The Feedback Arc Set Problem with Triangle Inequality is a Vertex Cover Problem

We consider the (precedence constrained) Minimum Feedback Arc Set problem with triangle inequalities on the weights, which finds important applications in problems of ranking with inconsistent information. We present a surprising structural insight showing that the problem is a special case of the minimum vertex cover in hypergraphs with edges of size at most 3. This result leads to combinatorial approximation algorithms for the problem and opens the road to studying the problem as a vertex cover problem

Bounds on Ramsey Games via Alterations

This note contains a refined alteration approach for constructing H-free graphs: we show that removing all edges in H-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdos and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers.Comment: 9 page

On vertex independence number of uniform hypergraphs

Abstract Let H be an r-uniform hypergraph with r ≥ 2 and let α(H) be its vertex independence number. In the paper bounds of α(H) are given for different uniform hypergraphs: if H has no isolated vertex, then in terms of the degrees, and for triangle-free linear H in terms of the order and average degree.</jats:p

The Feedback Arc Set Problem with Triangle Inequality Is a Vertex Cover Problem

We consider the (precedence constrained) Minimum Feedback Arc Set problem with triangle inequalities on the weights, which finds important applications in problems of ranking with inconsistent information. We present a surprising structural insight showing that the problem is a special case of the minimum vertex cover in hypergraphs with edges of size at most 3

Reconstruction of Kauffman networks applying trees

AbstractAccording to Kauffman’s theory [S. Kauffman, The Origins of Order, Self-Organization and Selection in Evolution, Oxford University Press, New York, 1993], enzymes in living organisms form a dynamic network, which governs their activity. For each enzyme the network contains:•a collection of enzymes affecting the enzyme and•a Boolean function prescribing next activity of the enzyme as a function of the present activity of the affecting enzymes.Kauffman’s original pure random structure of the connections was criticized by Barabasi and Albert [A.-L. Barabasi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509–512]. Their model was unified with Kauffman’s network by Aldana and Cluzel [M. Aldana, P. Cluzel, A natural class of robust networks, Proc. Natl. Acad. Sci. USA 100 (2003) 8710–8714]. Kauffman postulated that the dynamic character of the network determines the fitness of the organism. If the network is either convergent or chaotic, the chance of survival is lessened. If, however, the network is stable and critical, the organism will proliferate. Kauffman originally proposed a special type of Boolean functions to promote stability, which he called the property canalyzing. This property was extended by Shmulevich et al. [I. Shmulevich, H. Lähdesmäki, E.R. Dougherty, J. Astola, W. Zhang, The role of certain Post classes in Boolean network models of genetic networks, Proc. Natl. Acad. Sci. USA 100 (2003) 10734–10739] using Post classes. Following their ideas, we propose decision tree functions for enzymatic interactions. The model is fitted to microarray data of Cogburn et al. [L.A. Cogburn, W. Wang, W. Carre, L. Rejtő, T.E. Porter, S.E. Aggrey, J. Simon, System-wide chicken DNA microarrays, gene expression profiling, and discovery of functional genes, Poult. Sci. Assoc. 82 (2003) 939–951; L.A. Cogburn, X. Wang, W. Carre, L. Rejtő, S.E. Aggrey, M.J. Duclos, J. Simon, T.E. Porter, Functional genomics in chickens: development of integrated-systems microarrays for transcriptional profiling and discovery of regulatory pathways, Comp. Funct. Genom. 5 (2004) 253–261]. In microarray measurements the activity of clones is measured. The problem here is the reconstruction of the structure of enzymatic interactions of the living organism using microarray data. The task resembles summing up the whole story of a film from unordered and perhaps incomplete collections of its pieces. Two basic ingredients will be used in tackling the problem. In our earlier works [L. Rejtő, G. Tusnády, Evolution of random Boolean NK-models in Tierra environment, in: I. Berkes, E. Csaki, M. Csörgő (Eds.), Limit Theorems in Probability an Statistics, Budapest, vol. II, 2002, pp. 499–526] we used an evolutionary strategy called Tierra, which was proposed by Ray [T.S. Ray, Evolution, complexity, entropy and artificial reality, Physica D 75 (1994) 239–263] for investigating complex systems. Here we apply this method together with the tree–structure of clones found in our earlier statistical analysis of microarray measurements [L. Rejtő, G. Tusnády, Clustering methods in microarrays, Period. Math. Hungar. 50 (2005) 199–221]

NFA reduction via hypergraph vertex cover approximation

In this thesis, we study in minimum vertex cover problem on the class of k-partite k-uniform hypergraphs. This problem arises when reducing the size of nondeterministic finite automata (NFA) using preorders, as suggested by Champarnaud and Coulon. It has been shown that reducing NFAs using preorders is at least as hard as computing a minimal vertex cover on 3-partite 3-uniform hypergraphs, which is NP-hard. We present several classes of regular languages for which NFAs that recognize them can be optimally reduced via preorders. We introduce an algorithm for approximating vertex cover on k-partite k-uniform hypergraphs based on a theorem by Lovász and explore the use of fractional cover algorithms to improve the running time at the expense of a small increase in the approximation ratio

Algorithmic approaches to problems in probabilistic combinatorics

The probabilistic method is one of the most powerful tools in combinatorics; it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method: 1. We study the structural properties of the triangle-free graphs generated by a semi-random variant of triangle-free process and obtain a packing extension of Kim's famous R(3,t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse. 2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n). 3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory. 4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.Ph.D