Improved Local Search Algorithms with Multi-Cycle Reduction for Minimum Concave Cost Network Flow Problems

The minimum concave cost network flow problem (MCCNFP) has many applications in areas such as telecommunication network design, facility location, production and inventory planning, and traffic scheduling and control. However, it is a well known NP-hard problem, and all existing search based exact algorithms are not practical for networks with even moderate numbers of vertices. Therefore, the research community also focuses on approximation algorithms to tackle the problems in practice. In this paper, we present an improved local search algorithm for the minimum concave cost network flow problem based on multi-cycle reduction. The original cycle reduction local search algorithm as proposed by Gallo and Sodini considers only negative cost single cycles; however, we find that such cycle reduction is not complete. We show that negative cost multi-cycles may exist in a network with concave edge costs that has no negative cost cycles, and an existing flow can be reduced to an adjacent neighboring flow with lower cost by redirecting flows along these negative multi-cycles. In this paper, we present an improved local search algorithm based on multi-cycle reduction.We evaluate our proposed algorithm in networks with a simple concave edge cost in different topologies and sizes. The experimental results show that the original cycle reduction algorithms can improve the quality of solutions obtained from a simple minimum cost augmentation approximation heuristic (LDF), and that a multi-cycle Type of Report: Other Department of Computer Science & Engineering Washington University in St. Louis Campus Box 1045 St. Louis, MO 63130 ph: (314) 935-6160 Improved Local Search Algorithm with Multi-Cycle Reduction for Minimum Concave Cost Network Flow Problems Ruibiao Qiu and Jon Turner WUCSE-2004-74 December 9, 2004 Department of Computer Science and Engineering Campus Box 1045 Washington University One Brookings Drive St. Louis, MO 63130-4899, USA Abstract The minimum concave cost network flow problem (MCCNFP) has many applications in areas such as telecommunication network design, facility location, production and inventory planning, and traffic scheduling and control. However, it is a well known NP-hard problem, and all existing search based exact algorithms are not practical for networks with even moderate numbers of vertices. Therefore, the research community also focuses on approximation algorithms to tackle the problems in practice. In this paper, we present an improved local search algorithm for the minimum concave cost network flow problem based on multi-cycle reduction. The original cycle reduction local search algorithm as proposed by Gallo and Sodini considers only negative cost single cycles; however, we find that such cycle reduction is not complete. We show that negative cost multi-cycles may exist in a network with concave edge costs that has no negative cost cycles, and an existing flow can be reduced to an adjacent neighboring flow with lower cost by redirecting flows along these negative multi-cycles. In this paper, we present an improved local search algorithm based on multi-cycle reduction. We evaluate our proposed algorithm in networks with a simple concave edge cost in different topologies and sizes. The experimental results show that the original cycle reduction algorithms can improve the quality of solutions obtained from a simple minimum cost augmentation approximation heuristic (LDF), and that a multi-cycle reduction yields more improvements; however, it reaches a point of diminished returns when we attempt to reduce more than bicycles.The minimum concave cost network flow problem (MCCNFP) has many applications in areas such as telecommunication network design, facility location, production and inventory planning, and traffic scheduling and control. However, it is a well known NP-hard problem, and all existing search based exact algorithms are not practical for networks with even moderate numbers of vertices. Therefore, the research community also focuses on approximation algorithms to tackle the problems in practice. In this paper, we present an improved local search algorithm for the minimum concave cost network flow problem based on multi-cycle reduction. The original cycle reduction local search algorithm as proposed by Gallo and Sodini considers only negative cost single cycles; however, we find that such cycle reduction is not complete. We show that negative cost multi-cycles may exist in a network with concave edge costs that has no negative cost cycles, and an existing flow can be reduced to an adjacent neighboring flow with lower cost by redirecting flows along these negative multi-cycles. In this paper, we present an improved local search algorithm based on multi-cycle reduction. We evaluate our proposed algorithm in networks with a simple concave edge cost in different topologies and sizes. The experimental results show that the original cycle reduction algorithms can improve the quality of solutions obtained from a simple minimum cost augmentation approximation heuristic (LDF), and that a multi-cycle reduction yields more improvements; however, it reaches a point of diminished returns when we attempt to reduce more than bicycles. Improved Local Search Algorithm with Multi-Cycle Reduction for Minimum Concave Cost Network Flow Problems Ruibiao Qiu Jonathan S. Turner {ruibiao,jst}@arl.wustl.edu Applied Research Laboratory Department of Computer Science and Engineering Washington University St. Louis, MO 63130, USA