14,549 research outputs found
Generalized vertex coloring problems using split graphs
Graph theory experienced a remarkable increase of interest among the scientific community during the last decades. The vertex coloring problem (Min Coloring) deserves a particular attention rince it has been able to capture a wide variety of applications. For mathematicians, it is interesting for an additional reason: it is extremely hard to solve it in an efficient way. In this thesis, we introduce several problems generalizing the usual vertex coloring problem, and hence, extending its application domain. We say that a graph is (p, k)-colorable if its vertex set can be partitioned into p cliques and k stable sets. Then, for a given p (respectively k), one may ask the following questions: how to choose p cliques (respectively k stable sets) to be removed from the graph such that the number of stable sets (respectively cliques) partitioning the remaining vertices is minimized? These are called (p, k)-coloring problems. We also introduce Min Split-coloring which is, given a graph G, the problem of minimizing k such that G is (k, k)-colorable. Along the saine line, given a graph G, the objective of the problem Min Cocoloring is to minimize p + k such that G is (p, k)-colorable. All these problems, called together generalized coloring problems, are obviously at least as difficult as Min Coloring. The purpose of this dissertation is to study generalized coloring problems in nome restricted classes of graphs in order to bring a new insight on the relative difficulties of these problems. To this end, we detect in a more precise way the limits between NP-hard and polynomially solvable problems. Chapter 1 introduces generalized coloring problems by emphasizing nome preliminary results which will guide the questions to handle in the following chapters. Chapter 2 exposes the first clans of graphs, namely cacti, where Min Split-coloring is shown to be polynomially solvable. We also observe that generalized coloring problems can be polynomially solved in triangulated graphs. The main result of Chapter 3 is a new characterization of cographs: it is equivalent to say that G is a cograph, and to state that, for every subgraph G' ⊆ G, G' is (p, k)-colorable if and only if G' [V \ K] is (p – 1, k)-colorable, where K induces a maximum clique of G'. This result implies simple combinatorial algorithme to solve all generalized coloring problems; the one for Min Cocoloring improves the best time complexity known so far. In Chapter 4, we handle the recognition of polar graphs which can be seen as a particular (p, k)-coloring, where p cliques are independent (i.e., not linked at all) and k stable sets form a complete k-partite graph. It is known that the recognition of polar graphs is NP-complete. Here, we determine the first clans of graphs, namely cographs, where the polar graphs can be recognized in polynomial time, more precisely in time O(n log n). We also give a characterization by forbidden subgraphs. In the came manner, we characterize monopolar cographs, i.e., cographs admitting a polar partition with at most one clique or at most one stable set. Chapter 5 is devoted to generalized coloring problems in line graphs. Here, we detect the first classes of graphs, namely line graphs of trees, line graphs of bipartite graphs and line graphs of line-perfect graphs, where generalized coloring problems diverge in terms of NP-hardness. In Chapter 6 we study the approximability of generalized coloring problems in line graphs, in comparability graphs and in general graphs. We derive approximation algorithms with a performance guarantee using both the standard approximation ratio and the differential approximation ratio. We show that both Min Split-coloring and Min Cocoloring are at least as hard as Min Coloring to approximate from the standard approximation ratio point of view, whereas, they admit a polynomial time differential approximation scheme and Min Coloring only a constant differential approximation ratio. We also show that Min Cocoloring reduces to Min Split-coloring in all classes of graphs closed under addition of disjoint cliques and under join of a complete k-partite graph. In Chapter 7, we handle two different applications of Min Split-coloring in permutation graphs. They give birth to a new problem, called Min Threshold-coloring, that we study in the came spirit as the other generalized coloring problems. In the last chapter, we present several open questions arising from this thesis
Parameterized (in)approximability of subset problems
We discuss approximability and inapproximability in FPT-time for a large
class of subset problems where a feasible solution is a subset of the input
data and the value of is . The class handled encompasses many
well-known graph, set, or satisfiability problems such as Dominating Set,
Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first
time, we introduce the notion of intersective approximability that generalizes
the one of safe approximability and show strong parameterized inapproximability
results for many of the subset problems handled. Then, we study approximability
of these problems with respect to the dual parameter where is the
size of the instance and the standard parameter. More precisely, we show
that under such a parameterization, many of these problems, while
W[]-hard, admit parameterized approximation schemata.Comment: 7 page
Ramsey games with giants
The classical result in the theory of random graphs, proved by Erdos and
Renyi in 1960, concerns the threshold for the appearance of the giant component
in the random graph process. We consider a variant of this problem, with a
Ramsey flavor. Now, each random edge that arrives in the sequence of rounds
must be colored with one of R colors. The goal can be either to create a giant
component in every color class, or alternatively, to avoid it in every color.
One can analyze the offline or online setting for this problem. In this paper,
we consider all these variants and provide nontrivial upper and lower bounds;
in certain cases (like online avoidance) the obtained bounds are asymptotically
tight.Comment: 29 pages; minor revision
Coloring Random Triangulations
We introduce and solve a two-matrix model for the tri-coloring problem of the
vertices of a random triangulation. We present three different solutions: (i)
by orthogonal polynomial techniques (ii) by use of a discrete Hirota bilinear
equation (iii) by direct expansion. The model is found to lie in the
universality class of pure two-dimensional quantum gravity, despite the
non-polynomiality of its potential.Comment: 50 pages, 4 figures, Tex, uses harvmac, eps
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
On some varieties associated with trees
This article considers some affine algebraic varieties attached to finite
trees and closely related to cluster algebras. Their definition involves a
canonical coloring of vertices of trees into three colors. These varieties are
proved to be smooth and to admit sometimes free actions of algebraic tori. Some
results are obtained on their number of points over finite fields and on their
cohomology.Comment: 37 pages, 7 figure
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