Heuristic Coloring Algorithm for the Composite Graph Coloring Problem

A composite graph is a finite undirected graph in which a positive integer known as a chromaticity is associated with each vertex of the graph. The composite graph coloring problem (CGCP) is the problem of finding the chromatic number of a composite graph, i.e., the minimum number of colors (positive integers) required to assign a sequence of consecutive colors to each vertex of the graph in a manner such that adjacent vertices are not assigned sequences with colors in common and the sequence assigned to a vertex has the number of colors indicated by the chromaticity of the vertex. The CGCP problem is an NP-complete problem that has applications to scheduling and resource allocation problems in which the tasks to be scheduled are of unequal durations. The pigeonhole principle gives rise to a problem reduction technique for the CGCP and a vertex ordering used in the vertex-sequentia1-with-interchange (VSI) algorithm. LFPHI. An upper bound on the chromatic number of a composite graph is obtained from the definition of a color-sequential coloring algorithm for the CGCP. The performances of twelve heuristic coloring algorithms are compared on a variety of random composite graphs. Three VSI algorithms (LF1I, LFPHI, and LFCDI) performed superior to the other algorithms on graphs having the lower numbers of vertices and low edge densities while two color-sequential algorithms (RLF1 and RLFD1) were superior on graphs having the higher numbers of vertices and high edge densities

Marchenko-Pastur Theorem and Bercovici-Pata bijections for heavy-tailed or localized vectors

The celebrated Marchenko-Pastur theorem gives the asymptotic spectral distribution of sums of random, independent, rank-one projections. Its main hypothesis is that these projections are more or less uniformly distributed on the first grassmannian, which implies for example that the corresponding vectors are delocalized, i.e. are essentially supported by the whole canonical basis. In this paper, we propose a way to drop this delocalization assumption and we generalize this theorem to a quite general framework, including random projections whose corresponding vectors are localized, i.e. with some components much larger than the other ones. The first of our two main examples is given by heavy tailed random vectors (as in a model introduced by Ben Arous and Guionnet or as in a model introduced by Zakharevich where the moments grow very fast as the dimension grows). Our second main example is given by vectors which are distributed as the Brownian motion on the unit sphere, with localized initial law. Our framework is in fact general enough to get new correspondences between classical infinitely divisible laws and some limit spectral distributions of random matrices, generalizing the so-called Bercovici-Pata bijection.Comment: 40 pages, 10 figures, some minor mistakes correcte

Renormalization: an advanced overview

We present several approaches to renormalization in QFT: the multi-scale analysis in perturbative renormalization, the functional methods \`a la Wetterich equation, and the loop-vertex expansion in non-perturbative renormalization. While each of these is quite well-established, they go beyond standard QFT textbook material, and may be little-known to specialists of each other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added, minor improvements; v3: some changes to sect. 5, refs. adde

A Parallel Implementation of the Glowinski-Pironneau Algorithm for the Modified Stokes Problem

In this dissertation we consider a parallel implementation of the Glowinski-Pironneau algorithm for the modified Stokes problem. In particular, we motivate this effort by demonstrating the occurrence of the modified Stokes problem in the time dependent viscoelastic Oldroyd flow setting using Saramito\u27s splitting. We then present an analysis of the Glowinski-Pironneau pressure decomposition for the modified Stokes problem - including numerical error estimates. Next we discuss our parallel finite element method implementation of the pressure decomposition approach. Finally, we present numerical results including errors and performance measures. These measures are also compared with results for a coupled velocity-pressure modified Stokes solver using a publicly available parallel solver

Markov chains and applications in the generation of combinatorial designs

This Thesis deals with discrete Markov chains and their applications in the generation of combinatorial designs. A conjecture on the generation of proper edge colorings of the complete graph K_n, for n even, is tackled. A proper edge coloring is an edge coloring such that no two adjacent edges have the same color. Proper edge colorings are characterized by minimizing the potential and maximizing the entropy. We implement an algorithm in the software R to generate proper colorings from any arbitrary coloring of K_n, by identifying the colorings as nodes in a Markov chain, where transition probabilities are defined so that the potential decreases or, alternatively, the entropy increases. The conjecture states that the algorithm converges in polynomial time. We give original proofs of the conjecture in K_4 and K_6, and we provide new results and ideas that could be used to prove the conjecture in the general case K_n

Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number

A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that traverses all of its vertices. The problems of deciding their existence in an input graph are well-known to be NP-complete, in fact, they belong to the first problems shown to be computationally hard when the theory of NP-completeness was being developed. A lot of research has been devoted to the complexity of Hamiltonian path and Hamiltonian cycle problems for special graph classes, yet only a handful of positive results are known. The complexities of both of these problems have been open even for $4K_1$-free graphs, i.e., graphs of independence number at most $3$. We answer this question in the general setting of graphs of bounded independence number. We also consider a newly introduced problem called \emph{Hamiltonian-$\ell$-Linkage} which is related to the notions of a path cover and of a linkage in a graph. This problem asks if given $\ell$ pairs of vertices in an input graph can be connected by disjoint paths that altogether traverse all vertices of the graph. For $\ell=1$, Hamiltonian-1-Linkage asks for existence of a Hamiltonian path connecting a given pair of vertices. Our main result reads that for every pair of integers $k$ and $\ell$, the Hamiltonian-$\ell$-Linkage problem is polynomial time solvable for graphs of independence number not exceeding $k$. We further complement this general polynomial time algorithm by a structural description of obstacles to Hamiltonicity in graphs of independence number at most $k$ for small values of $k$