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The complexity of unions of disjoint sets
 In Proceedings 24th Symposium on Theoretical Aspects of Computer Science
, 2007
"... This paper is motivated by the open question whether the union of two disjoint NPcomplete sets always is NPcomplete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we app ..."
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This paper is motivated by the open question whether the union of two disjoint NPcomplete sets always is NPcomplete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions of mequivalent disjoint sets. Under the hypothesis that NE � = coNE, we construct degrees in NP where our main question has a positive answer, i.e., these degrees are closed under unions of disjoint sets. 1
Planar graph coloring is not selfreducible, assuming
 P �= NP. Theoretical Computer Science
, 1991
"... We show that obtaining the lexicographically first four coloring of a planar graph is NP –hard. This shows that planar graph fourcoloring is not selfreducible, assuming P ̸= NP. One consequence of our result is that the schema of [JVV 86] cannot be used for approximately counting the number of fou ..."
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Cited by 1 (0 self)
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We show that obtaining the lexicographically first four coloring of a planar graph is NP –hard. This shows that planar graph fourcoloring is not selfreducible, assuming P ̸= NP. One consequence of our result is that the schema of [JVV 86] cannot be used for approximately counting the number of four colorings of a planar graph. These results extend to planar graph kcoloring, for k ≥ 4.
Unions of Disjoint NPComplete Sets
"... Abstract. We study the following question: if A and B are disjoint NPcomplete sets, then is A ∪ B NPcomplete? We provide necessary and sufficient conditions under which the union of disjoint NPcomplete sets remain complete. 1 ..."
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Abstract. We study the following question: if A and B are disjoint NPcomplete sets, then is A ∪ B NPcomplete? We provide necessary and sufficient conditions under which the union of disjoint NPcomplete sets remain complete. 1
On Computing the Smallest FourColoring of Planar Graphs and NonSelfReducible Sets in P
, 2006
"... We show that computing the lexicographically first fourcoloring for planar graphs is ∆ p 2hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NPhard, and conclude that it is not selfreducible in the sense of Schnorr, assuming P � = NP. We discuss thi ..."
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We show that computing the lexicographically first fourcoloring for planar graphs is ∆ p 2hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NPhard, and conclude that it is not selfreducible in the sense of Schnorr, assuming P � = NP. We discuss this application to nonselfreducibility and provide a general related result. We also discuss when raising a problem’s NPhardness lower bound to ∆ p 2hardness can be valuable.
Integer Polyhedra: Combinatorial Properties and Complexity
, 2001
"... A polyhedron having vertices is called integer if all of its vertices are integer. This property is coNPcomplete in general. Recognizing integral setpacking polyhedra is one of the biggest challenges of graph theory (perfectness test). Various other special cases are major problems of discrete ..."
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A polyhedron having vertices is called integer if all of its vertices are integer. This property is coNPcomplete in general. Recognizing integral setpacking polyhedra is one of the biggest challenges of graph theory (perfectness test). Various other special cases are major problems of discrete mathematics.