U153592.pdf (4.47 MB)
The complexity of greedy algorithms on ordered graphs
thesis
posted on 2014-12-15, 10:40 authored by Antonio. PuricellaLet p be any fixed polynomial time testable, non-trivial, hereditary property of graphs. Suppose that the vertices of a graph G are not necessarily linearly ordered but partially ordered, where we think of this partial order as a collection of (possibly exponentially many) linear orders in the natural way. In the first part of this thesis, we prove that the problem of deciding whether a lexicographically first maximal (with respect to one of these linear orders) subgraph of G satisfying p, contains a specified vertex is NP-complete. For some of these properties p we then show that by applying certain restrictions the problem still remains NP-complete, and show how the problem can be solved in deterministic polynomial time if the restrictions imposed become more severe.;Let H be a fixed undirected graph. An H-colouring of an undirected graph G is a homomorphism from G to H. In the second part of the thesis, we show that, if the vertices of G are partially ordered then the complexity of deciding whether a given vertex of G is in a lexicographically first maximal H-colourable subgraph of G is NP-complete, if H is bipartite, and Sp2-complete, if H is non-bipartite. We then show that if the vertices of G are linearly, as opposed to partially, ordered then the complexity of deciding whether a given vertex of G is in the lexicographically first maximal H-colourable subgraph of G is P-complete, if H is bipartite, and DP2-complete, if H is non-bipartite.;In the final part of the thesis we show that the results obtained can be paralleled in the setting of graphs where orders are given by degrees of the vertices.
History
Date of award
2002-01-01Author affiliation
MathematicsAwarding institution
University of LeicesterQualification level
- Doctoral
Qualification name
- PhD