### 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.

Original language | English |
---|---|

Pages (from-to) | 639-654 |

Number of pages | 16 |

Journal | Journal of Combinatorial Optimization |

Volume | 28 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 Oct |

### Keywords

- Approximation algorithm
- PTAS
- Reachability on solution space
- Subset sum

### ASJC Scopus subject areas

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Approximability of the subset sum reconfiguration problem'. Together they form a unique fingerprint.

## Cite this

*Journal of Combinatorial Optimization*,

*28*(3), 639-654. https://doi.org/10.1007/s10878-012-9562-z