Carlos A. J. Martinhon

Paths and trails in edge-colored graphs

This paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s and t in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s-t path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s and t for a particular class of graphs and characterize edge-colored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k pairwise vertex/edge disjoint properly edge-colored s-t paths/trails in a c-edge-colored graph G^c is NP-complete even for k=2 and c=@W(n^2), where n denotes the number of vertices in G^c. Moreover, we prove that these problems remain NP-complete for c-edge-colored graphs containing no properly edge-colored cycles and c=@W(n). We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.

Stronger K-tree relaxations for the vehicle routing problem☆

Abstract A Lagrangian based exact solution algorithm for the vehicle routing problem (VRP), defined on an undirected graph, is introduced in this paper. Lower bounds are obtained by allowing exponentially many inequalities as candidates to Lagrangian dualization. Three different families of strong valid inequalities (each one with exponentially many elements) are used within VRP formulations. For each of them, separation procedures are proposed for points that define incidence vectors of K -trees. Violated inequalities identified in this way are then dualized in a relax and cut framework. Upper bounds are generated through a Lagrangian Clarke and Wright heuristic. A variable fixation test based on (approximating) linear programming reduced costs, is also implemented. Computational results are presented for the proposed algorithm.

The Minimum Reload s-t Path/Trail/Walk Problems

This paper deals with problems on non-oriented edge-colored graphs. The aim is to find a route between two given vertices s and t . This route can be a walk, a trail or a path. Each time a vertex is crossed by a walk there is an associated non-negative reload cost r i ,j , where i and j denote, respectively, the colors of successive edges in this walk. The goal is to find a route whose total reload cost is minimum. Polynomial algorithms and proofs of NP-hardness are given for particular cases: when the triangle inequality is satisfied or not, when reload costs are symmetric (i.e. r i ,j = r j ,i ) or asymmetric. We also investigate bounded degree graphs and planar graphs.

Complexity of trails, paths and circuits in arc-colored digraphs

We deal with different algorithmic questions regarding properly arc-colored s-t trails, paths and circuits in arc-colored digraphs. Given an arc-colored digraph D^c with c>=2 colors, we show that the problem of determining the maximum number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time. Surprisingly, we prove that the determination of a properly arc-colored s-t path is NP-complete even for planar digraphs containing no properly arc-colored circuits and c=@W(n), where n denotes the number of vertices in D^c. If the digraph is an arc-colored tournament, we show that deciding whether it contains a properly arc-colored circuit passing through a given vertex x (resp., properly arc-colored Hamiltonian s-t path) is NP-complete for c>=2. As a consequence, we solve a weak version of an open problem posed in Gutin et al. (1998) [23], whose objective is to determine whether a 2-arc-colored tournament contains a properly arc-colored circuit.

Complexity of paths, trails and circuits in arc-colored digraphs

We deal with different algorithmic questions regarding properly arc-colored s-t paths, trails and circuits in arc-colored digraphs Given an arc-colored digraph Dc with c≥2 colors, we show that the problem of maximizing the number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time Surprisingly, we prove that the determination of one properly arc-colored s-t path is NP-complete even for planar digraphs containing no properly arc-colored circuits and c=Ω(n), where n denotes the number of vertices in Dc If the digraph is an arc-colored tournament, we show that deciding whether it contains a properly arc-colored circuit passing through a given vertex x (resp., properly arc-colored Hamiltonian s-t path) is NP-complete, even if c=2 As a consequence, we solve a weak version of an open problem posed in Gutin et al. [17].

On s-t paths and trails in edge-colored graphs

In this paper we deal from an algorithmic perspective with different questions regarding monochromatic and properly edge-colored s-t paths/trails on edge-colored graphs. Given a c-edge-colored graph Gc without properly edge-colored closed trails, we present a polynomial time procedure for the determination of properly edge-colored s-t trails visiting all vertices of Gc a prescribed number of times. As an immediate consequence, we polynomially solve the Hamiltonian path (resp., Eulerian trail) problem for this particular class of graphs. In addition, we prove that to check whether Gc contains 2 properly edge-colored s-t paths/trails with length at most L>0 is NP-complete in the strong sense. Finally, we prove that, if Gc is a general c-edge-colored graph, to find 2 monochromatic vertex disjoint s-t paths with different colors is NP-complete.

Gerando orientações acíclicas com algoritmos probabilísticos distribuídos

This paper presents a new randomized distributed algorithm for the generation of acyclic orientations upon anonymous distributed systems of arbitrary topology. This algorithm is analyzed in terms of correctness and complexity as well as its convergence rate. In particular, it is shown that this new algorithm, called Alg-Arestas, is able to produce, with high probability, acyclic orientations quasi instantaneously, i.e., in less than two steps. Two applications of this form of symmetry breaking will be discussed: (i) initialization of Scheduling by Edge Reversal (SER), a simple and powerful distributed scheduling algorithm, and (ii) a strategy for distributed uploading in computer networks.

An Improved Derandomized Approximation Algorithm for the Max-Controlled Set Problem

A vertex i of a graph G=(V,E) is said to be controlled by M ⊆ V if the majority of the elements of the neighborhood of i (including itself) belong to M. The set M is a monopoly in G if every vertex i ∈ V is controlled by M. Given a set M ⊆ V and two graphs G 1=(V,E 1) and G 2=(V,E 2) where E 1 ⊆ E 2, the monopoly verification problem (mvp) consists of deciding whether there exists a sandwich graph G=(V,E) (i.e., a graph where E 1 ⊆ E ⊆ E 2) such that M is a monopoly in G=(V,E). If the answer to the mvp is No, we then consider the max-controlled set problem (mcsp), whose objective is to find a sandwich graph G=(V,E) such that the number of vertices of G controlled by M is maximized. The mvp can be solved in polynomial time; the mcsp, however, is NP-hard. In this work, we present a deterministic polynomial time approximation algorithm for the mcsp with ratio \(\frac{1}{2}+ \frac{1+\sqrt{n}}{2n-2}\), where n=|V|>4. (The case n ≤ 4 is solved exactly by considering the parameterized version of the mcsp.) The algoritm is obtained through the use of randomized rounding and derandomization techniques, namely the method of conditional expectations. Additionally, we show how to improve this ratio if good estimates of expectation are obtained in advance.

A variable fixing heuristic with Local Branching for the fixed charge uncapacitated network design problem with user-optimal flow

This paper presents an iterated local search for the fixed-charge uncapacitated network design problem with user-optimal flow (FCNDP-UOF), which concerns routing multiple commodities from its origin to its destination by designing a network through selecting arcs, with an objective of minimizing the sum of the fixed costs of the selected arcs plus the sum of variable costs associated with the flows on each arc. Besides that, since the FCNDP-UOF is a bilevel problem, each commodity has to be transported through a shortest path, concerning the edges length, in the built network. The proposed algorithm generates an initial solution using a variable fixing heuristic. Then a local branching strategy is applied to improve the quality of the solution. At last, an efficient perturbation strategy is presented to perform cycle-based moves to explore different parts of the solution space. Computational experiments show that the proposed solution method consistently produces high-quality solutions in reasonable computational times. HighlightsWe propose an Iterated Local Search based heuristic for the Fixed Charge Uncapacitated Network Design Problem with User-optimal Flow.The developed algorithm relies on very few parameters.A novel perturbation mechanism based on cycle-based moves is proposed.Extensive computational experiments illustrate the robustness of our algorithm in terms of solution quality.First Hybrid Method for this problem.

The edge-recoloring cost of monochromatic and properly edge-colored paths and cycles

We introduce a number of problems regarding edge-color modifications in edge-colored graphs and digraphs. Consider a property π, a c-edge-colored graph G c not satisfying π, and an edge-recoloring cost matrix R = r i j c × c where r i j ? 0 denotes the cost of changing color i of edge e to color j. Basically, in this kind of problem the idea is to change the colors of one or more edges of G c in order to construct a new edge-colored graph such that the total edge-recoloring cost is minimized and property π is satisfied. We also consider the destruction of potentially undesirable structures with the minimum edge-recoloring cost. In this paper, we are especially concerned with the construction and destruction of properly edge-colored and monochromatic paths, trails and cycles in graphs and digraphs. Some related problems and future directions are presented.