Reconfiguration of Satisfying Assignments and Subset Sums: Easy to Find, Hard to Connect

  • Jean Cardinal
  • Erik D. Demaine
  • David Eppstein
  • Robert A. Hearn
  • Andrew WinslowEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10976)


We consider the computational complexity of reconfiguration problems, in which one is given two combinatorial configurations satisfying some constraints, and is asked to transform one into the other using elementary transformations, while satisfying the constraints at all times. Such problems appear naturally in many contexts, such as model checking, motion planning, enumeration and sampling, and recreational mathematics. We provide hardness results for problems in this family, in which the constraints and operations are particularly simple.

More precisely, we prove the \(\mathsf {PSPACE}\)-completeness of the following decision problems:
  • Given two satisfying assignments to a planar monotone instance of Not-All-Equal 3-SAT, can one assignment be transformed into the other by single variable “flips” (assignment changes), preserving satisfiability at every step?

  • Given two subsets of a set S of integers with the same sum, can one subset be transformed into the other by adding or removing at most three elements of S at a time, such that the intermediate subsets also have the same sum?

  • Given two points in \(\{0,1\}^n\) contained in a polytope P specified by a constant number of linear inequalities, is there a path in the n-hypercube connecting the two points and contained in P?

These problems can be interpreted as reconfiguration analogues of standard problems in \(\mathsf {NP}\). Interestingly, the instances of the \(\mathsf {NP}\) problems that appear as input to the reconfiguration problems in our reductions can be shown to lie in \(\mathsf {P}\). In particular, the elements of S and the coefficients of the inequalities defining P can be restricted to have logarithmic bit-length.


Boolean satisfiability Subset sum Combinatorial reconfiguration \(\mathsf {PSPACE}\)-completeness 



This work was initiated at the 32nd Bellairs Winter Workshop on Computational Geometry, January 27–February 3, 2017. We thank the other participants of the workshop for a productive and positive atmosphere.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jean Cardinal
    • 1
  • Erik D. Demaine
    • 2
  • David Eppstein
    • 3
  • Robert A. Hearn
    • 4
  • Andrew Winslow
    • 5
    Email author
  1. 1.Université libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.University of CaliforniaIrvineUSA
  4. 4.Portola ValleyUSA
  5. 5.University of Texas Rio Grande ValleyEdinburgUSA

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