# ŠTAJNEROVI SISTEMI I NOVE KONSTRUKCIJE (v,k,t)-POKRIVANJA

Title: | ŠTAJNEROVI SISTEMI I NOVE KONSTRUKCIJE (v,k,t)-POKRIVANJA |

Author: | Nikolić, Nebojša |

Abstract: | A Steiner system S(t; k; v) is a set which contains v elements (v-set) and a family of k-subsets (blocks), such that each t-subset appears in exactly one block (v > k > t > 1; v; k; t 2 N). In the case of a (v; k; t)¡covering, each t-subset appears in at least one block of a given family. A Steiner system S(t; k; v) exists if and only if C(v; k; t) = ¡v t ¢±¡k t ¢ , where C(v; k; t) is the cardinality of minimal (v; k; t)¡covering. As the existence of Steiner system S(t; k; v) and the determination of the minimal (v; k; t)¡covering are still open problems, their solutions are known only in some special cases. Besides the review of the previous results related to the problem of the existence of Steiner systems and the problem of determining the minimal (v; k; t)¡covering, several new constructions of (v; k; t)¡covering are given in this paper. Since the number of blocks in (v; k; t)¡covering represents the upper bound on C(v; k; t), a large number of upper bounds are also obtained by using these constructions. In many cases, the obtained upper bounds are better than the best known upper bounds on C(v; k; t). This dissertation gives a new combinatorial construction of minimal (v; 3; 2)¡ coverings, which represents a generalization of Bose and Skolem constructions of the Steiner triple systems STS(6n + 3) and STS(6n + 1). In each of the 6 cases (v = 6n; : : : ; 6n+5), (v; 3; 2)¡covering is obtained by applying certain permutation p to the initial set of blocks. The obtained construction also represents a new proof of the statement that the values of C(v; 3; 2) are equal to SchÄonheim lower bound L(v; 3; 2). Other constructions of (v; k; t)¡coverings, given in this paper, are heuristic. First, we give improved implementation of the well known greedy algorithm. Then, a new greedy algorithm, as well as the theorem which provides a su±cient con- dition for equality of greedy lex and greedy colex coverings are given. Finally, by v using so called LR procedure, three other heuristics are developed and implemented: Large neighbourhood search, Variable neighborhood descent and General variable neighborhood search. Large neighbourhood search is the procedure of alternately destroying and re- pairing a solution in order to improve the incumbent solution. In the proposed algorithm, this is the procedure for systematic removing and adding blocks to the covering, based on LR procedure. By using LR procedure, the blocks which exclu- sively cover the minimal number of t-subsets are removed from (v; k; t)¡covering, and then the uncovered t-subsets are covered with as few blocks as possible. The greedy algorithm is used for the covering of the uncovered t-subsets. Variable neighborhood search is based on the idea of systematic change of neigh- borhood within a local search algorithm in order to avoid the convergence to a local minimum. A variant of the method where changes of neighborhood is performed in a deterministic way is called Variable neighborhood descent. A variant where the local search procedure is replaced by the Variable neighborhood descent method is General variable neighborhood search. The basis of the local search procedure applied in these two heuristics is also LR procedure. Regarding quality of the obtained results and the performance of the methods, Large neighbourhood search, Variable neighborhood descent, and General variable neighborhood search overcome two heuristics for solving the problem of minimal (v; k; t)¡covering known from the literature: Simulated annealing and Tabu search. Unlike the existing heuristics, the proposed heuristics are applicable to arbitrarily (v; k; t)¡covering. By applying aforementioned heuristics, 23 new best known upper bounds on C(v; k; t) are established. |

URI: | http://hdl.handle.net/123456789/4285 |

Date: | 2015 |

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