Sparsification of two-variable valued constraint satisfaction problems

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A valued constraint satisfaction problem (VCSP) instance (V;Πω) is a set of variables V with a set of constraints Π weighted by ω. Given a VCSP instance, we are interested in a reweighted subinstance (V;Π'⊃ Π, ω') that preserves the value of the given instance (under every assignment to the variables) within factor 1 ± ∈. A well-studied special case is cut sparsification in graphs, which has found various applications. We show that a VCSP instance consisting of a single boolean predicate P(x, y) (e.g., for cut, P = XOR) can be sparsified into O(|V|=∈2) constraints iff the number of inputs that satisfy P is anything but one (i.e., |P-1(1)| ≠ 1). Furthermore, this sparsity bound is tight unless P is a relatively trivial predicate. We conclude that also systems of 2SAT (or 2LIN) constraints can be sparsified.

Original languageEnglish
Pages (from-to)1263-1276
Number of pages14
JournalSIAM Journal on Discrete Mathematics
Issue number2
StatePublished - 1 Jan 2017


  • Boolean predicates
  • Cut sparsification
  • Valued constraint satisfaction problem

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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