The K-median Problem on a Tree
نویسندگان
چکیده
The polyhedral structure of the K-median problem on a tree is examined. Even for very small connected graphs, we show that additional constraints are needed to describe the integer polytope. We completely characterize those trees for which an optimal integer LP solution is guaranteed to exist. Also, a new and simpler proof of the worst case bound of an LP solution is provided. We find a new class of constraints that restrict the feasible region and a new class of valid inequalities. The valid inequalities are lifted to find facets for the K-median problem on a general graph. Also, we provide polyhedral descriptions for several types of trees. As part of this work, we summarize most known results for the K-median problem on a tree.
منابع مشابه
An efficient algorithm for finding the semi-obnoxious $(k,l)$-core of a tree
In this paper we study finding the $(k,l)$-core problem on a tree which the vertices have positive or negative weights. Let $T=(V,E)$ be a tree. The $(k,l)$-core of $T$ is a subtree with at most $k$ leaves and with a diameter of at most $l$ which the sum of the weighted distances from all vertices to this subtree is minimized. We show that, when the sum of the weights of vertices is negative, t...
متن کاملPolyhedral Properties of the K -median Problem on a Tree
The polyhedral structure of the K-median problem on a tree is examined. Even for very small connected graphs, we show that additional constraints are needed to describe the integer polytope. A complete description is given of those trees for which an optimal integer LP solution is guaranteed to exist. We present a new and simpler demonstration that an LP characterization of the 2-median problem...
متن کاملSOLVING A STEP FIXED CHARGE TRANSPORTATION PROBLEM BY A SPANNING TREE-BASED MEMETIC ALGORITHM
In this paper, we consider the step fixed-charge transportation problem (FCTP) in which a step fixed cost, sometimes called a setup cost, is incurred if another related variable assumes a nonzero value. In order to solve the problem, two metaheuristic, a spanning tree-based genetic algorithm (GA) and a spanning tree-based memetic algorithm (MA), are developed for this NP-hard problem. For compa...
متن کاملA Novel Hybrid Modified Binary Particle Swarm Optimization Algorithm for the Uncertain p-Median Location Problem
Here, we investigate the classical p-median location problem on a network in which the vertex weights and the distances between vertices are uncertain. We propose a programming model for the uncertain p-median location problem with tail value at risk objective. Then, we show that it is NP-hard. Therefore, a novel hybrid modified binary particle swarm optimization algorithm is presented to obtai...
متن کاملA Metaheuristic Algorithm for the Minimum Routing Cost Spanning Tree Problem
The routing cost of a spanning tree in a weighted and connected graph is defined as the total length of paths between all pairs of vertices. The objective of the minimum routing cost spanning tree problem is to find a spanning tree such that its routing cost is minimum. This is an NP-Hard problem that we present a GRASP with path-relinking metaheuristic algorithm for it. GRASP is a multi-start ...
متن کامل