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)

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

Combinatorial Optimization

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry

A Characterization of Minimizable Metrics in the Multifacility Location Problem

In the minimum 0-extension problem (a version of the multifacility location problem), one is given a metric m on a subset X of a finite set V and a nonnegative function c on the unordered pairs of elements of V . The objective is to find a semimetric m 0 on V that minimizes the inner product c \Delta m 0 , provided that m 0 coincides with m within X and each element of V is at zero distance from X . For m fixed, this problem is solvable in strongly polynomial time if m is minimizable, which means that for any superset V and function c, the minimum objective value is equal to that in the corresponding linear relaxation. In [9] A. Karzanov showed that the path metric of a graph H is minimizable if and only if all isometric cycles of H have length four and the edges of H can be oriented so that non-adjacent edges in each 4--cycle have opposite orientations along the cycle (such graphs are called frames in [9]). Extending this result to general metrics m, we show that m is minimiza..

A characterization of minimizable metrics in the multifacility location problem

In the minimum 0-extension problem (a version of the multifacility location problem), one is given a metric m on a subset X of a finite set V and a non-negative function c on the unordered pairs of elements of V. The objective is to find a semimetric m ′ on V that minimizes the inner product c · m ′, provided that m ′ coincides with m within X and each element of V is at zero distance from X. For m fixed, this problem is solvable in strongly polynomial time if m is minimizable, which means that for any superset V and function c, the minimum objective value is equal to that in the corresponding linear relaxation. In [9], Karzanov showed that the path metric of a graph H is minimizable if and only if all isometric cycles of H have length four and the edges of H can be oriented so that non-adjacent edges in each 4-cycle have opposite orientations along the cycle (such graphs are called frames in [9]). Extending this result to general metrics m, we show that m is minimizable if and only if m is modular and its underlying graph is a frame. c ○ 2000 Academic Pres