Efficient algorithms for decomposing graphs under degree constraints

AbstractStiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321–324] proved that if every vertex v in a graph G has degree d(v)⩾a(v)+b(v)+1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A,B) such that dA(v)⩾a(v) for every v∈A and dB(v)⩾b(v) for every v∈B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7–9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237–239] strengthened this result, proving that it suffices to assume d(v)⩾a+b (a,b⩾1) or just d(v)⩾a+b-1 (a,b⩾2) if G contains no cycles shorter than 4 or 5, respectively.The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented

The satisfactory partition problem

AbstractThe SATISFACTORY PARTITION problem consists in deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [Partitioning a graph to satisfy all vertices, Technical report, Swiss Federal Institute of Technology, Lausanne, 1998; Algorithmic approach to the satisfactory graph partitioning problem, European J. Oper. Res. 125 (2000) 283–291] and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. However, for graphs with maximum degree at most 4 the problem is polynomially solvable. We also study generalizations and variants of this problem where a partition into k nonempty parts (k⩾3) is requested

Lexicographic

In the last decade, several robustness approaches have been developed to deal with uncertainty. In decision problems, and particularly in location problems, the most used robustness approach rely either on maximal cost or on maximal regret criteria. However, it is well known that these criteria are too conservative. In this paper, we present a new robustness approach, called lexicographic α-robustness, which compensates for the drawbacks of criteria based on the worst case. We apply this approach to the 1-median location problem under uncertainty on node weights and we give a specific algorithm to determine robust solutions in the case of a tree. We also show that this algorithm can be extended to the case of a general network

Aiding decisions with multiple criteria: essays in honor of Bernard Roy

info:eu-repo/semantics/publishe

Multiobjective decision support in IT-risk management

Strauss C, Stummer C. Multiobjective decision support in IT-risk management. International Journal of Information Technology and Decision Making. 2002;1(2):251-268