Computer Science

We present a constraint model for the problem of producing a tree decomposition of a graph. The inputs to the model are a simple graph G, the number of nodes in the desired tree decomposition and the maximum cardinality of each node in that decomposition. Via a sequence of decision problems, the model allows us to find the tree width of a graph whilst delivering a tree decomposition of that width, i.e. a witness.

Almost 10 years ago, Impagliazzo and Kabanets (2010) gave a new combinatorial proof of Chernoff's bound for sums of bounded independent random variables. Unlike previous methods, their proof is constructive. This means that it provides an efficient randomized algorithm for the following task: given a set of Boolean random variables whose sum is not concentrated around its expectation, find a subset of statistically dependent variables. However, the algorithm of Impagliazzo and Kabanets (2010) is given only for the Boolean case. On the other hand, the general proof technique works also for real-valued random variables, even though for this case, Impagliazzo and Kabanets (2010) obtain a concentration bound that is slightly suboptimal. Herein, we revisit both these issues and show that it is relatively easy to extend the Impagliazzo-Kabanets algorithm to real-valued random variables and to improve the corresponding concentration bound by a constant factor.

We consider random graphs in which the edges are allowed to be dependent. In our model the edge dependence is quite general, we call it p -robust random graph. It means that every edge is present with probability at least p , regardless of the presence/absence of other edges. This is more general than independent edges with probability p , as we illustrate with examples. Our main result is that for any monotone graph property, the p -robust random graph has at least as high probability to have the property as an Erdos-Renyi random graph with edge probability p . This is very useful, as it allows the adaptation of many results from classical Erdos-Renyi random graphs to a non-independent setting, as lower bounds.

Discrete orthogonal matrices have several applications in information technology, such as in coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize continuous orthogonal functions that have been discovered. The need of certain continuous functions is restrictive. To simplify the process while improving the efficiency and flexibility, we present a general method for generating orthogonal matrices directly through the construction of certain even and odd polynomials from a set of distinct positive values, bypassing the need of continuous orthogonal functions. We provide a constructive proof by induction that not only asserts the existence of such polynomials, but also tells how to iteratively construct them. Besides the derivation of the method as simple as a few nested loops, we discuss two well-known discrete transforms, the Discrete Cosine Transform and the Discrete Tchebichef Transform. How they can be achieved using our method with the specific values, and show how to embed them into the transform module of video coding. By the same token, we also show some examples of how to generate new orthogonal matrices from arbitrarily chosen values.

We study the stabilization time of a wide class of processes on graphs, in which each node can only switch its state if it is motivated to do so by at least a 1+λ 2 fraction of its neighbors, for some 0<λ<1 . Two examples of such processes are well-studied dynamically changing colorings in graphs: in majority processes, nodes switch to the most frequent color in their neighborhood, while in minority processes, nodes switch to the least frequent color in their neighborhood. We describe a non-elementary function f(λ) , and we show that in the sequential model, the worst-case stabilization time of these processes can completely be characterized by f(λ) . More precisely, we prove that for any ϵ>0 , O( n 1+f(λ)+ϵ ) is an upper bound on the stabilization time of any proportional majority/minority process, and we also show that there are graph constructions where stabilization indeed takes Ω( n 1+f(λ)−ϵ ) steps.

Let L be any finite distributive lattice and B be any boolean predicate defined on L such that the set of elements satisfying B is a sublattice of L . Consider any subset M of L of size k of elements of L that satisfy B . Then, we show that k generalized median elements generated from M also satisfy B . We call this result generalized median theorem on finite distributive lattices. When this result is applied to the stable matching, we get Teo and Sethuraman's median stable matching theorem. Our proof is much simpler than that of Teo and Sethuraman. When the generalized median theorem is applied to the assignment problem, we get an analogous result for market clearing price vectors.

Inspired by the increasing popularity of Swiss-system tournaments in sports, we study the problem of predetermining the number of rounds that can be guaranteed in a Swiss-system tournament. Matches of these tournaments are usually determined in a myopic round-based way dependent on the results of previous rounds. Together with the hard constraint that no two players meet more than once during the tournament, at some point it might become infeasible to schedule a next round. For tournaments with n players and match sizes of k≥2 players, we prove that we can always guarantee ⌊ n k(k−1) ⌋ rounds. We show that this bound is tight. This provides a simple polynomial time constant factor approximation algorithm for the social golfer problem. We extend the results to the Oberwolfach problem. We show that a simple greedy approach guarantees at least ⌊ n+4 6 ⌋ rounds for the Oberwolfach problem. This yields a polynomial time 1 3+ϵ -approximation algorithm for any fixed ϵ>0 for the Oberwolfach problem. Assuming that El-Zahar's conjecture is true, we improve the bound on the number of rounds to be essentially tight.

Constraint Satisfaction Problem (CSP) is a framework for modeling and solving a variety of real-world problems. Once the problem is expressed as a finite set of constraints, the goal is to find the variables' values satisfying them. Even though the problem is in general NP-complete, there are some approximation and practical techniques to tackle its intractability. One of the most widely used techniques is the Constraint Propagation. It consists in explicitly excluding values or combination of values for some variables whenever they make a given subset of constraints unsatisfied. In this paper, we deal with a CSP subclass which we call 4-CSP and whose constraint network infers relations of the form: {x∼α,x−y∼β,(x−y)−(z−t)∼λ} , where x,y,z and t are real variables, α,β and λ are real constants and ∼∈{≤,≥} . The paper provides the first graph-based proofs of the 4-CSP tractability and elaborates algorithms for 4-CSP resolution based on the positive linear dependence theory, the hypergraph closure and the constraint propagation technique. Time and space complexities of the resolution algorithms are proved to be polynomial.

In this paper we present an algorithm for finding a minimum dominator coloring of orientations of paths. To date this is the first algorithm for dominator colorings of digraphs in any capacity. We prove that the algorithm always provides a minimum dominator coloring of an oriented path and show that it runs in O(n) time. The algorithm is available at this https URL.

The eternal vertex cover problem is a dynamic variant of the classical vertex cover problem. It is NP-hard to compute the eternal vertex cover number of graphs and known algorithmic results for the problem are very few. This paper presents a linear time recursive algorithm for computing the eternal vertex cover number of cactus graphs. Unlike other graph classes for which polynomial time algorithms for eternal vertex cover number are based on efficient computability of a known lower bound directly derived from minimum vertex cover, we show that it is a certain substructure property that helps the efficient computation of eternal vertex cover number of cactus graphs. An extension of the result to graphs in which each block is an edge, a cycle or a biconnected chordal graph is also presented.