fixed, but remains NP-hard in directed acyclic graphs and APX-hard in bounded tree-width and bounded degree unweighted digraphs. The vertex version, although tractable in trees, is proved to be NP-hard in unweighted cacti of bounded degree and bounded path-width.     

Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree

Given an undirected graph with edge weights, we are asked to find an orientation, that is, an assignment of a direction to each edge, so as to minimize the weighted maximum outdegree in the resulted directed graph. The problem is called MMO, and is a restricted variant of the well-known minimum makespan problem. As in previous studies, it is shown that MMO is in P for trees, weak NP-hard for planar bipartite graphs, and strong NP-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tighter thresholds of complexity: We show that MMO is (i) in P for cactus graphs, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for graphs which are both planar and bipartite. This implies the NP-hardness for P₄-bipartite, diamond-free or house-free graphs, each of which is a superclass of cactus. We also show (iv) the NP-hardness for series-parallel graphs and multi-outerplanar graphs, and (v) present a pseudo-polynomial time algorithm for graphs with bounded treewidth.     

An efficient representation of Benes networks and its applications

The most popular bounded-degree derivative network of the hypercube is the butterfly network. The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly—like architectures. We identify a new topological representation of butterfly and Benes networks.The minimum metric dimension problem is to find a minimum set of vertices of a graph G(V,E) such that for every pair of vertices u and v of G, there exists a vertex w with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. It is NP-hard in the general sense. We show that it remains NP-hard for bipartite graphs. The algorithmic complexity status of this NP-hard problem is not known for butterfly and Benes networks, which are subclasses of bipartite graphs. By using the proposed new representations, we solve the minimum metric dimension problem for butterfly and Benes networks. The minimum metric dimension problem is important in areas such as robot navigation in space applications.     

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P₈-free bipartite graphs, but polynomial for P₅-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.     

Inverse <Emphasis Type="Italic">p</Emphasis>-median problems with variable edge lengths

The inverse p-median problem with variable edge lengths on graphs is to modify the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified set of p vertices becomes a p-median with respect to the new edge lengths. The problem is shown to be strongly NP{\mathcal{NP}}-hard on general graphs and weakly NP{\mathcal{NP}}-hard on series-parallel graphs. Therefore, the special case on a tree is considered: It is shown that the inverse 2-median problem with variable edge lengths on trees is solvable in polynomial time. For the special case of a star graph we suggest a linear time algorithm.     

Intersection graphs of curves in the plane

Let V be a set of curves in the plane. The corresponding intersection graph has V as the set of vertices, and two vertices are connected by an edge if and only if the two corresponding curves intersect in the plane.It is shown that the set of intersection graphs of curves in the plane is a proper subset of the set of all undirected graphs. Furthermore, the set of intersection graphs of straight line-segments is a proper subset of the set of the intersection graphs of curves in the plane. Finally, it is shown that for every k ≥ 3, the problem of determining whether an intersection graph of straight line-segments is k-colorable is NP-complete.     

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S⊆V, such that, for each u∈V, |N[u]∩S|?|N[u]|/2 in a given graph G=(V,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k?5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k-1)-regular graphs, where k?3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.     

How to find Nash equilibria with extreme total latency in network congestion games?

We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion games and analyse how it is influenced by the graph topology and the number of users. In our context best and worst equilibria are those with minimum or maximum total latency, respectively. We establish that both problems can be solved by a Greedy type algorithm equipped with a suitable tie breaking rule on extension-parallel graphs. On series-parallel graphs finding a worst Nash equilibrium is NP-hard for two or more users while finding a best one is solvable in polynomial time for two users and NP-hard for three or more. additionally we establish NP-hardness in the strong sense for the problem of finding a worst Nash equilibrium on a general acyclic graph.     

Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints.We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge.Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.     

Colouring Clique-Hypergraphs of Circulant Graphs

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, ${\mathcal{H}(G)}$ , of a graph G has V(G) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of ${\mathcal{H}(G)}$ is a clique-colouring of G. Determining the clique-chromatic number, the least number of colours for which a graph G admits a clique-colouring, is known to be NP-hard. In this work, we establish that the clique-chromatic number of powers of cycles is equal to two, except for odd cycles of size at least five, that need three colours. For odd-seq circulant graphs, we show that their clique-chromatic number is at most four, and determine the cases when it is equal to two. Similar bounds for the chromatic number of these graphs are also obtained.     

On Polygons Excluding Point Sets

By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that ${B\cup R}$ is in general position, and the convex hull of B contains k interior blue points and l interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points R. We show that there is a minimal number K = K(l), which is bounded from above by a polynomial in l, such that one can always find a blue polygonization excluding all red points, whenever k ≥ K. Some other related problems are also considered.     

Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation ratio known so far for these problems has ratio 3/2+?, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver [On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM J. Discrete Math. 2(1) (1989) 68-72]. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.     

Maximum weight independent set in trees

Computing a maximum independent set, weighted or unweighted, isNP-hard for general as well as planar graphs. However, polynomial time algorithms do exist for solving this problem on special classes of graphs. In this paper we present an efficient algorithm for computing a maximum weight independent set in trees. A divide and conquer approach based on centroid decomposition of trees is used to compute a maximum weight independent set withinO(n logn) time, wheren is the number of vertices in the tree. We introduce a notion of analternating tree which is crucial in obtaining a new independent set from the previous one.     

Geometric three-dimensional assignment problems

We investigate two geometric special cases of the three-dimensional assignment problem: Given are three sets B, R and G (blue, red and green) each containing n grid points in the Euclidean plane. We want to find a partition of B ∪ R ∪ G into n three0colored triangles such that (a) the total circumference of all triangles or (b) the total area of all triangles becomes minimum. Both versions of the problem are proved to be NP-hard.     

<Emphasis Type="Bold">NP</Emphasis> -Hardness of Largest Contained and Smallest Containing Simplices for <Emphasis Type="Italic">V</Emphasis>- and <Emphasis Type="Italic">H</Emphasis>-Polytopes

   Abstract. The problem of finding a d -simplex of maximum volume in an arbitrary d -dimensional V -polytope, for arbitrary d , was shown by Gritzmann et al. [GKL] in 1995 to be NP-hard. They conjectured that the corresponding problem for H -polytopes is also NP-hard. This paper presents a unified way of proving the NP-hardness of both these problems. The approach also yields NP-hardness proofs for the problems of finding d -simplices of minimum volume containing d -dimensional V - or H -polytopes. The polytopes that play the key role in the hardness proofs are truncations of simplices. A construction is presented which associates a truncated simplex to a given directed graph, and the hardness results follow from the hardness of detecting whether a directed graph has a partition into directed triangles.     

NP -Hardness of Largest Contained and Smallest Containing Simplices for V - and H -Polytopes

Abstract. The problem of finding a d -simplex of maximum volume in an arbitrary d -dimensional V -polytope, for arbitrary d , was shown by Gritzmann et al. [GKL] in 1995 to be NP-hard. They conjectured that the corresponding problem for H -polytopes is also NP-hard. This paper presents a unified way of proving the NP-hardness of both these problems. The approach also yields NP-hardness proofs for the problems of finding d -simplices of minimum volume containing d -dimensional V - or H -polytopes. The polytopes that play the key role in the hardness proofs are truncations of simplices. A construction is presented which associates a truncated simplex to a given directed graph, and the hardness results follow from the hardness of detecting whether a directed graph has a partition into directed triangles.     

A subexponential algorithm for the coloured tree partition problem

Given a tree of n vertices and a list of feasible colours for each vertex, the coloured tree partition problem (CTPP) consists in partitioning the tree into p vertex-disjoint subtrees of minimum total cost, and assigning to each subtree a different colour, which must be feasible for all of its vertices. The problem is strongly NP-hard on general graphs, as well as on grid and bipartite graphs. This paper deals with the previously open case of tree graphs, showing that it is strongly NP-complete to determine whether a feasible solution exists. It presents reduction, decomposition and bounding procedures to simplify the problem and an exact algorithm of complexity (with ) for the special case in which a vertex of each subtree is given.     

Decision-theoretic troubleshooting: Hardness of approximation

Decision-theoretic troubleshooting is one of the areas to which Bayesian networks can be applied. Given a probabilistic model of a malfunctioning man-made device, the task is to construct a repair strategy with minimal expected cost. The problem has received considerable attention over the past two decades. Efficient solution algorithms have been found for simple cases, whereas other variants have been proven NP-complete. We study several variants of the problem found in literature, and prove that computing approximate troubleshooting strategies is NP-hard. In the proofs, we exploit a close connection to set-covering problems.     

Path problems in generalized stars, complete graphs, and brick wall graphs

Path problems such as the maximum edge-disjoint paths problem, the path coloring problem, and the maximum path coloring problem are relevant for resource allocation in communication networks, in particular all-optical networks. In this paper, it is shown that maximum path coloring can be solved optimally in polynomial time for bidirected generalized stars, even in the weighted case. Furthermore, the maximum edge-disjoint paths problem is proved NP-hard for complete graphs (undirected or bidirected), a constant-factor off-line approximation algorithm is presented for the weighted case, and an on-line algorithm with constant competitive ratio is given for the unweighted case. Finally, an open problem concerning the existence of routings that simultaneously minimize the maximum load and the number of colors is solved: an example for a graph and a set of requests is given such that any routing that minimizes the maximum load requires strictly more colors for path coloring than a routing that minimizes the number of colors.