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Nonnegative partial s-goodness for the equivalence of a 0-1 linear program to weighted linear programming

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  • Meijia Han

    (Fuzhou University)

  • Wenxing Zhu

    (Fuzhou University)

Abstract

The 0-1 linear programming problem with non-negative constraint matrix and objective vector $${\textbf{e}}$$ e origins from many NP-hard combinatorial optimization problems. In this paper, we consider under what condition an optimal solution of the 0-1 problem can be obtained from a weighted linear programming. To this end, we first formulate the 0-1 problem as a sparse minimization problem. Any optimal solution of the 0-1 linear programming problem can be obtained by rounding up an optimal solution of the sparse minimization problem. Then, we establish a condition under which the sparse minimization problem and the weighted linear programming problem have the same optimal solution. The condition is based on the defined non-negative partial s-goodness of the constraint matrix and the weight vector. Further, we use two quantities to characterize a sufficient condition and necessary condition for the non-negative partial s-goodness. However, the two quantities are difficult to calculate, therefore, we provide a computable upper bound for one of the two quantities to verify the non-negative partial s-goodness. Furthermore, we propose two operations of the constraint matrix and weight vector that still preserve non-negative partial s-goodness. Finally, we give some examples to illustrate that our theory is effective and verifiable.

Suggested Citation

  • Meijia Han & Wenxing Zhu, 2023. "Nonnegative partial s-goodness for the equivalence of a 0-1 linear program to weighted linear programming," Journal of Combinatorial Optimization, Springer, vol. 45(5), pages 1-37, July.
  • Handle: RePEc:spr:jcomop:v:45:y:2023:i:5:d:10.1007_s10878-023-01054-1
    DOI: 10.1007/s10878-023-01054-1
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    References listed on IDEAS

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    4. Weili Zhang & Charles D. Nicholson, 2020. "Objective scaling ensemble approach for integer linear programming," Journal of Heuristics, Springer, vol. 26(1), pages 1-19, February.
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