55 research outputs found
A multifacility location problem on median spaces
AbstractThis paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum-cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with ¦V¦points is O(¦V¦3 + ¦V¦ψ(n)), where ψ(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N + k(logN + logk + ψ(n))) time and O(N + kψ(n)) time in the case of the vertex-restricted multifacility location problem
Hard cases of the multifacility location problem
AbstractLet μ be a rational-valued metric on a finite set T. We consider (a version of) the multifacility location problem: given a finite set V⊇T and a function c:V2→Z+, attach each element x∈V−T to an element γ(x)∈T minimizing ∑c(xy)μ(γ(x)γ(y)):xy∈V2, letting γ(t)≔t for each t∈T. Large classes of metrics μ have been known for which the problem is solvable in polynomial time. On the other hand, Dalhaus et al. (SIAM J. Comput. 23 (4) (1994) 864) showed that if T={t1,t2,t3} and μ(titj)=1 for all i≠j, then the problem (turning into the minimum 3-terminal cut problem) becomes strongly NP-hard. Extending that result and its generalization in (European J. Combin. 19 (1998) 71), we prove that for μ fixed, the problem is strongly NP-hard if the metric μ is nonmodular or if the underlying graph of μ is nonorientable (in a certain sense)
A generalized model of equality measures in network location problems
In this paper, the concept of the ordered weighted averaging operator is applied to define a model which unifies and generalizes several inequality measures. For a location x, the value of the new objective function is the ordered weighted average of the absolute deviations from the average distance from the facilities to the location x. Several kinds of networks are studied: cyclic, tree and path networks and, for each of them, the properties of the objective function are analyzed in order to identify a finite dominating set for optimal locations. Polynomial-time algorithms are proposed for these problems, and the corresponding complexity is discussed.Future and Emerging Technologies Unit (European Commission)Ministerio de Educación y Cienci
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
Minimum 0-Extension Problems on Directed Metrics
For a metric on a finite set , the minimum 0-extension problem
0-Ext is defined as follows: Given and , minimize
subject to , where the
sum is taken over all unordered pairs in . This problem generalizes several
classical combinatorial optimization problems such as the minimum cut problem
or the multiterminal cut problem. The complexity dichotomy of 0-Ext was
established by Karzanov and Hirai, which is viewed as a manifestation of the
dichotomy theorem for finite-valued CSPs due to Thapper and \v{Z}ivn\'{y}.
In this paper, we consider a directed version -Ext
of the minimum 0-extension problem, where and are not assumed to be
symmetric. We extend the NP-hardness condition of 0-Ext to
-Ext: If cannot be represented as the shortest
path metric of an orientable modular graph with an orbit-invariant ``directed''
edge-length, then -Ext is NP-hard. We also show a
partial converse: If is a directed metric of a modular lattice with an
orbit-invariant directed edge-length, then -Ext is
tractable. We further provide a new NP-hardness condition characteristic of
-Ext, and establish a dichotomy for the case where
is a directed metric of a star
Polynomially solvable cases of multifacility distance constraints on cyclic networks
Ankara : The Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent Univ., 1993.Thesis (Master's) -- Bilkent University, 1993.Includes bibliographical references leaves 79-81Distance Constraints Problem is to locate one or more new facilities on a
network so that the distances between new and existing facilities as well as
between pairs of new facilities do not exceed given upper bounds. The problem
is AfV-Complete on cyclic networks and polynomially solvable on trees.
Although theory for tree networks is well-developed, there is virtually no theory
for cyclic networks. In this thesis, we identify a special class of instances
for which we develop theory and algorithms that are applicable to any metric
space defining the location space. We require that the interaction between
new facilities has a tree structure. The method is based on successive applications
of EXPANSION and INTERSECTION operations defined on subsets
of the location space. Application of this method to general networks yields
strongly polynomial algorithms. Finally, we give an algorithm that constructs
an e-optimal solution to a related minimax problem.Yeşilkökçen, Naile GülcanM.S
On the multisource hyperplanes location problem to fitting set of points
In this paper we study the problem of locating a given number of hyperplanes
minimizing an objective function of the closest distances from a set of points.
We propose a general framework for the problem in which norm-based distances
between points and hyperplanes are aggregated by means of ordered median
functions. A compact Mixed Integer Linear (or Non Linear) programming
formulation is presented for the problem and also an extended set partitioning
formulation with an exponential number of variables is derived. We develop a
column generation procedure embedded within a branch-and-price algorithm for
solving the problem by adequately performing its preprocessing, pricing and
branching. We also analyze geometrically the optimal solutions of the problem,
deriving properties which are exploited to generate initial solutions for the
proposed algorithms. Finally, the results of an extensive computational
experience are reported. The issue of scalability is also addressed showing
theoretical upper bounds on the errors assumed by replacing the original
datasets by aggregated versions.Comment: 30 pages, 5 Tables, 3 Figure
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