We consider the problem of spanning the nodes of a given colored graph G=(N,A) by a set of node-disjoint cycles at minimum reload cost, where a non-negative reload cost is paid whenever passing through a node where the two consecutive arcs have different colors. We call this problem Minimum Reload Cost Cycle Cover (MinRC3 for short). We prove that it is strongly NP-hard and not approximable within 1ϵ for any ϵ>0 even when the number of colors is 2, the reload costs are symmetric and satisfy the triangle inequality. Some IP models for MinRC3 are then presented, one well suited for a Column Generation approach. The corresponding pricing subproblem is also proved strongly NP-hard. Primal bounds for MinRC3 are obtained via local search based heuristics exploiting 2-opt and 3-opt neighborhoods. Computational results are presented comparing lower and upper bounds obtained by the above mentioned approaches.

Let M = (E, F) be a matroid on a set E, B one of its bases and M-B the base matroid associated to B. In this paper we determine a characterization of simple binary matroids M which are not isomorphic to M-B, for every base B of M. We also extend to matroids some graph notions.

We consider the problem of finding k-bipartite subgraphs, called clusters, in a bipartite graph G=(S,T,E), such that each vertex i of S appears in exactly one of the subgraphs, every vertex j of T appears in each cluster in which at least one of its neighbors appears, and the total number of edges needed to complete each cluster (i.e. to become a biclique) is minimized. This problem has been shown to be strongly NP-hard and is known as the k-clustering minimum biclique completion problem. It has been applied to bundling channels for multicast transmissions. The application consists of finding k-multicast sessions that partition the set of demands, given a set of demands of services from clients. Each service should belong to a single multicast session, while each client can appear in more than one session. We extend previous work by developing a branch-and-price algorithm that embeds a new metaheuristic based on variable neighborhood infeasible search and a branching rule that exploits the problem structure. The metaheuristic can also efficiently solve the pricing subproblem. In addition to the random instances used in the literature, we present structured instances generated using the MovieLens dataset collected by the GroupLens Research Project. Extensive computational results show that our branch-and-price algorithm outperforms the approaches proposed in the literature.

Equations are everywhere! Between the atmosphere and the wing of a spaceship, in the blood flowing in an artery, on the demarcation line between ice and water at the poles, in the motion of the tides, in the charge density of a semiconductor, in the compression algorithms of a signal sending images from outer space. Such equations represent real problems. The Mathematical Engineer can see and understand the nature of these equations, and can develop models in order to understand their relevant qualities and solve real problems. This PhD program aims at training young researchers by providing them with a strong mathematical background and with ability to apply their knowledge to the solution of real-world problems that arise in various areas of science, technology, industry, finance, management, whenever advanced methods are required in analysis, design, planning, decision and control activities. PhD students carry our their research both in the development of new mathematical methods and in the implementation and improvement of advanced techniques in connection with specific contexts and applications. The Faculty of the PhD program is responsible for the organization of the training and research activities of the PhD students. Decisions of the Faculty comply with the requirements and standards of the Doctoral School of the Politecnico di Milano. A Chairman is elected within the Faculty, for representative and coordination activities. Admission of students to the PhD program is decided after examination of the candidates. Students applying to our program must provide their CV, along with reference and motivation letters. After admission, each student is assigned a tutor. The tutor is a member of the Faculty who assists the student in the early stages of his career, especially in the choice of the courses and in identifying a thesis advisor. The overall activity of the PhD students can be quantified in 180 credits. The PhD program has a duration of three years. Activity can be classified

Health care delivery has a dramatic impact on the quality of life of every community. Hence, in the past years in many countries, a great effort has been spent in order to rationalize the management of limited resources, such as operating rooms, ambulances or workforce. Nevertheless, to the best of our knowledge, the idea of using demand for driving the workforce optimization process has received little attention in the literature. In this work we address the problem of managing workforce taking into account forecasted demand. We focus on the management of the operators working at the Operations Center of the Emergency Medical Service of Milano, Italy, for which a huge amount of accurate data is available.

We are given a digraph G = (N,A), where each arc is colored with one among k given colors. We look for a spanning arborescence T of G rooted (wlog) at node 1 and having minimum changeover cost. We call this the Minimum Changeover Cost Arborescence problem. To the authors' knowledge, it is a new problem. The concept of changeover costs is similar to the one, already considered in the literature, of reload costs, but the latter depend also on the amount of commodity flowing in the arcs and through the nodes, whereas this is not the case for the changeover costs. Here, given any node j ≠ 1, if a is the color of the single arc entering node j in arborescence T, and b is the color of an arc (if any) leaving node j, then these two arcs contribute to the total changeover cost of T by the quantity d ab , an entry of a k-dimensional square matrix D. We first prove that our problem is NPO-complete and very hard to approximate. Then we present Integer Programming formulations together with a combinatorial lower bound, a greedy heuristic and an exact solution approach. Finally, we report extensive computational results and exhibit a set of challenging instances.

The concept of reload cost, that is of a cost incurred when two consecutive arcs along a path are of different types, naturally arises in a variety of applications related to transportation, telecommunication, and energy networks. Previous work on reload costs is devoted to the problem of finding a spanning tree of minimum reload cost diameter (with no arc costs) or of minimum reload cost. In this article, we investigate the complexity and approximability of the problems of finding optimum paths, tours, and flows under a general cost model including reload costs as well as regular arc costs. Some of these problems, such as shortest paths and minimum cost flows, turn out to be polynomially solvable while others, such as minimum shortest path tree and minimum unsplittable multicommodity flows, are NP -hard to approximate within any polynomial-time computable function. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(3), 254–260 2011 © 2011 Wiley Periodicals, Inc.

In the Maximum Cut with Limited Unbalance problem, we want to partition the vertices of a weighted graph into two sets of sizes differing at most by a given threshold B , so that the sum of the weights of the crossing edges is maximum. This problem has been introduced in (Galbiati and Maffioli, Theor Comput Sci 385 (2007), 78–87) where polynomial time randomized approximation algorithms are proposed and their performance guarantees are analyzed in the case of non-negative integer weights. In this article, we present extensive computational experience with these algorithms on a large number of different graphs. We then extend the analysis of these algorithms to integer weights not restricted in sign, and continue the computational testing. It turns out that the approximation ratios obtained are always substantially better than those guaranteed by the theoretical analysis. © 2010 Wiley Periodicals, Inc. NETWORKS 2010

Given an undirected, connected network G = ( V , E ) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as another graph theoretic problem closely related to it, namely, the cycle basis problem. We consider two versions of the problem: the unconstrained and the fundamental cut basis problem. For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem, and is thus solvable in strongly polynomial time. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each from a spanning tree T , is shown to be NP-hard. In this proof, we also show that a tree which induces the minimum fundamental cycle basis is also an optimal solution for the minimum fundamental cut basis problem in unweighted graphs. We present heuristics, integer programming formulations and summarize first experiences with numerical tests.

Let $M = (E, \mathcal{F})$ be a matroid on a set $E$ and $B$ one of its bases. A closed set $\theta \subseteq E$ is saturated with respect to $B$ when $|\theta \cap B | \leq r(\theta)$, where $r(\theta)$ is the rank of $\theta$. The collection of subsets $I$ of $E$ such that $| I \cap \theta| \leq r(\theta)$ for every closed saturated set $\theta$ turns out to be the family of independent sets of a new matroid on $E$, called base-matroid and denoted by $M_B$. In this paper we prove that a graphic matroid $M$, isomorphic to a cycle matroid $M(G)$, is isomorphic to $M_B$, for every base $B$ of $M$, if and only if $M$ is direct sum of uniform graphic matroids or, in equivalent way, if and only if $G$ is disjoint union of cacti. Moreover we characterize simple binary matroids $M$ isomorphic to $M_B$, with respect to an assigned base $B$.

We consider the problem of finding a fundamental cycle basis with minimum total cost in an undirected graph. This problem is NP-hard and has several interesting applications. Since fundamental cycle bases correspond to spanning trees, we propose a local search algorithm, a tabu search and variable neighborhood search in which edge swaps are iteratively applied to a current spanning tree. We also present a mixed integer programming formulation of the problem whose linear relaxation yields tighter lower bounds than other known formulations. Computational results obtained with our algorithms are compared with those from the best available constructive heuristic on several types of graphs.

We consider the problem of partitioning the vertices of a weighted graph into two sets of sizes that differ at most by a given threshold B , so as to maximize the weight of the crossing edges. For B equal to 0 this problem is known as Max Bisection, whereas for B equal to the number n of nodes it is the maximum cut problem. We present polynomial time randomized approximation algorithms with non trivial performance guarantees for its solution. The approximation results are obtained by extending the methodology used by Y. Ye for Max Bisection and by combining this technique with another one that uses the algorithm of Goemans and Williamson for the maximum cut problem. When B is equal to zero the approximation ratio achieved coincides with the one obtained by Y. Ye; otherwise it is always above this value and tends to the value obtained by Goemans and Williamson as B approaches the number n of nodes.

We present polynomial time randomized approximation algorithms with non trivial performance guarantees for the problem of partitioning the vertices of a weighted graph into two sets of sizes that differ at most by a given threshold B, so as to maximize the weight of the crossing edges. For B equal to 0 this problem is known as Max Bisection, whereas for B equal to the number n of nodes it is the Maximum Cut problem. The approximation results are obtained by extending the methodology used by Y. Ye for Max Bisection and by combining this technique with another one that uses the algorithm of Goemans and Williamson for the Maximum Cut problem. When B is equal to zero the approximation ratio achieved coincides with the one obtained by Y. Ye; otherwise it is always above this value and tends to the value obtained by Goemans and Williamson as B approaches the number n of nodes.

Accreditation of the degree programmes in Engineering is surely an argument which stimulates a great interest not only in the Italian level but ab ove all in the European perspective. It appears strategic that Europe is equipped with a system which permits to compare the degree programmes in Engineering offered by various universities in Europe also in view of a major competition in the area of higher education in the European Union, in comparison with third countries. This appears the principal basis of different actio ns financed by the European Commission, which have among their own objectives also the study of an accreditation system of the degree programmes in Engineering in Europe. In this article 3 SOCRATES Thematic Networks are presented, which, one after the other, starting from 1998, have been operational in the European panorama. Among their objectives there is a recurrent motive: accreditation of the degree programmes in Engineering in Europe.

Consider a matroid M = ( E , B ) , where B denotes the family of bases of M , and assign a color c ( e ) to every element e ∈ E (the same color can go to more than one element). The palette of a subset F of E , denoted by c ( F ) , is the image of F under c . Assume also that colors have prices (in the form of a function π ( ℓ ) , where ℓ is the label of a color), and define the chromatic price as: π ( F ) = ∑ ℓ ∈ c ( F ) π ( ℓ ) . We consider the following problem: find a base B ∈ B such that π ( B ) is minimum. We show that the greedy algorithm delivers a ln r ( M ) -approximation of the unknown optimal value, where r ( M ) is the rank of matroid M . By means of a reduction from S ET C OVER , we prove that the ln r ( M ) ratio cannot be further improved, even in the special case of partition matroids, unless NP ⊆ DTIME ( n log log n ) . The results apply to the special case where M is a graphic matroid and where the prices π ( ℓ ) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the ln ( n - 1 ) + 1 ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function π ( F ) , where F is a common independent set on matroids M 1 , … , M k and, more generally, to independent systems characterized by the k -for-1 property.

Let M = ( E , F) be a rank- n matroid on a set E and B one of its bases. A closed set θ ⊆ E is saturated with respect to B , or B -saturated, when |θ∩ B | = r (θ), where r (θ) is the rank of θ.The collection of subsets I of E such that | I ∩θ| ≤ r (θ), for every closed B -saturated set θ, turns out to be the family of independent sets of a new matroid on E , called base-matroid and denoted by M B . In this paper we prove some properties of M B , in particular that it satisfies the base-axiom of a matroid.Moreover, we determine a characterization of the matroids M which are isomorphic to M B for every base B of M .Finally, we prove that the poset of the closed B -saturated sets ordered by inclusion is isomorphic to the Boolean lattice B n .

In this paper, we consider combinatorial optimization problems with additional cardinality constraints. In k-cardinality combinatorial optimization problems, a cardinality constraint requires feasible solutions to contain exactly k elements of a finite set E. Problems of this type have applications in many areas, e.g. in the mining and oil industry, telecommunications, circuit layout, and location planning. We formally define the problem, mention some examples and summarize general results. We provide an annotated bibliography of combinatorial optimization problems of which versions with cardinality constraint have been considered in the literature.

The present work tackles a recent problem in the class of cardinality constrained combinatorial optimization problems for the planar graph case: the minimum k -cardinality cut problem. Given an undirected edge-weighted connected graph the min k -cardinality cut problem consists in finding a partition of the vertex set V in two sets V 1 , V 2 such that the number of the edges between V 1 and V 2 is exactly k and the sum of the weights of these edges is minimal. Although for general graphs the problem is already strongly 𝒩 𝒫 -hard, we have found a pseudopolynomial algorithm for the planar graph case. This algorithm is based on the fact that the min k -cardinality cut problem in the original graph is equivalent to a bi-weighted exact perfect matching problem in a suitable transformation of the geometric dual graph. Because the Lagrangian relaxation of cardinality constraint yields a max cut problem and max cut is polynomially solvable in planar graphs, we also develop a Lagrangian heuristic for the min k -cardinality cut in planar graphs. We compare the performance of this heuristic with the performance of a more general heuristic based on a Semidefinite Programming relaxation and on the Goemans and Williamson's random hyperplane technique. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(4), 195–208 2006