A tabu search heuristic based on k-diamonds for the weighted feedback vertex set problem

Given an undirected and vertex weighted graph G = (V,E,w), the Weighted Feedback Vertex Problem (WFVP) consists of finding a subset F ⊆ V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The WFVP on general graphs is known to be NP-hard and to be polynomially solvable on some special classes of graphs (e.g., interval graphs, co-comparability graphs, diamond graphs). In this paper we introduce an extension of diamond graphs, namely the k-diamond graphs, and give a dynamic programming algorithm to solve WFVP in linear time on this class of graphs. Other than solving an open question, this algorithm allows an efficient exploration of a neighborhood structure that can be defined by using such a class of graphs. We used this neighborhood structure inside our Iterated Tabu Search heuristic. Our extensive experimental show the effectiveness of this heuristic in improving the solution provided by a 2-approximate algorithm for the WFVPon general graphs

an evolutionary approach for the offsetting inventory cycle problem

AbstractIn inventory management, a fundamental issue is the rational use of required space. Among the numerous techniques adopted, an important role is played by the determination of the replenishment cycle offsetting which minimizes the warehouse space within a considered time horizon. The NP-completeness of the Offsetting Inventory Cycle Problem (OICP) has led the researchers towards the development and the comparison of specific heuristics. We propose and implement a genetic algorithm for the OICP, whose effectiveness is validated by comparing its solutions with those found by a mixed integer programming model. The algorithm, tested on realistic instances, shows a high reduction of the maximum space and a more regular warehouse saturation with negligible increase of the total cost. This paper, unlike other papers currently available in literature, provides instances data and results necessary for reproducibility, aiming to become a benchmark for future comparisons with other OICP algorithms

Minimum Weighted Feedback Vertex Set on Diamonds

Given a vertex weighted graph G, a minimum Weighted Feedback Vertex Set (MWFVS) is a subset F ? V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The MWFVS on general graph is known to be NP-hard. In this paper we introduce a new class of graphs, namely the diamond graphs, and give a linear time algorithm to solve MWFVS on it. We will discuss, moreover, how this result could be used to effectively improve the approximated solution of any known heuristic to solve MWFVS on a general graph

A linear time algorithm for the minimum Weighted Feedback Vertex Set on diamonds

Given an undirected and vertex weighted graph G, the Weighted Feedback Vertex Problem (WFVP) consists in finding a subset Fsubset of or equal toV of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The WFVP on general graphs is known to be NP-hard. In this paper we introduce a new class of graphs, namely the diamond graphs, and give a linear time algorithm to solve WFVP on it

A Mathematical Programming Approach for the Maximum Labeled Clique Problem

This paper addresses a variant of the classical clique problem in which the edges of the graph are labeled. The problem consists of finding a clique as large as possible whose edge set contains at most b ∈ Z+ different labels. Moreover, in case of more feasible cliques of the same maximum size, we look for the one with the minimum number of labels. We study the time complexity of the problem, also in special cases, and we propose a mathematical programming approach for its solution by introducing two different formulations: the basic and the enforced. We experimentally evaluate the performance of the proposed approach on a set of benchmark instances (DIMACS) suitably adapted to the problem

A reduction heuristic for the all-colors shortest path problem

The All-Colors Shortest Path is a recently introduced NP-Hard optimization problem, in which a color is assigned to each vertex of an edge weighted graph, and the aim is to find the shortest path spanning all colors. The solution path can be not simple, that is it is possible to visit multiple times the same vertices if it is a convenient choice. The starting vertex can be constrained (ACSP) or not (ACSP-UE). We propose a reduction heuristic based on the transformation of any ACSP-UE instance into an Equality Generalized Traveling Salesman Problem one. Computational results show the algorithm to outperform the best previously known one

The monochromatic set partitioning problem.Presentato al convegno Airo 2010, 07-10 Settembre 2010, Villa San Giovanni (RC).

On the last few years several problems have been studied on a particular class of graphs, where each edge has a label (color) assigned to it. Real applications for this class of problems arise in fields such as telecommunication or multimodal transport networks (edges of the same color can model transportation lines of the same type, or connections belonging to the same company). Moreover, the edge-labeled graphs can be of interest whenever we need some measure of homogeneity (or heterogeneity) regarding the edges in the solution we are looking for. In this context we focalize our attention on the "monochromatic set partitioning problem" (MSP). Let G=(V,E) be an edge-colored graph. A sub-graph H of G is said to be monochromatic if all the edges of H have the same color and multicolored if no two edges of H have the same color. A feasible solution for the MSP is a partitioning of G in monochromatic sub-graph. We look for a feasible solution containing the minimum number of such monochromatic sub-graph. In our work we first prove the complexity of this problem. Then we propose a mathematical formulation and a polynomial case. Finally we present a meta-heuristic approach to solve the problem and show some preliminary computational results