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Approximability of the subset sum reconfiguration problem

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Abstract

The subset sum problem is a well-known NP-complete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NP-hard, and is PSPACE-complete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in a reconfiguration. We show that this maximization problem admits a polynomial-time approximation scheme, while the problem is APX-hard if we are given a conflict graph.

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Notes

  1. Note that subset sum in Garey and Johnson (1979) is slightly different from our definition: subset sum in Garey and Johnson (1979) is defined as the problem of finding a packing whose total size is exactly \(k\).

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Acknowledgments

We thank the referees for their helpful comments and suggestions. The first author’s work is partially supported by JSPS KAKENHI Grant No. 22700001. The second author’s work is supported in part by NSF Grant CCF-1161626 and DARPA/AFOSR Grant FA9550-12-1-0423.

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Aoba-yama 6-6-05, Sendai, 980-8579, Japan

    Takehiro Ito

  2. MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA, 02139, USA

    Erik D. Demaine

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  1. Takehiro Ito
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  2. Erik D. Demaine
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Correspondence to Takehiro Ito.

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Ito, T., Demaine, E.D. Approximability of the subset sum reconfiguration problem. J Comb Optim 28, 639–654 (2014). https://doi.org/10.1007/s10878-012-9562-z

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