NP is as easy as detecting unique solutions
Theoretical Computer Science
On unique satisfiability and the threshold behavior of randomized reductions
Journal of Computer and System Sciences
Improving known solutions is hard
Computational Complexity
Defining sets in vertex colorings of graphs and latin rectangles
Discrete Mathematics
On the complexity of unique solutions
Journal of the ACM (JACM)
Structure of Complexity Classes: Separations, Collapses, and Completeness
MFCS '88 Proceedings of the Mathematical Foundations of Computer Science 1988
A Short Guide To Approximation Preserving Reductions
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
The computational complexity of equivalence and isomorphism problems
The computational complexity of equivalence and isomorphism problems
On the computational complexity of defining sets
Discrete Applied Mathematics - Special issue: Boolean and pseudo-boolean funtions
On the computational complexity of defining sets
Discrete Applied Mathematics - Special issue: Boolean and pseudo-boolean funtions
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Let χ(G) denote the chromatic number of a graph G. A colored set of vertices of G is called forcing if its coloring is extendable to a proper χ(G)-coloring of the whole graph in a unique way. The forcing chromatic number Fχ(G) is the smallest cardinality of a forcing set of G. We estimate the computational complexity of Fχ(G) relating it to the complexity class US introduced by Blass and Gurevich. We prove that recognizing if Fχ(G) ≤ 2 is US-hard with respect to polynomial-time many-one reductions. Furthermore, this problem is coNP-hard even under the promises that Fχ(G) ≤ 3 and G is 3-chromatic. On the other hand, recognizing if Fχ(G) ≤ k, for each constant k, is reducible to a problem in US via a disjunctive truth-table reduction. Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.