Matching interdiction

We introduce two interdiction problems involving matchings, one dealing with edge removals and the other dealing with vertex removals. Given is an undirected graph G with positive weights on its edges. In the edge interdiction problem, every edge of G has a positive cost and the task is to remove a subset of the edges constrained to a given budget, such that the weight of a maximum matching in the resulting graph is minimized. The vertex interdiction problem is analogous to the edge interdiction problem, with the difference that vertices instead of edges are removed. Hardness results are presented for both problems under various restrictions on the weights, interdiction costs and graph classes. Furthermore, we study the approximability of the edge and vertex interdiction problem on different graph classes. Several approximation-hardness results are presented as well as two constant-factor approximations, one of them based on iterative rounding. A pseudo-polynomial algorithm for solving the edge interdiction problem on graphs with bounded treewidth is proposed which can easily be adapted to the vertex interdiction problem. The algorithm presents a general framework to apply dynamic programming for solving a large class of problems in graphs with bounded treewidth. Additionally, we present a method to transform pseudo-polynomial algorithms for the edge interdiction problem into fully polynomial approximation schemes, using a scaling and rounding technique.     

Planar acyclic oriented graphs

A plane Hasse representation of an acyclic oriented graph is a drawing of the graph in the Euclidean plane such that all arcs are straight-line segments directed upwards and such that no two arcs cross. We characterize completely those oriented graphs which have a plane Hasse representation such that all faces are bounded by convex polygons. From this we derive the Hasse representation analogue, due to Kelly and Rival of Fary's theorem on straight-line representations of planar graphs and the Kuratowski type theorem of Platt for acyclic oriented graphs with only one source and one sink. Finally, we describe completely those acyclic oriented graphs which have a vertex dominating all other vertices and which have no plane Hasse representation, a problem posed by Trotter.     

Survivable network design under optimal and heuristic interdiction scenarios

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer’s network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.     

Optimizing cost flows by edge cost and capacity upgrade

We examine a network upgrade problem for cost flows. A budget can be distributed among the arcs of the network. An investment on each single arc can be used either to decrease the arc flow cost, or to increase the arc capacity, or both. The goal is to maximize the flow through the network while not exceeding bounds on the budget and on the total flow cost.

The problems are NP-hard even on series-parallel graphs. We provide an approximation algorithm on series-parallel graphs which, for arbitrary δ,>0, produces a solution which exceeds the bounds on the budget and the flow cost by factors of at most 1+δ and 1+, respectively, while the amount of flow is at least that of an optimum solution. The running time of the algorithm is polynomial in the input size and 1/(δ). In addition we give an approximation algorithm on general graphs applicable to problem instances with small arc capacities.     

Pointed drawings of planar graphs

We study the problem how to draw a planar graph crossing-free such that every vertex is incident to an angle greater than π. In general a plane straight-line drawing cannot guarantee this property. We present algorithms which construct such drawings with either tangent-continuous biarcs or quadratic Bézier curves (parabolic arcs), even if the positions of the vertices are predefined by a given plane straight-line drawing of the graph. Moreover, the graph can be drawn with circular arcs if the vertices can be placed arbitrarily. The topic is related to non-crossing drawings of multigraphs and vertex labeling.     

Anticoloring and separation of graphs

An anticoloring of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. The anticoloring problem seeks, roughly speaking, such colorings with many vertices colored in each color. We deal with the anticoloring problem for planar graphs and, using Lipton and Tarjan’s separation algorithm, provide an algorithm with some bound on the error. We also show that, to solve the anticoloring problem for general graphs, it suffices to solve it for connected graphs.     

Heuristics for Multi-Stage Interdiction of Stochastic Networks

We describe and compare heuristic solution methods for a multi-stage stochastic network interdiction problem. The problem is to maximize the probability of sufficient disruption of the flow of information or goods in a network whose characteristics are not certain. In this formulation, interdiction subject to a budget constraint is followed by operation of the network, which is then followed by a second interdiction subject to a second budget constraint. Computational results demonstrate and compare the effectiveness of heuristic algorithms. This problem is interesting in that computing an objective function value requires tremendous effort. We exhibit classes of instances in our computational experiments where local search based on a transformation neighborhood is dominated by a constructive neighborhood.     

A note on transitive orientations with maximum sets of sources and sinks

Given a transitive orientation of a comparability graph G, a vertex of G is a source (sink) if it has indegree (outdegree) zero in , respectively. A source set of G is a subset of vertices formed by sources of some transitive orientation . A pair of subsets S,TV(G) is a source–sink pair of G when the vertices of S and T are sources and sinks, of some transitive orientation , respectively. We describe algorithms for finding a transitive orientation with a maximum source–sink pair in a comparability graph. The algorithms are applications of modular decomposition and are all of linear-time complexity.     

Polynomial algorithms for (integral) maximum two-flows in vertex edge-capacitated planar graphs

In this paper we study the maximum two-flow problem in vertex- and edge-capacitated undirected ST₂-planar graphs, that is, planar graphs where the vertices of each terminal pair are on the same face. For such graphs we provide an O(n) algorithm for finding a minimum two-cut and an O(n log n) algorithm for determining a maximum two-flow and show that the value of a maximum two-flow equals the value of a minimum two-cut. We further show that the flow obtained is half-integral and provide a characterization of edge and vertex capacitated ST₂-planar graphs that guarantees a maximum two-flow that is integral. By a simple variation of our maximum two-flow algorithm we then develop, for ST₂-planar graphs with vertex and edge capacities, an O(n log n) algorithm for determining an integral maximum two-flow of value not less than the value of a maximum two-flow minus one.     

Solving the vehicle routing problem with time windows and multiple routes exactly using a pseudo-polynomial model

In this paper, we address a variant of the vehicle routing problem called the vehicle routing problem with time windows and multiple routes. It considers that a given vehicle can be assigned to more than one route per planning period. We propose a new exact algorithm for this problem. Our algorithm is iterative and it relies on a pseudo-polynomial network flow model whose nodes represent time instants, and whose arcs represent feasible vehicle routes. This algorithm was tested on a set of benchmark instances from the literature. The computational results show that our method is able to solve more instances than the only other exact method described so far in the literature, and it clearly outperforms this method in terms of computing time.     

Sources and Sinks in Comparability Graphs

We prove that a subset S of vertices of a comparability graph G is a source set if and only if each vertex of S is a source and there is no odd induced path in G between two vertices of S. We also characterize pairs of subsets corresponding to sources and sinks, respectively. Finally, an application to interval graphs is obtained.     

The quickest path problem with batch constraints

The quickest path problem has been proposed to cope with flow problems through networks whose arcs are characterized by both travel times and flowrate constraints. Basically, it consists in finding a path in a network to transmit a given amount of items from a source node to a sink in as little time as possible, when the transmission time depends on both the traversal times of the arcs and the rates of flow along arcs. This paper is focused on the solution procedure when the items transmission must be partitioned into batches with size limits. For this problem we determine how many batches must be made and what the sizes should be.     

Protection of flows under targeted attacks

We present a new robust optimization model for the problem of maximizing the amount of flow surviving the attack of an interdictor. Given some path flow, our model allows the interdictor to specify the amount of flow removed from each path individually. In contrast to previous models, for which no efficient algorithms are known, the most important basic variants of our model can be solved in poly-time. We also consider extensions where there is a budget to set the interdiction costs.     

Analysis of budget for interdiction on multicommodity network flows

In this paper, we concentrate on computing several critical budgets for interdiction of the multicommodity network flows, and studying the interdiction effects of the changes on budget. More specifically, we first propose general interdiction models of the multicommodity flow problem, with consideration of both node and arc removals and decrease of their capacities. Then, to perform the vulnerability analysis of networks, we define the function F(R) as the minimum amount of unsatisfied demands in the resulted network after worst-case interdiction with budget R. Specifically, we study the properties of function F(R), and find the critical budget values, such as \(R_a\), the largest value under which all demands can still be satisfied in the resulted network even under the worst-case interdiction, and \(R_b\), the least value under which the worst-case interdiction can make none of the demands be satisfied. We prove that the critical budget \(R_b\) for completely destroying the network is not related to arc or node capacities, and supply or demand amounts, but it is related to the network topology, the sets of source and destination nodes, and interdiction costs on each node and arc. We also observe that the critical budget \(R_a\) is related to all of these parameters of the network. Additionally, we present formulations to estimate both \(R_a\) and \(R_b\). For the effects of budget increasing, we present the conditions under which there would be extra capabilities to interdict more arcs or nodes with increased budget, and also under which the increased budget has no effects for the interdictor. To verify these results and conclusions, numerical experiments on 12 networks with different numbers of commodities are performed.     

An algorithm to solve the proportional network flow problem

The proportional network flow problem is a generalization of the equal flow problem on a generalized network in which the flow on arcs in given sets must all be proportional. This problem appears in several natural contexts, including processing networks and manufacturing networks. This paper describes a transformation on the underlying network that reduces the problem to the equal flow problem; this transformation is used to show that algorithms that solve the equal flow problem can be directly applied to the proportional network flow problem as well, with no increase in asymptotic running time. Additionally, computational results are presented for the proportional network flow problem demonstrating equivalent performance to the same algorithm for the equal flow problem.     

A specialized network simplex algorithm for the constrained maximum flow problem

The constrained maximum flow problem is to send the maximum flow from a source to a sink in a directed capacitated network where each arc has a cost and the total cost of the flow cannot exceed a budget. This problem is similar to some variants of classical problems such as the constrained shortest path problem, constrained transportation problem, or constrained assignment problem, all of which have important applications in practice. The constrained maximum flow problem itself has important applications, such as in logistics, telecommunications and computer networks. In this research, we present an efficient specialized network simplex algorithm that significantly outperforms the two widely used LP solvers: CPLEX and lp_solve. We report CPU times of an average of 27 times faster than CPLEX (with its dual simplex algorithm), the closest competitor of our algorithm.     

The complexity of the majority rule on planar graphs

We study the complexity of the majority rule on planar automata networks. We reduce a special case of the Monotone Circuit Value Problem to the prediction problem of determining if a vertex of a planar graph will change its state when the network is updated with the majority rule.     

Comparing and Simplifying Distinct-Cluster Phylogenetic Networks

Phylogenetic networks are rooted acyclic directed graphs in which the leaves are identified with members of a set X of species. The cluster of a vertex is the set of leaves that are descendants of the vertex. A network is “distinct-cluster” if distinct vertices have distinct clusters. This paper focuses on the set DC(X) of distinct-cluster networks whose leaves are identified with the members of X. For a fixed X, a metric on DC(X) is defined. There is a “cluster-preserving” simplification process by which vertices or certain arcs may be removed without changing the clusters of any remaining vertices. Many of the resulting networks may be uniquely determined without regard to the order of the simplifying operations.     

List precoloring extension in planar graphs

A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding palette for each vertex so that the resulting coloring is proper. This result is referred to as 5-choosability of planar graphs. Albertson asked whether Thomassen’s theorem can be extended by precoloring some vertices which are at a large enough distance apart in a graph. Here, among others, we answer the question in the case when the graph does not contain short cycles separating precolored vertices and when there is a “wide” Steiner tree containing all the precolored vertices.     

Directed faces in planar digraphs and unicoloured faces in edge 2-coloured planar Eulerian maps

A theorem is proved on the existence of directed faces and sources or sinks in directed planar maps. This is applied to obtain a result about the existence of unicoloured faces and alternately coloured edges at a vertex in planar Eulerian maps. A theorem of the same type on planar directed four-valent graphs is also given. It is shown that the three theorems are equivalent.