Isabella Lari

Finding the ℓ-core of a tree

An l-core of a tree T = (V,E) with |V|= n, is a path P with length at most l that is central with respect to the property of minimizing the sum of the distances from the vertices in P to all the vertices of T not in P. The distance between two vertices is the length of the shortest path joining them. In this paper we present efficient algorithms for finding the l-core of a tree. For unweighted trees we present an O(nl) time algorithm, while for weighted trees we give a procedure with time complexity of O(nlog2n). The algorithms use two different types of recursive principle in their operation.

Max‐min partitioning of grid graphs into connected components

The partitioning of a rectangular grid graph with weighted vertices into p connected components such that the component of smallest weight is as heavy as possible (the max-min problem) is considered. It is shown that the problem is NP-hard for rectangles with at least three rows. A shifting algorithm is given which approximates the optimal solution. Bounds for the relative error are determined under a posteriori hypotheses. A further shifting algorithm is also given which allows for error estimates under a priori hypotheses and for asymptotic error estimates. A similar approach can be taken with the problem of finding the partition whose largest component is as small as possible (the min-max problem). The case of rectangles with two rows has a polynomial algorithm and is dealt with in another paper.

Comparing different metaheuristic approaches for the median path problem with bounded length

In this paper we consider the Bounded Length Median Path Problem which can be defined as the problem of locating a path-shaped facility that departures from a given origin and arrives at a given destination in a network. The length of the path is assumed to be bounded by a given maximum length. At each vertex of the network (customer-point) the demand for the service is given and the cost to reach the closest service-point is computed. The objective is to minimize the sum of these costs over all the customer-points in the network. We consider two local search metaheuristics, the well known Tabu search and the Old bachelor acceptance and we propose a comparative analysis of the performances of the two procedures. We implement streamlined versions of Tabu search and Old bachelor acceptance. Our main concern is to provide knowledge about the intrinsic strategy adopted by each metaheuristic and to know whether one is more appropriate than the other for our problem.

The location of median paths on grid graphs

In this paper we consider the location of a path shaped facility on a grid graph. In the literature this problem was extensively studied on particular classes of graphs as trees or series-parallel graphs. We consider here the problem of finding a path which minimizes the sum of the (shortest) distances from it to the other vertices of the grid, where the path is also subject to an additional constraint that takes the form either of the length of the path or of the cardinality. We study the complexity of these problems and we find two polynomial time algorithms for two special cases, with time complexity of O(n) and O(nℓ) respectively, where n is the number of vertices of the grid and ℓ is the cardinality of the path to be located.The literature about locating dimensional facilities distinguishes between the location of extensive facilities in continuous spaces and network facility location. We will show that the problems presented here have a close connection with continuous dimensional facility problems, so that the procedures provided can also be useful for solving some open problems of dimensional facilities location in the continuous case.

Sharp bounds for the maximum of the chi-square index in a class of contingency tables with given marginals

Our research is motivated by the problem of finding a joint frequency distribution with given marginals which is ‘as far as possible’ from the independent distribution with the same marginals. If the distance between these two distributions is measured by Pearsons chi-square index, the problem can be formulated as maximizing a quadratic convex separable function subject to transportation constraints. We present three heuristics for this problem: two of them are greedy heuristics; the third one amounts to solve a linear relaxation of the problem, and yields also an upper bound of the optimum; thus one can estimate the relative error E of any given heuristic. The lower bounds given by the three heuristics may be improved via Frank-Wolfes algorithm. Numerical experiments on 600 randomly generated test problems with up to 50 rows and 100 columns show that the above heuristics provide sharp bounds on the optimum (often one has E < 0.01). Even more interestingly, these bounds become sharper and sharper as the problem size increases.

Efficient algorithms for finding the (k, l)‐core of tree networks

Given a tree T = (V, E), with |V| = n, we consider the problem of selecting a subtree with at most k leaves and with a diameter of at most l which minimizes the sum of the distances of the vertices from the selected subtree. We call such a subtree the (k, l)-core of T. We provide two algorithms; the first one for unweighted trees has time complexity of O(n2), whereas the second one for weighted trees has time complexity of O(n2log n). The idea for both the algorithms is that, by starting from the tree T, we construct new rooted trees where the maximum length of a path is at most l. Then, for each new tree, we can apply a greedy-type procedure to find a subtree containing the root with at most k leaves and which minimizes the sum of the distances.

Bidimensional allocation of seats via zero-one matrices with given line sums

In some proportional electoral systems with more than one constituency the number of seats allotted to each constituency is pre-specified, as well as, the number of seats that each party has to receive at a national level. “Bidimensional allocation” of seats to parties within constituencies consists of converting the vote matrix V into an integer matrix of seats “as proportional as possible” to V, satisfying constituency and party totals and an additional “zero-vote zero-seat” condition. In the current Italian electoral law this Bidimensional Allocation Problem (or Biproportional Apportionment Problem—BAP) is ruled by an erroneous procedure that may produce an infeasible allocation, actually one that is not able to satisfy all the above conditions simultaneously.In this paper we focus on the feasibility aspect of BAP and, basing on the theory of (0,1)-matrices with given line sums, we formulate it for the first time as a “Matrix Feasibility Problem”. Starting from some previous results provided by Gale and Ryser in the 60’s, we consider the additional constraint that some cells of the output matrix must be equal to zero and extend the results by Gale and Ryser to this case. For specific configurations of zeros in the vote matrix we show that a modified version of the Ryser procedure works well, and we also state necessary and sufficient conditions for the existence of a feasible solution. Since our analysis concerns only special cases, its application to the electoral problem is still limited. In spite of this, in the paper we provide new results in the area of combinatorial matrix theory for (0,1)-matrices with fixed zeros which have also a practical application in some problems related to graphs.

The Forest Wrapping Problem on Outerplanar Graphs

In this paper we study the Forest Wrapping Problem (FWP) which can be stated as follows: given a connected graph G = (V, E), with |V| = n, let ?0 be a partition of G into K (not necessarily connected) components, find a connected partition ?* of G that wraps ?0 and has maximum number of components.The Forest Wrapping problem is NP-complete on grid graphs while is solvable in O(n log n) time on ladder graphs. We provide a two-phase O(n2) time algorithm for solving FWP on outerplanar graphs.

Locating median paths on connected outerplanar graphs

In this article, we study the median path problem without length restrictions on the class of connected outerplanar graphs, assuming that weights equal to 1 are assigned to the edges of a graph G, and nonnegative weights are associated to its vertices. We provide an

The Cent-dian Path Problem on Tree Networks

O(kn)