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.     

Rainbow Connection and Graph Products

A path in an edge colored graph G is called a rainbow path if all its edges have pairwise different colors. Then G is rainbow connected if there exists a rainbow path between every pair of vertices of G and the least number of colors needed to obtain a rainbow connected graph is the rainbow connection number. If we demand that there must exist a shortest rainbow path between every pair of vertices, we speak about strongly rainbow connected graph and the strong rainbow connection number. In this paper we study the (strong) rainbow connection number on the direct, strong, and lexicographic product and present several upper bounds for these products that are attained by many graphs. Several exact results are also obtained.     

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.     

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.     

Edge cover by connected bipartite subgraphs

We consider the problem of covering the edge set of an unweighted, undirected graph with the minimum number of connected bipartite subgraphs (where the subgraphs are not necessarily bicliques). We show that this is an NP-hard problem, provide lower bounds through an integer programming formulation, propose several constructive heuristics and a local search, and discuss computational results. Finally, we consider a constrained variant of the problem which we show to be NP-hard, and provide an integer programming formulation for the variant.     

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.     

The strong metric dimension of graphs and digraphs

Let G be a connected (di)graph. A vertex w is said to strongly resolve a pair u,v of vertices of G if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong resolving set for G is called the strong dimension of G. It is shown that the problem of finding the strong dimension of a connected graph can be transformed to the problem of finding the vertex covering number of a graph. Moreover, it is shown that computing this invariant is NP-hard. Related invariants for directed graphs are defined and studied.     

The shortest path problem with forbidden paths

We consider a variant of the constrained shortest path problem, where the constraints come from a set of forbidden paths (arc sequences) that cannot be part of any feasible solution. Two solution approaches are proposed for this variant. The first uses Aho and Corasick's keyword matching algorithm to filter paths produced by a k-shortest paths algorithm. The second generalizes Martins' deviation path approach for the k-shortest paths problem by merging the original graph with a state graph derived from Aho and Corasick's algorithm. Like Martins' approach, the second method amounts to a polynomial reduction of the shortest path problem with forbidden paths to a classic shortest path problem. Its significant advantage over the first approach is that it allows considering forbidden paths in more general shortest path problems such as the shortest path problem with resource constraints.     

Generating 3-vertex connected spanning subgraphs

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K²log(K)m²+K²m³) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.     

Models of adding relations to an organization structure of a complete K-ary tree

This paper proposes three models of adding relations to an organization structure which is a complete K-ary tree of height H: (i) a model of adding an edge between two nodes with the same depth N, (ii) a model of adding edges between every pair of nodes with the same depth N and (iii) a model of adding edges between every pair of siblings with the same depth N. For each of the three models, an optimal depth N¹ is obtained by maximizing the total shortening path length which is the sum of shortening lengths of shortest paths between every pair of all nodes.     

The parameterized complexity of the induced matching problem

Given a graph G and an integer k≥0, the NP-complete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs, as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W[1]-hard on general graphs, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide first-time fixed-parameter tractability results for planar graphs, bounded-degree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linear-size problem kernel for planar graphs.     

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.     

Local and variable neighborhood search for the <Emphasis Type="Italic">k</Emphasis>-cardinality subgraph problem

The minimum weighted k-cardinality subgraph problem consists of finding a connected subgraph of a given graph with exactly k edges whose sum of weights is minimum. For this NP-hard combinatorial problem, only constructive types of heuristics have been suggested in the literature. In this paper we propose a new heuristic based on variable neighborhood search metaheuristic rules. This procedure uses a new local search developed by us. Extensive numerical results that include graphs with up to 5,000 vertices are reported. It appears that VNS outperforms all previous methods.     

Revisiting dynamic programming for finding optimal subtrees in trees

In this paper we revisit an existing dynamic programming algorithm for finding optimal subtrees in edge weighted trees. This algorithm was sketched by Maffioli in a technical report in 1991. First, we adapt this algorithm for the application to trees that can have both node and edge weights. Second, we extend the algorithm such that it does not only deliver the values of optimal trees, but also the trees themselves. Finally, we use our extended algorithm for developing heuristics for the k-cardinality tree problem in undirected graphs G with node and edge weights. This NP-hard problem consists of finding in the given graph a tree with exactly k edges such that the sum of the node and the edge weights is minimal. In order to show the usefulness of our heuristics we conduct an extensive computational analysis that concerns most of the existing problem instances. Our results show that with growing problem size the proposed heuristics reach the performance of state-of-the-art metaheuristics. Therefore, this study can be seen as a cautious note on the scaling of metaheuristics.     

The computational complexity of bilevel assignment problems

In bilevel optimization problems there are two decision makers, the leader and the follower, who act in a hierarchy. Each decision maker has his own objective function, but there are common constraints. This paper deals with bilevel assignment problems where each decision maker controls a subset of edges and each edge has a leader’s and a follower’s weight. The edges selected by the leader and by the follower need to form a perfect matching. The task is to determine which edges the leader should choose such that his objective value which depends on the follower’s optimal reaction is maximized. We consider sum- and bottleneck objective functions for the leader and follower. Moreover, if not all optimal reactions of the follower lead to the same leader’s objective value, then the follower either chooses an optimal reaction which is best (optimistic rule) or worst (pessimistic rule) for the leader. We show that all the variants arising if the leader’s and follower’s objective functions are sum or bottleneck functions are NP-hard if the pessimistic rule is applied. In case of the optimistic rule the problem is shown to be NP-hard if at least one of the decision makers has a sum objective function.     

Note on maximal split-stable subgraphs

A multigraph G=(V,R∪B) with red and blue edges is an R/B-split graph if V is the union of a red and a blue stable set. Gavril has shown that R/B-split graphs yield a common generalization of split graphs and König-Egerváry graphs. Moreover, R/B-split graphs can be recognized in linear time. In this note, we address the corresponding optimization problem: identify a set of vertices of maximal cardinality that decomposes into a red and a blue stable set. This problem is NP-hard in general. We investigate the complexity of special and related cases (e.g., (anti-)chains in partial orders and stable matroid bases) and exhibit some NP-hard cases as well as polynomial ones.     

Min–max and min–max regret versions of combinatorial optimization problems: A survey

Min–max and min–max regret criteria are commonly used to define robust solutions. After motivating the use of these criteria, we present general results. Then, we survey complexity results for the min–max and min–max regret versions of some combinatorial optimization problems: shortest path, spanning tree, assignment, min cut, min s–t cut, knapsack. Since most of these problems are NP-hard, we also investigate the approximability of these problems. Furthermore, we present algorithms to solve these problems to optimality.     

Distances in orientations of graphs

We prove that there is a function h(k) such that every undirected graph G admits an orientation H with the following property: If an edge uv belongs to a cycle of length k in G, then uv or vu belongs to a directed cycle of length at most h(k) in H. Next, we show that every undirected bridgeless graph of radius r admits an orientation of radius at most r² + r, and this bound is best possible. We consider the same problem with radius replaced by diameter. Finilly, we show that the problem of deciding whether an undirected graph admits an orientation of diameter (resp. radius) 2 belongs to a class of problems called NP-hard.     

A trust region method for solving the decentralized static output feedback design problem

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n ²) time sequential algorithm and anO(n ²/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.