Given a set of timetabled tasks, the multi-depot vehicle scheduling problem is a well-known probl... more Given a set of timetabled tasks, the multi-depot vehicle scheduling problem is a well-known problem that consists of determining leastcost schedules for vehicles assigned to several depots such that each task is accomplished exactly once by a vehicle. In this paper, we propose to compare the performance of five different heuristic approaches for this problem, namely, a heuristic mip solver, a Lagrangian heuristic, a column generation heuristic, a large neighborhood search heuristic using column generation for neighborhood evaluation, and a tabu search heuristic. The first three methods are adaptations of existing methods, while the last two are novel approaches for this problem. Computational results on randomly generated instances show that the column generation heuristic performs the best when enough computational time is available and stability is required, while the large neighborhood search method is the best alternative when looking for a compromise between computational time and solution quality.
Les textes publiés dans la série des rapports de recherche Les Cahiers du GERAD n’engagent que ... more Les textes publiés dans la série des rapports de recherche Les Cahiers du GERAD n’engagent que la responsabilite ́ de leurs auteurs. La publication de ces rapports de recherche est rendue possible grâce au soutien de HEC Montréal, Polytechnique Montréal, Universite ́ McGill, Universite ́ du Québec a ̀ Montréal, ainsi que du Fonds de recherche du Québec – Nature et technologies.
Abstract. It is well known that each tree metric M has a unique realization as a tree, and that t... more Abstract. It is well known that each tree metric M has a unique realization as a tree, and that this realization minimizes the total length of the edges among all other realizations of M. We extend this result to the class of symmetric matrices M with zero diagonal, positive entries, and such that mij + mkl ≤ max{mik + mjl, mil + mjk} for all distinct i, j, k, l.
Let G = (V, E, w) be a graph with vertex and edge sets V and E, respectively, and w : E → IR + a ... more Let G = (V, E, w) be a graph with vertex and edge sets V and E, respectively, and w : E → IR + a function which assigns a positive weigth or length to each edge of G. G is called a realization of a finite metric space (M, d), with M = {1, ..., n} if and only if {1, ..., n} ⊆ V and d(i, j) is equal to the length of the shortest chain linking i and j in G ∀i, j = 1, ..., n. A realization G of (M, d), is said optimal if the sum of its weights is minimal among all the realizations of (M, d). Consider a partition of M into two nonempty subsets K and L, and let e be an edge in a realization G of (M, d); we say that e is a bridge linking K with L if e belongs to all chains in G linking a vertex of K with a vertex of L. The Metric Bridge Partition Problem is to determine if the elements of a finite metric space (M, d) can be partitioned into two nonempty subsets K and L such that all optimal realizations of (M, d) contain a bridge linking K with L. We prove in this paper that this problem is polynomially solvable. We also describe an algorithm that constructs an optimal realization of (M, d) from optimal realizations of (K, d| K) and (L, d| L).
Two vertex colorings of a graph G are equivalent if they induce the same partition of the vertex ... more Two vertex colorings of a graph G are equivalent if they induce the same partition of the vertex set into color classes. The graphical Bell number B(G) is the number of non-equivalent vertex colorings of G. We determine a sharp lower bound on B(G) for graphs G of order n and maximum degree n − 3, and we characterize the graphs for which the bound is attained.
We describe a tabu search algorithm for the vehicle routing problem with split deliveries. At eac... more We describe a tabu search algorithm for the vehicle routing problem with split deliveries. At each iteration, a neighbor solution is obtained by removing a customer from a set of routes where it is currently visited and inserting it either into a new route or into an existing route that has enough residual capacity. The algorithm also considers the possibility of inserting a customer into a route without removing it from another route. The insertion of a customer into a route is done by means of the cheapest insertion method. Computational experiments are reported for a set of benchmark problems, and the results are compared with those obtained by the algorithm proposed by Dror and Trudeau.
... [10] A. Hertz and D. de Werra, "Using tabu search techniques for graph coloring", C... more ... [10] A. Hertz and D. de Werra, "Using tabu search techniques for graph coloring", Computing 39 (1987) 345351. ... [14] DW Matula, G. Marble, JD Isaacson, "Graph coloring algorithms", in: RC Read (ed.), Graph Theo' and Com puting Academic Press, New York, 1972, 108122. ...
In this paper, we study the capacitated team orienteering and profitable tour problems (CTOP and ... more In this paper, we study the capacitated team orienteering and profitable tour problems (CTOP and CPTP). The interest in these problems comes from recent developments in the use of the Internet for a better matching of demand and offer of transportation services. We propose exact and heuristic procedures for the CTOP and the CPTP. The computational results show that the heuristic procedures often find the optimal solution and in general cause very limited errors.
We consider the problem of orienting the edges of a graph so that the length of a longest path in... more We consider the problem of orienting the edges of a graph so that the length of a longest path in the resulting digraph is minimum. As shown by Gallai, Roy and Vitaver, this edge orienting problem is equivalent to finding the chromatic number of a graph. We study various properties of edge orienting methods in the context of local search for graph coloring. We then exploit these properties to derive four tabu search algorithms, each based on a different neighborhood. We compare these algorithms numerically to determine which are the most promising and to give potential research directions.
A total dominating set in a digraph G is a subset W of its vertices such that every vertex of G h... more A total dominating set in a digraph G is a subset W of its vertices such that every vertex of G has an immediate successor in W . The total domination number of G is the size of the smallest total dominating set. We consider several lower bounds on the total domination number and conjecture that these bounds are strictly larger than g(G) -1, where g(G) is the number of vertices of the smallest directed cycle contained in G. We prove that these new conjectures are equivalent to the Caccetta-Häggkvist conjecture which asserts that g(G) -1 < n r in every digraph on n vertices with minimum outdegree at least r > 0.
We consider the problem of determining the size of a maximum clique in a graph, also known as the... more We consider the problem of determining the size of a maximum clique in a graph, also known as the clique number. Given any method that computes an upper bound on the clique number of a graph, we present a sequential elimination algorithm which is guaranteed to improve upon that upper bound. Computational experiments on DIMACS instances show that, on average, this algorithm can reduce the gap between the upper bound and the clique number by about 60%. We also show how to use this sequential elimination algorithm to improve the computation of lower bounds on the clique number of a graph.
Journal of Graph Algorithms and Applications, 2021
Let G = (V, E) be a graph and let S ⊆ V be a subset of its vertices. If the subgraph of G induced... more Let G = (V, E) be a graph and let S ⊆ V be a subset of its vertices. If the subgraph of G induced by V \ S is acyclic, then S is said to be a decycling set of G. The size of a smallest decycling set of G is called the decycling number of G. Determining the decycling number of a graph G is NP-hard, even if G is bipartite. We describe a tabu search procedure that generates decycling sets of small size for arbitrary bipartite graphs. Tests on challenging families of graphs show that the proposed algorithm improves many best-known solutions, thus closing or narrowing the gap to the best-known lower bounds.
We consider the problem of assigning clients to nurses for home care services. The aim is to bala... more We consider the problem of assigning clients to nurses for home care services. The aim is to balance the work load of the nurses while avoiding long travels to visit the clients. We analyze the case of the CSSS Côte-des-Neiges, Métro and Parc Extension for which a previous analysis has shown that demand fluctuations may create work overload for the nursing staff. We present two models, one with linear constraints and a quadratic objective function which we optimize using CPLEX, and a more complex model with non linear constraints that we optimize using a tabu search algorithm.
The following is a chapter of the book entitled Fourmis artificielles, des bases algorithmiques a... more The following is a chapter of the book entitled Fourmis artificielles, des bases algorithmiques aux concepts et réalisations avancés, Nicolas Monmarché, Frédéric Guinand and Patrick Siarry, editors, which will be published by Hermès Science Publishing Ltd in 2008. Les Cahiers du GERAD G–2008–29 1
A profit and a demand are associated with each arc of a set of profitable arcs of a given graph. ... more A profit and a demand are associated with each arc of a set of profitable arcs of a given graph. A travel time is associated with each arc of the graph. A fleet of capacitated vehicles is given to serve the profitable arcs. A maximum duration of the route of each vehicle is also given. The profit of an arc can be collected by one vehicle only that also serves the demand of the arc. The objective of this problem, that is called the Capacitated Arc Routing Problem with Profits (CARPP), is to find a set of routes that satisfy the constraints on the duration of the route and on the capacity of the vehicle and maximize the total collected profit. We propose a branch-and-price algorithm and several heuristics. We can solve exactly instances with up to 97 profitable arcs. The best heuristics find the optimal solution on most of instances where it is available.
Distance metric learning algorithms aim to appropriately measure similarities and distances betwe... more Distance metric learning algorithms aim to appropriately measure similarities and distances between data points. In the context of clustering, metric learning is typically applied with the assist of side-information provided by experts, most commonly expressed in the form of cannot-link and must-link constraints. In this setting, distance metric learning algorithms move closer pairs of data points involved in must-link constraints, while pairs of points involved in cannot-link constraints are moved away from each other. For these algorithms to be effective, it is important to use a distance metric that matches the expert knowledge, beliefs, and expectations, and the transformations made to stick to the side-information should preserve geometrical properties of the dataset. Also, it is interesting to filter the constraints provided by the experts to keep only the most useful and reject those that can harm the clustering process. To address these issues, we propose to exploit the dual...
Given a set of timetabled tasks, the multi-depot vehicle scheduling problem is a well-known probl... more Given a set of timetabled tasks, the multi-depot vehicle scheduling problem is a well-known problem that consists of determining leastcost schedules for vehicles assigned to several depots such that each task is accomplished exactly once by a vehicle. In this paper, we propose to compare the performance of five different heuristic approaches for this problem, namely, a heuristic mip solver, a Lagrangian heuristic, a column generation heuristic, a large neighborhood search heuristic using column generation for neighborhood evaluation, and a tabu search heuristic. The first three methods are adaptations of existing methods, while the last two are novel approaches for this problem. Computational results on randomly generated instances show that the column generation heuristic performs the best when enough computational time is available and stability is required, while the large neighborhood search method is the best alternative when looking for a compromise between computational time and solution quality.
Les textes publiés dans la série des rapports de recherche Les Cahiers du GERAD n’engagent que ... more Les textes publiés dans la série des rapports de recherche Les Cahiers du GERAD n’engagent que la responsabilite ́ de leurs auteurs. La publication de ces rapports de recherche est rendue possible grâce au soutien de HEC Montréal, Polytechnique Montréal, Universite ́ McGill, Universite ́ du Québec a ̀ Montréal, ainsi que du Fonds de recherche du Québec – Nature et technologies.
Abstract. It is well known that each tree metric M has a unique realization as a tree, and that t... more Abstract. It is well known that each tree metric M has a unique realization as a tree, and that this realization minimizes the total length of the edges among all other realizations of M. We extend this result to the class of symmetric matrices M with zero diagonal, positive entries, and such that mij + mkl ≤ max{mik + mjl, mil + mjk} for all distinct i, j, k, l.
Let G = (V, E, w) be a graph with vertex and edge sets V and E, respectively, and w : E → IR + a ... more Let G = (V, E, w) be a graph with vertex and edge sets V and E, respectively, and w : E → IR + a function which assigns a positive weigth or length to each edge of G. G is called a realization of a finite metric space (M, d), with M = {1, ..., n} if and only if {1, ..., n} ⊆ V and d(i, j) is equal to the length of the shortest chain linking i and j in G ∀i, j = 1, ..., n. A realization G of (M, d), is said optimal if the sum of its weights is minimal among all the realizations of (M, d). Consider a partition of M into two nonempty subsets K and L, and let e be an edge in a realization G of (M, d); we say that e is a bridge linking K with L if e belongs to all chains in G linking a vertex of K with a vertex of L. The Metric Bridge Partition Problem is to determine if the elements of a finite metric space (M, d) can be partitioned into two nonempty subsets K and L such that all optimal realizations of (M, d) contain a bridge linking K with L. We prove in this paper that this problem is polynomially solvable. We also describe an algorithm that constructs an optimal realization of (M, d) from optimal realizations of (K, d| K) and (L, d| L).
Two vertex colorings of a graph G are equivalent if they induce the same partition of the vertex ... more Two vertex colorings of a graph G are equivalent if they induce the same partition of the vertex set into color classes. The graphical Bell number B(G) is the number of non-equivalent vertex colorings of G. We determine a sharp lower bound on B(G) for graphs G of order n and maximum degree n − 3, and we characterize the graphs for which the bound is attained.
We describe a tabu search algorithm for the vehicle routing problem with split deliveries. At eac... more We describe a tabu search algorithm for the vehicle routing problem with split deliveries. At each iteration, a neighbor solution is obtained by removing a customer from a set of routes where it is currently visited and inserting it either into a new route or into an existing route that has enough residual capacity. The algorithm also considers the possibility of inserting a customer into a route without removing it from another route. The insertion of a customer into a route is done by means of the cheapest insertion method. Computational experiments are reported for a set of benchmark problems, and the results are compared with those obtained by the algorithm proposed by Dror and Trudeau.
... [10] A. Hertz and D. de Werra, "Using tabu search techniques for graph coloring", C... more ... [10] A. Hertz and D. de Werra, "Using tabu search techniques for graph coloring", Computing 39 (1987) 345351. ... [14] DW Matula, G. Marble, JD Isaacson, "Graph coloring algorithms", in: RC Read (ed.), Graph Theo' and Com puting Academic Press, New York, 1972, 108122. ...
In this paper, we study the capacitated team orienteering and profitable tour problems (CTOP and ... more In this paper, we study the capacitated team orienteering and profitable tour problems (CTOP and CPTP). The interest in these problems comes from recent developments in the use of the Internet for a better matching of demand and offer of transportation services. We propose exact and heuristic procedures for the CTOP and the CPTP. The computational results show that the heuristic procedures often find the optimal solution and in general cause very limited errors.
We consider the problem of orienting the edges of a graph so that the length of a longest path in... more We consider the problem of orienting the edges of a graph so that the length of a longest path in the resulting digraph is minimum. As shown by Gallai, Roy and Vitaver, this edge orienting problem is equivalent to finding the chromatic number of a graph. We study various properties of edge orienting methods in the context of local search for graph coloring. We then exploit these properties to derive four tabu search algorithms, each based on a different neighborhood. We compare these algorithms numerically to determine which are the most promising and to give potential research directions.
A total dominating set in a digraph G is a subset W of its vertices such that every vertex of G h... more A total dominating set in a digraph G is a subset W of its vertices such that every vertex of G has an immediate successor in W . The total domination number of G is the size of the smallest total dominating set. We consider several lower bounds on the total domination number and conjecture that these bounds are strictly larger than g(G) -1, where g(G) is the number of vertices of the smallest directed cycle contained in G. We prove that these new conjectures are equivalent to the Caccetta-Häggkvist conjecture which asserts that g(G) -1 < n r in every digraph on n vertices with minimum outdegree at least r > 0.
We consider the problem of determining the size of a maximum clique in a graph, also known as the... more We consider the problem of determining the size of a maximum clique in a graph, also known as the clique number. Given any method that computes an upper bound on the clique number of a graph, we present a sequential elimination algorithm which is guaranteed to improve upon that upper bound. Computational experiments on DIMACS instances show that, on average, this algorithm can reduce the gap between the upper bound and the clique number by about 60%. We also show how to use this sequential elimination algorithm to improve the computation of lower bounds on the clique number of a graph.
Journal of Graph Algorithms and Applications, 2021
Let G = (V, E) be a graph and let S ⊆ V be a subset of its vertices. If the subgraph of G induced... more Let G = (V, E) be a graph and let S ⊆ V be a subset of its vertices. If the subgraph of G induced by V \ S is acyclic, then S is said to be a decycling set of G. The size of a smallest decycling set of G is called the decycling number of G. Determining the decycling number of a graph G is NP-hard, even if G is bipartite. We describe a tabu search procedure that generates decycling sets of small size for arbitrary bipartite graphs. Tests on challenging families of graphs show that the proposed algorithm improves many best-known solutions, thus closing or narrowing the gap to the best-known lower bounds.
We consider the problem of assigning clients to nurses for home care services. The aim is to bala... more We consider the problem of assigning clients to nurses for home care services. The aim is to balance the work load of the nurses while avoiding long travels to visit the clients. We analyze the case of the CSSS Côte-des-Neiges, Métro and Parc Extension for which a previous analysis has shown that demand fluctuations may create work overload for the nursing staff. We present two models, one with linear constraints and a quadratic objective function which we optimize using CPLEX, and a more complex model with non linear constraints that we optimize using a tabu search algorithm.
The following is a chapter of the book entitled Fourmis artificielles, des bases algorithmiques a... more The following is a chapter of the book entitled Fourmis artificielles, des bases algorithmiques aux concepts et réalisations avancés, Nicolas Monmarché, Frédéric Guinand and Patrick Siarry, editors, which will be published by Hermès Science Publishing Ltd in 2008. Les Cahiers du GERAD G–2008–29 1
A profit and a demand are associated with each arc of a set of profitable arcs of a given graph. ... more A profit and a demand are associated with each arc of a set of profitable arcs of a given graph. A travel time is associated with each arc of the graph. A fleet of capacitated vehicles is given to serve the profitable arcs. A maximum duration of the route of each vehicle is also given. The profit of an arc can be collected by one vehicle only that also serves the demand of the arc. The objective of this problem, that is called the Capacitated Arc Routing Problem with Profits (CARPP), is to find a set of routes that satisfy the constraints on the duration of the route and on the capacity of the vehicle and maximize the total collected profit. We propose a branch-and-price algorithm and several heuristics. We can solve exactly instances with up to 97 profitable arcs. The best heuristics find the optimal solution on most of instances where it is available.
Distance metric learning algorithms aim to appropriately measure similarities and distances betwe... more Distance metric learning algorithms aim to appropriately measure similarities and distances between data points. In the context of clustering, metric learning is typically applied with the assist of side-information provided by experts, most commonly expressed in the form of cannot-link and must-link constraints. In this setting, distance metric learning algorithms move closer pairs of data points involved in must-link constraints, while pairs of points involved in cannot-link constraints are moved away from each other. For these algorithms to be effective, it is important to use a distance metric that matches the expert knowledge, beliefs, and expectations, and the transformations made to stick to the side-information should preserve geometrical properties of the dataset. Also, it is interesting to filter the constraints provided by the experts to keep only the most useful and reject those that can harm the clustering process. To address these issues, we propose to exploit the dual...
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Papers by Alain Hertz