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<p>It is well known in the OR literature that the advantage that specialized algorithms have over general-purpose integer programming software for optimally solving knapsack problems with conflicts (KPC) decreases as the conflict graph density decreases. However, it is not generally known which KPCs cannot be solved to optimality in a reasonable time using general-purpose integer programming software on a standard PC. In this paper, using 4800 KPC instances from the OR literature, we determine how well the general-purpose integer programming software Gurobi can solve these KPC instances. Specifically, we show that, using default parameter settings on a standard PC, Gurobi determines optimal solutions for 82.1% of these KPCs in less than 60&#xa0;s (average of 3.2&#xa0;s), and optimal solutions for 8.2% in less than 1800s (average of 399&#xa0;s). Only 9.7% of these KPCs cannot be solved to proven optimality in 1800s. It is these 9.7% that we denote as hard to solve and use classification trees to characterize these KPCs.</p>

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Identifying Knapsack Problems with Conflicts that Are Difficult to Solve Optimally Using General-Purpose Integer Programming Software

  • 期刊论文

  • 发表日期：

    2025年6月25日

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  • Myung Soon Song,
  • Pei Hua Lin,
  • Yun Lu,
  • Francis J. Vasko

摘要

It is well known in the OR literature that the advantage that specialized algorithms have over general-purpose integer programming software for optimally solving knapsack problems with conflicts (KPC) decreases as the conflict graph density decreases. However, it is not generally known which KPCs cannot be solved to optimality in a reasonable time using general-purpose integer programming software on a standard PC. In this paper, using 4800 KPC instances from the OR literature, we determine how well the general-purpose integer programming software Gurobi can solve these KPC instances. Specifically, we show that, using default parameter settings on a standard PC, Gurobi determines optimal solutions for 82.1% of these KPCs in less than 60 s (average of 3.2 s), and optimal solutions for 8.2% in less than 1800s (average of 399 s). Only 9.7% of these KPCs cannot be solved to proven optimality in 1800s. It is these 9.7% that we denote as hard to solve and use classification trees to characterize these KPCs.