Problem liczby skoków w zbiorach częściowo uporządkowanych. Kombinatoryczne algorytmy aproksymacyjne, przeszukiwanie wyczerpujące i złożoność obliczeniowa

Głównym problemem rozprawy doktorskiej jest minimalizacja liczby skoków posetu. Problem skoków dla danego posetu P polega na znalezieniu rozszerzenia liniowego, które minimalizuje liczbę sąsiadujących par elementów, nieporównywalnych w P. NP-trudność tego problemu została najpierw wykazana przez Pulleyblanka [56], a nastepnie na posetach przedziałowych przez Mitas [52]. Proponujemy kilka nowych algorytmów dla tego problemu. Najwięcej uwagi poświęcamy posetom przedziałowym. Dla posetów przedziałowych w latach 90-tych zaproponowano trzy wielomianowe algorytmy aproksymacyjne o współczynniku 3/2. Głównym rezultatem rozprawy jest przełamanie tego współczynnika. Poprawiamy algorytm podany przez Mitas i otrzymujemy aproksymację ze współczynnikiem 1.484. Ponadto przedstawiamy algorytm genetyczny dla problemu skoków na posetach przedziałowych, a także szybki algorytm dokładny dla tej klasy. W przypadku ogólnym, prezentujemy adaptację algorytmu przeszukiwania z zakazami działającą w oparciu o półsilnie zachłanne rozszerzenia liniowe, sformułowane przez Sysłę. Podejmujemy również temat posetów dwuwymiarowych. W klasie dwuwymiarowych posetów przedziałowych otrzymujemy algorytm aproksymacyjny dla problemu skoków ze współczynnikiem 4/3. Praktyczna część pracy obejmuje eksperymentalną analizę wydajności algorytmów przybliżajacych liczbę skoków.The main problem considered in this thesis is to minimize the jump number of a poset. The jump number problem for a given poset P is to find a linear extension minimizing the number of adjacent pairs which are incomparable in P. NP-hardness of this problem was first established by Pulleyblank [56], and later for interval orders by Mitas [52]. In the thesis, some new algorithms for this problem are proposed. We focus mainly on interval orders. In the 1990’s, three polynomial-time approximation algorithms have been given for interval orders with approximation ratio of 3/2. The main result of this thesis is an improvement of this approximation ratio. We enhance the algorithm given by Mitas and we obtain a 1.484-approximation algorithm. Moreover, we present a genetic algorithm for the jump number problem on interval orders, and a fast exact algorithm for this class. In the general case, we present an adaptation of the tabu search technique, based on semi-strongly greedy linear extensions, defined by Sysło. We also undertake the jump number of two-dimensional orders. We obtain a 4/3-approximation algorithm for the class of two-dimensional interval orders. In addition, the thesis contains an experimental analysis of efficiency of algorithms to approximate the jump number

A tabu search approach to the jump number problem

We consider algorithmics for the jump number problem, which is to generate a linear extension of a given poset, minimizing the number of incomparable adjacent pairs. Since this problem is NP-hard on interval orders and open on two-dimensional posets, approximation algorithms or fast exact algorithms are in demand. In this paper, succeeding from the work of the second named author on semi-strongly greedy linear extensions, we develop a metaheuristic algorithm to approximate the jump number with the tabu search paradigm. To benchmark the proposed procedure, we infer from the previous work of Mitas [Order 8 (1991), 115--132] a new fast exact algorithm for the case of interval orders, and from the results of Ceroi [Order 20 (2003), 1--11] a lower bound for the jump number of two-dimensional posets. Moreover, by other techniques we prove an approximation ratio of n/ log(log(n)) for 2D orders

A tabu search approach to the jump number problem

We consider algorithmics for the jump numberproblem, which is to generate a linear extension of a given poset,minimizing the number of incomparable adjacent pairs. Since thisproblem is NP-hard on interval orders and open on two-dimensionalposets, approximation algorithms or fast exact algorithms are indemand.In this paper, succeeding from the work of the second namedauthor on semi-strongly greedy linear extensions, we develop ametaheuristic algorithm to approximate the jump number with thetabu search paradigm. To benchmark the proposed procedure, weinfer from the previous work of Mitas [Order 8 (1991), 115–132] anew fast exact algorithm for the case of interval orders, and from theresults of Ceroi [Order 20 (2003), 1–11] a lower bound for the jumpnumber of two-dimensional posets. Moreover, by other techniqueswe prove an approximation ratio ofn/log lognfor 2D orders

Efficient computation of rank probabilities in posets

As the title of this work indicates, the central theme in this work is the computation of rank probabilities of posets. Since the probability space consists of the set of all linear extensions of a given poset equipped with the uniform probability measure, in first instance we develop algorithms to explore this probability space efficiently. We consider in particular the problem of counting the number of linear extensions and the ability to generate extensions uniformly at random. Algorithms based on the lattice of ideals representation of a poset are developed. Since a weak order extension of a poset can be regarded as an order on the equivalence classes of a partition of the given poset not contradicting the underlying order, and thus as a generalization of the concept of a linear extension, algorithms are developed to count and generate weak order extensions uniformly at random as well. However, in order to reduce the inherent complexity of the problem, the cardinalities of the equivalence classes is fixed a priori. Due to the exponential nature of these algorithms this approach is still not always feasible, forcing one to resort to approximative algorithms if this is the case. It is well known that Markov chain Monte Carlo methods can be used to generate linear extensions uniformly at random, but no such approaches have been used to generate weak order extensions. Therefore, an algorithm that can be used to sample weak order extensions uniformly at random is introduced. A monotone assignment of labels to objects from a poset corresponds to the choice of a weak order extension of the poset. Since the random monotone assignment of such labels is a step in the generation process of random monotone data sets, the ability to generate random weak order extensions clearly is of great importance. The contributions from this part therefore prove useful in e.g. the field of supervised classification, where a need for synthetic random monotone data sets is present. The second part focuses on the ranking of the elements of a partially ordered set. Algorithms for the computation of the (mutual) rank probabilities that avoid having to enumerate all linear extensions are suggested and applied to a real-world data set containing pollution data of several regions in Baden-Württemberg (Germany). With the emergence of several initiatives aimed at protecting the environment like the REACH (Registration, Evaluation, Authorisation and Restriction of Chemicals) project of the European Union, the need for objective methods to rank chemicals, regions, etc. on the basis of several criteria still increases. Additionally, an interesting relation between the mutual rank probabilities and the average rank probabilities is proven. The third and last part studies the transitivity properties of the mutual rank probabilities and the closely related linear extension majority cycles or LEM cycles for short. The type of transitivity is translated into the cycle-transitivity framework, which has been tailor-made for characterizing transitivity of reciprocal relations, and is proven to be situated between strong stochastic transitivity and a new type of transitivity called delta*-transitivity. It is shown that the latter type is situated between strong stochastic transitivity and a kind of product transitivity. Furthermore, theoretical upper bounds for the minimum cutting level to avoid LEM cycles are found. Cutting levels for posets on up to 13 elements are obtained experimentally and a theoretic lower bound for the cutting level to avoid LEM cycles of length 4 is computed. The research presented in this work has been published in international peer-reviewed journals and has been presented on international conferences. A Java implementation of several of the algorithms presented in this work, as well as binary files containing all posets on up to 13 elements with LEM cycles, can be downloaded from the website http://www.kermit.ugent.be

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D