Exact solution approaches for the discrete $α$-neighbor $p$-center problem

TL;DR

Two integer programming formulations for the discrete α$$ \alpha $$ ‐ p$$ p $$ CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures are presented.

Abstract

The discrete $α$-neighbor $p$-center problem (d-$α$-$p$CP) is an emerging variant of the classical $p$-center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate $p$ facilities on these points in such a way that the maximum distance between each point where no facility is located and its $α$-closest facility is minimized. The only existing algorithms in literature for solving the d-$α$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$α$-$p$CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&C algorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances.

Figures (4)

• Figure 1: Illustration of Example \ref{['ex:DaskinBasic']}, in which$p=3$ and $\alpha=2$. The value in the nodes in Figure \ref{['ex:daskinLRxa']} is the index of the node and the values near the arcs are the distances. The values in the nodes in Figures \ref{['ex:daskinLRxb']} and \ref{['ex:daskinLRxc']} are the values of the $y$-variables in the optimal solution, and the values near the arcs are the values of the $x$-variables in the optimal solution. If an arc is not drawn in a solution, this means the corresponding $x$-variable takes value zero.
• Figure 2: Illustration of Example \ref{['ex:daskinLRbetter']}, in which$p=2$ and $\alpha=2$. The value in the nodes in Figure \ref{['ex:daskinLRbetter1']} is the index of the node and the values near the arcs are the distances. The values in the nodes in Figures \ref{['ex:daskinLRbetter2']}, \ref{['ex:daskinLRbetter3']} and \ref{['ex:daskinLRbetter4']} are the values of the $y$-variables in the optimal solution, and the values near the arcs in Figure \ref{['ex:daskinLRbetter4']} are the values of the $x$-variables in the optimal solution.
• Figure 3: Illustration of Example \ref{['ex:elloumiLRbetter']}, in which$p=2$ and $\alpha=2$. The value in the nodes in Figure \ref{['ex:elloumiLRbetter1']} is the index of the node and the values near the arcs are the distances. The values in the nodes in Figures \ref{['ex:elloumiLRbetter2']}, \ref{['ex:elloumiLRbetter3']} and \ref{['ex:elloumiLRbetter4']} are the values of the $y$-variables in the optimal solution, and the values near the arcs in Figure \ref{['ex:elloumiLRbetter2']} and \ref{['ex:elloumiLRbetter3']} are the values of the $x$-variables in the optimal solution.
• Figure 4: Runtime for different settings of our B&C algorithms on a subset of the instances.

Theorems & Definitions (37)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Example 5
• Theorem 6