Graph decomposition with constraints on the minimum degree
Discrete Mathematics
Balanced graphs with minimum degree constraints
Discrete Mathematics
Selected papers of the 13th British Combinatorial Conference on British combinatorial conference
The hardness of approximation: gap location
Computational Complexity
Graph partitions with minimum degree constraints
Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The satisfactory partition problem
Discrete Applied Mathematics
The most vital nodes with respect to independent set and vertex cover
Discrete Applied Mathematics
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The Satisfactory Bisection problem means to decide whether a given graph has a partition of its vertex set into two parts of the same cardinality such that each vertex has at least as many neighbors in its part as in the other part. A related variant of this problem, called Co-Satisfactory Bisection, requires that each vertex has at most as many neighbors in its part as in the other part. A vertex satisfying the degree constraint above in a partition is called 'satisfied' or 'co-satisfied,' respectively. After stating the NP-completeness of both problems, we study approximation results in two directions. We prove that maximizing the number of (co-)satisfied vertices in a bisection has no polynomial-time approximation scheme (unless P=NP), whereas constant approximation algorithms can be obtained in polynomial time. Moreover, minimizing the difference of the cardinalities of vertex classes in a bipartition that (co-)satisfies all vertices has no polynomial-time approximation scheme either.