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A Metaheuristic Algorithm for Large Maximum Weight Independent Set Problems

Yuanyuan Dong, Andrew V. Goldberg, Alexander Noe, Nikos Parotsidis, Mauricio G. C. Resende, Quico Spaen

TL;DR

A new local search algorithm is developed, which is a metaheuristic in the greedy randomized adaptive search framework, which outperforms this openly available code on large vehicle routing instances and is compared with state‐of‐the‐art openly available code on public benchmark sets.

Abstract

Motivated by a real-world vehicle routing application, we consider the maximum-weight independent set problem: Given a node-weighted graph, find a set of independent (mutually nonadjacent) nodes whose node-weight sum is maximum. Some of the graphs airsing in this application are large, having hundreds of thousands of nodes and hundreds of millions of edges. To solve instances of this size, we develop a new local search algorithm, which is a metaheuristic in the greedy randomized adaptive search (GRASP) framework. This algorithm, which we call METAMIS, uses a wider range of simple local search operations than previously described in the literature. We introduce data structures that make these operations efficient. A new variant of path-relinking is introduced to escape local optima and so is a new alternating augmenting-path local search move that improves algorithm performance. We compare an implementation of our algorithm with a state-of-the-art openly available code on public benchmark sets, including some large instances with hundreds of millions of vertices. Our algorithm is, in general, competitive and outperforms this openly available code on large vehicle routing instances. We hope that our results will lead to even better MWIS algorithms.

A Metaheuristic Algorithm for Large Maximum Weight Independent Set Problems

TL;DR

A new local search algorithm is developed, which is a metaheuristic in the greedy randomized adaptive search framework, which outperforms this openly available code on large vehicle routing instances and is compared with state‐of‐the‐art openly available code on public benchmark sets.

Abstract

Motivated by a real-world vehicle routing application, we consider the maximum-weight independent set problem: Given a node-weighted graph, find a set of independent (mutually nonadjacent) nodes whose node-weight sum is maximum. Some of the graphs airsing in this application are large, having hundreds of thousands of nodes and hundreds of millions of edges. To solve instances of this size, we develop a new local search algorithm, which is a metaheuristic in the greedy randomized adaptive search (GRASP) framework. This algorithm, which we call METAMIS, uses a wider range of simple local search operations than previously described in the literature. We introduce data structures that make these operations efficient. A new variant of path-relinking is introduced to escape local optima and so is a new alternating augmenting-path local search move that improves algorithm performance. We compare an implementation of our algorithm with a state-of-the-art openly available code on public benchmark sets, including some large instances with hundreds of millions of vertices. Our algorithm is, in general, competitive and outperforms this openly available code on large vehicle routing instances. We hope that our results will lead to even better MWIS algorithms.
Paper Structure (20 sections, 5 equations, 7 figures, 9 tables, 2 algorithms)

This paper contains 20 sections, 5 equations, 7 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. ILP Formulation
  3. High Level Description
  4. Greedy Algorithm
  5. Local Search
  6. Using the Relaxed LP Solution
  7. Adaptive Path Relinking
  8. Data Structures and Optimizations
  9. Input Graph
  10. Interstate Graph
  11. Efficient Implementation of $(x, y)$ Moves
  12. AAP Moves
  13. Maintaining the Interstate Graph
  14. Experimental results
  15. Algorithms and Computational Environment
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Time-quality plot for the standard CR-S-L-4 instance with $95\%$ confidence intervals, initial solution and LP bound. Note that the plots for METAMIS+Init and METAMIS+Init+LP are very close
  • Figure 2: Time-quality plot for as-Skitter with $95\%$ confidence intervals
  • Figure 3: Time-quality plot for web-BerkSt with $95\%$ confidence intervals
  • Figure 4: Time-quality plot for roadNet-TX with $95\%$ confidence intervals
  • Figure 5: Time-quality plot for greenland_AM3 with $95\%$ confidence intervals
  • ...and 2 more figures