[Paper Review] Fixed-Parameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs

Gregory Gutin, Daniel Karapetyan+1 more|arXiv (Cornell University)|Jun 7, 2009

Advanced Graph Theory Research12 references3 citations

TL;DR

This paper evaluates the SGA heuristic for extracting maximum-size reflected networks in linear programs using fixed-parameter algorithms to verify solution quality. By solving most test instances to optimality, it confirms SGA produces high-quality solutions and identifies depth-first search (DFS) for spanning tree construction as the key factor in SGA's success, leading to a superior SGA80(DFS) variant that achieves optimality on 50 out of 54 solvable instances.

ABSTRACT

We consider the problem of extracting a maximum-size reflected network in a linear program. This problem has been studied before and a state-of-the-art SGA heuristic with two variations have been proposed. In this paper we apply a new approach to evaluate the quality of SGA\@. In particular, we solve majority of the instances in the testbed to optimality using a new fixed-parameter algorithm, i.e., an algorithm whose runtime is polynomial in the input size but exponential in terms of an additional parameter associated with the given problem. This analysis allows us to conclude that the the existing SGA heuristic, in fact, produces solutions of a very high quality and often reaches the optimal objective values. However, SGA contain two components which leave some space for improvement: building of a spanning tree and searching for an independent set in a graph. In the hope of obtaining even better heuristic, we tried to replace both of these components with some equivalent algorithms. We tried to use a fixed-parameter algorithm instead of a greedy one for searching of an independent set. But even the exact solution of this subproblem improved the whole heuristic insignificantly. Hence, the crucial part of SGA is building of a spanning tree. We tried three different algorithms, and it appears that the Depth-First search is clearly superior to the other ones in building of the spanning tree for SGA. Thereby, by application of fixed-parameter algorithms, we managed to check that the existing SGA heuristic is of a high quality and selected the component which required an improvement. This allowed us to intensify the research in a proper direction which yielded a superior variation of SGA.

Motivation & Objective

• To assess the quality of the state-of-the-art SGA heuristic for extracting maximum-size reflected networks in linear programs.
• To determine whether the SGA heuristic consistently finds optimal or near-optimal solutions.
• To identify which components of SGA are most responsible for solution quality and thus prime for improvement.
• To evaluate the impact of replacing heuristic components with exact fixed-parameter algorithms.
• To develop a superior SGA variant by focusing on the most critical component—spanning tree construction.

Proposed method

• Applied a fixed-parameter algorithm to solve the maximum embedded reflected network (DMERN) problem to optimality for 54 out of 93 test instances.
• Used the fixed-parameter algorithm to evaluate the SGA heuristic by comparing its solutions to the optimal ones.
• Replaced the greedy independent set algorithm in SGA with a fixed-parameter vertex cover-based exact algorithm.
• Replaced the spanning tree construction method in SGA with three alternatives: depth-first search (DFS), breadth-first search (BFS), and random search (RS).
• Evaluated the performance of modified SGA variants (SGA(DFS), SGA3(DFS), SGA80(DFS), SGA(DFS)+VC) on the testbed.
• Used the complementarity between independent sets and vertex covers to implement an exact algorithm for the independent set subproblem.

Experimental results

Research questions

• RQ1 Does the SGA heuristic produce solutions close to optimal for the maximum reflected network extraction problem?
• RQ2 Which component of the SGA heuristic—spanning tree construction or independent set search—is most responsible for solution quality?
• RQ3 Can replacing the greedy independent set algorithm with an exact fixed-parameter algorithm significantly improve SGA's performance?
• RQ4 Does using depth-first search for spanning tree construction outperform breadth-first or random search in SGA?
• RQ5 Can a fixed-parameter algorithm be effectively used to evaluate and guide the improvement of heuristic methods in combinatorial optimization?

Key findings

• The fixed-parameter algorithm solved 54 out of 93 test instances to optimality, with 52 of them completed within one minute.
• SGA(DFS) achieved the optimal solution in 36 out of the 54 solvable instances, SGA3(DFS) in 44, and SGA80(DFS) in 50, demonstrating the superiority of DFS.
• Replacing the greedy independent set algorithm with an exact fixed-parameter algorithm (via vertex cover) improved the solution in only four instances and was not practical due to high runtime.
• SGA(DFS)+VC failed to terminate within 30 minutes for six instances, while SGA80(DFS) remained efficient and accurate.
• The SGA80(DFS) variant achieved optimal solutions on 50 out of 54 feasible instances, significantly outperforming other variants and approaching the performance of the exact algorithm in much less time.
• The study confirms that the spanning tree construction method is the dominant factor in SGA’s performance, and that DFS is the most effective choice among the tested methods.