Solving the maximum edge disjoint path problem using a modified Lagrangian particle swarm optimisation hybrid

Highlights

• Lagrangian Relaxation can be used effectively to solve the Maximum Edge Disjoint Path problem.

• Hybridising Lagrangian Relaxation with Particle Swarm Optimisation has significantly improved performance.

• The new method generates both high quality primal and dual solutions.

• A new synthetic dataset has been created for future researchers.

Abstract

This paper presents a new method to solve the Maximum Edge Disjoint Paths (MEDP) problem. Given a set of node pairs within a network, the MEDP problem is the task of finding the largest number of pairs that can be connected by paths, using each edge within the network at most once. We present a heuristic algorithm that builds a hybridisation of Lagrangian Relaxation and Particle Swarm Optimisation, referred to as LaPSO. This hybridisation is combined with a new repair heuristic, called Largest Violation Perturbation (LVP). We show that our LaPSO method produces better heuristic solutions than both current state-of-the-art heuristic methods, as well as the primal solution found by a standard Mixed Integer Programming (MIP) solver within a limited runtime. Significantly, when run with a limited runtime, our LaPSO method also produces strong bounds which are superior to a standard MIP solver for the larger instances tested, whilst being competitive for the remainder. This allows our LaPSO method to prove optimality for many instances and provide optimality gaps for the remainder, making it a “quasi-exact” method. In this way our LaPSO algorithm, which draws on ideas from both mathematical programming and evolutionary algorithms, is able to outperform both MIP and metaheuristic solvers that only use ideas from one of these areas.

Introduction

In communication networks there often exist multiple information requests that need to be facilitated simultaneously across the network. In many applications it is important to ensure that no two connection paths within the network interfere with one another, resulting in multiple disjoint paths. The task of connecting the largest number of requesting and transmitting node pairs, often termed commodities, without any two paths sharing any edges defines the Maximum Edge Disjoint Path (MEDP) problem. The MEDP problem can be found in a wide array of applications. In Optical Networks, different paths using the same wavelength are not allowed to share physical links (Raghavan & Upfal, 1994). In Very Large Scale Integration (VLSI) design, it is a requirement that wire paths do not interfere with each other (Chan, Chin, Ting, 2003, Gerez, 1998). Edge Disjoint Paths also play an important role in Ad Hoc Networks, to reduce energy consumption (Sumpter, Burson, Tang, & Chen, 2013) and reduce signal dropout rates (Jain, Das, 2005, Li, Cuthbert, 2004). In Virtual Circuit Routing and Admission control, disjoint-paths are often required when there is only limited bandwidth available (Awerbuch, Gawlick, Leighton, & Rabani, 1994)

As these problem types tend to be quite large, greedy based heuristics have traditionally been used to solve the MEDP problem. Examples of such algorithms include the Simple Greedy Algorithm (SGA) (Kleinberg, 1996), the bounded-length greedy algorithm (Kleinberg, 1996), and the greedy path algorithm (Kolliopoulos & Stein, 2004). In 2007 Blesa and Blum developed the first metaheuristic to solve the MEDP problem using an Ant Colony Optimisation (ACO) algorithm (Blesa & Blum, 2007). This ACO algorithm is still used as a benchmark for the MEDP problem. Since 2012, a variety of new methods have been developed, including a Constraint based Local Search (Pham, Deville, & Van Hentenryck, 2012), a Genetic Algorithm (Hsu & Cho, 2015), a Message Passing algorithm (Altarelli, Braunstein, Dall’Asta, De Bacco, & Franz, 2015) and a Two-Stage Hybrid Metaheuristic (Martín, Sánchez, Beltran-Royo, & Duarte, 2020). A limitation these existing techniques face is the inability to provide any performance guarantees in the context of optimality gaps. In this paper we attempt to outperform these previous methodologies in both primal solution qualities of the non-trivial, large problem instances, as well as being the first paper to provide good Upper Bounds for the instances tested. To accomplish both goals, we use a modified Lagrangian Heuristic and Particle Swarm Optimisation Hybrid (LaPSO) algorithm, originally developed by (Ernst, 2010, Ernst, Singh, 2012). Whilst this hybrid method has worked well on a limited number of problems, this paper provides a more extensive application test, carrying out further examination of the inner workings behind the LaPSO algorithm.

The motivation for using a Lagrangian Heuristic based technique arises after modelling the MEDP problem as an Integer Program (IP). By relaxing the edge capacity constraints for the MEDP problem, the problem can be decomposed into a series of independent shortest path problems, which are easily solvable using a well known shortest path algorithm such as Dijkstra’s method (Dijkstra, 1959). This relaxation approach, termed Lagrangian Relaxation (LR) (Fisher, 1981), has to the best of our knowledge not been attempted previously to solve the MEDP problem. Solving the relaxed problem provides us with Upper Bounds for the optimal solutions to the original MEDP problem, something traditional heuristic based techniques are unable to provide. Whilst a traditional Mixed Integer Programming (MIP) solver could be used to provide bounds for small scale problems, these solvers are unable to do so in a reasonable amount of time as problem sizes increase. Solutions to the Lagrangian Relaxed problem are often almost feasible, and are suitable candidates for a good repair heuristic. Another unique contribution of this paper is the creation of a novel repair heuristic termed Largest Violation Perturbation (LVP) for the MEDP problem. The LVP repair heuristic uses perturbation variables which are introduced in the LaPSO algorithm to increase exploitation of promising solutions, something which traditional LR techniques lack.

In addition to the creation of a new repair heuristic, we provide further analysis surrounding the effect that the perturbation variables have on both the primal solution and bound qualities, something which has not been done previously (Ernst, 2010, Ernst, Singh, 2012). We also provide further tests surrounding the particle swarm, showing the effect a population based metaheuristic can have when hybridised with a traditional exact technique. The adapted LaPSO method was found to produce significantly better results, becoming the new state-of-the-art for many of the problem instances tested, particularly on the larger problem instances. We also show that LaPSO can generate tighter bounds than a traditional MIP solver and the LR based Volume algorithm (Barahona & Anbil, 2000), especially for the largest instances tested when given a limited CPU runtime. This demonstrates that our hybrid method outperforms both base optimisation approaches (metaheuristics and Lagrangian Relaxation) from which it draws inspiration.

The rest of this paper is structured as follows. Section 2 introduces the background and related work. Our main approach is described in Section 3. Section 4 details the experimental design and presents the results. Section 5 then concludes the paper.