Approximability of the subset sum reconfiguration problem

Takehiro Ito, Erik D. Demaine

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


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 languageEnglish
Pages (from-to)639-654
Number of pages16
JournalJournal of Combinatorial Optimization
Issue number3
Publication statusPublished - 2014 Oct


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


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