COMBINATORIAL OPTIMIZATION APPROACHES TO DISCRETE PROBLEMS

Abstract

<p>As stressed by the Society for Industrial and Applied Mathematics (SIAM): Applied mathematics, in partnership with computational science, is essential in solving many real-world problems. Combinatorial optimization focuses on problems arising from discrete structures such as graphs and polyhedra. This thesis deals with extremal graphs and strings and focuses on two problems: the Erdos' problem on multiplicities of complete subgraphs and the maximum number of distinct squares in a string.<br />The first part of the thesis deals with strengthening the bounds for the minimum proportion of monochromatic t cliques and t cocliques for all 2-colourings of the edges of the complete graph on n vertices. Denote by k_t(G) the number of cliques of order t in a graph G. Let k_t(n) = min{k_t(G)+k_t(\overline{G})} where \overline{G} denotes the complement of G of order n. Let c_t(n) = {k_t(n)} / {\tbinom{n}{t}} and c_t be the limit of c_t(n) for n going to infinity. A 1962 conjecture of Erdos stating that c_t = 2^{1-\tbinom{t}{2}} was disproved by Thomason in 1989 for all t > 3. Tighter counterexamples have been constructed by Jagger, Stovicek and Thomason in 1996, by Thomason for t < 7 in 1997, and by Franek for t=6 in 2002. We present a computational framework to investigate tighter upper bounds for small t yielding the following improved upper bounds for t=6,7 and 8: c_6 \leq 0.7445 \times 2^{1- \tbinom{6}{2}}, c_7\leq 0.6869\times 2^{1- \tbinom{7}{2}}, and c_8 \leq 0.7002\times 2^{1- \tbinom{8}{2}}. The constructions are based on a large but highly regular variant of Cayley graphs for which the number of cliques and cocliques can be expressed in closed form. Considering the quantity e_t=2^{\tbinom{t}{2}-1} c_t, the new upper bound of 0.687 for e_7 is the first bound for any e_t smaller than the lower bound of 0.695 for e_4 due to Giraud in 1979.<br />The second part of the thesis deals with extremal periodicities in strings: we consider the problem of the maximum number of distinct squares in a string. The importance of considering as key variables both the length n and the size d of the alphabet is stressed. Let (d,n)-string denote a string of length n with exactly d distinct symbols. We investigate the function \sigma_d(n) = max {s(x) | x} where s(x) denotes the number of distinct primitively rooted squares in a (d,n)-string x. We discuss a computational framework for computing \sigma_d(n) based on the notion of density and exploiting the tightness of the available lower bound. The obtained computational results substantiate the hypothesized upper bound of n-d for \sigma_d(n). The structural similarities with the approach used for investigating the Hirsch bound for the diameter of a polytope of dimension d having n facets is underlined. For example, the role played by (d,2d)-polytope was presented in 1967 by Klee and Walkup who showed the equivalency between the Hirsch conjecture and the d-step conjecture.</p>