Computer Science

Given an undirected weighted graph G(V,E) , a constrained sketch over a terminal set T⊂V is a subgraph G ′ that connects the terminal vertices while satisfying a given set of constraints. Examples include Steiner trees (preserving connectivity among T ) and subsetwise spanners (preserving shortest path distances up to a stretch factor). Multi-level constrained terminal sketches are generalizations in which terminal vertices require different levels or grades of service and each terminal pair is connected with edges of at least the minimum required level of the two terminals. This paper gives a flexible procedure for computing a broad class of multi-level graph sketches, which encompasses multi-level graph spanners, Steiner trees, and k --connected subgraphs as a few special cases. The proposed procedure is modular, i.e., it relies on availability of a single-level solver module (be it an oracle or approximation algorithm). One key result is that an ℓ --level constrained terminal sketch can be computed with logℓ queries of the solver module while producing feasible solutions with approximation guarantees independent of ℓ . Additionally, a new polynomial time algorithm for computing a subsetwise spanner is proposed. We show that for $k\in\N$, $\eps>0$, and T⊂V , there is a subsetwise $(2k-1)(1+\eps)$--spanner with total weight $O(|T|^\frac1kW(\ST(G,T)))$, where $W(\ST(G,T))$ is the weight of the Steiner tree of G over the subset T . This is the first algorithm and weight guarantee for a multiplicative subsetwise spanner for nonplanar graphs. Numerical experiments are also done to illustrate the performance of the proposed algorithms.

Given a graph G=(V,E) , a subgraph H is an \emph{additive +β spanner} if $\dist_H(u,v) \le \dist_G(u,v) + \beta$ for all u,v?�V . A \emph{pairwise spanner} is a spanner for which the above inequality only must hold for specific pairs P?�V?V given on input, and when the pairs have the structure P=S?S for some subset S?�V , it is specifically called a \emph{subsetwise spanner}. Spanners in unweighted graphs have been studied extensively in the literature, but have only recently been generalized to weighted graphs. In this paper, we consider a multi-level version of the subsetwise spanner in weighted graphs, where the vertices in S possess varying level, priority, or quality of service (QoS) requirements, and the goal is to compute a nested sequence of spanners with the minimum number of total edges. We first generalize the +2 subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance of this and several other algorithms for weighted additive spanners, both in terms of runtime and sparsity of output spanner, when applied at each level of the multi-level problem. Spanner sparsity is compared to the sparsest possible spanner satisfying the given error budget, obtained using an integer programming formulation of the problem. We run our experiments with respect to input graphs generated by several different random graph generators: Erd?s--Rényi, Watts--Strogatz, Barabási--Albert, and random geometric models. By analyzing our experimental results we developed a new technique of changing an initialization parameter value that provides better performance in practice.

In this paper, we study a primal and dual relationship about triangles: For any graph G , let ν(G) be the maximum number of edge-disjoint triangles in G , and τ(G) be the minimum subset F of edges such that G∖F is triangle-free. It is easy to see that ν(G)≤τ(G)≤3ν(G) , and in fact, this rather obvious inequality holds for a much more general primal-dual relation between k -hyper matching and covering in hypergraphs. Tuza conjectured in 1981 that τ(G)≤2ν(G) , and this question has received attention from various groups of researchers in discrete mathematics, settling various special cases such as planar graphs and generalized to bounded maximum average degree graphs, some cases of minor-free graphs, and very dense graphs. Despite these efforts, the conjecture in general graphs has remained wide open for almost four decades. In this paper, we provide a proof of a non-trivial consequence of the conjecture; that is, for every k≥2 , there exist a (multi)-set F⊆E(G):|F|≤2kν(G) such that each triangle in G overlaps at least k elements in F . Our result can be seen as a strengthened statement of Krivelevich's result on the fractional version of Tuza's conjecture (and we give some examples illustrating this.) The main technical ingredient of our result is a charging argument, that locally identifies edges in F based on a local view of the packing solution. This idea might be useful in further studying the primal-dual relations in general and the Tuza's conjecture in particular.

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running k multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker, and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when k random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of Ω((n/k)logn) on the stationary cover time, holding for any graph G and any 1≤k=o(nlogn) . Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

A word is said to be \emph{bordered} if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words (u,v) is said to be \emph{mutually bordered} if there exists a word that is a non-empty proper prefix of u and suffix of v , and there exists a word that is a non-empty proper suffix of u and prefix of v . In other words, (u,v) is mutually bordered if u overlaps v and v overlaps u . We give a recurrence for the number of mutually bordered pairs of words. Furthermore, we show that, asymptotically, there are c⋅ k 2n mutually bordered words of length- n over a k -letter alphabet, where c is a constant. Finally, we show that the expected shortest overlap between pairs of words is bounded above by a constant.

We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order q d−1 , we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field F q are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree n over F q . Finally, we present a construction of MOLS generated by LBCA with irreducible polynomials and prove the maximality of the resulting sets, as well as a lower bound which is asymptotically close to their actual number.

N-PAT is a new model-checking tool that supports the verification of nested-models, i.e. models whose behaviour depends on the results of verification tasks. In this paper, we describe its operation and discuss mechanisms that are tailored to the efficient verification of nested-models. Further, we motivate the advantages of N-PAT over traditional model-checking tools through a network security case study.

We consider a search problem on a 2 -dimensional infinite grid with a single mobile agent. The goal of the agent is to find her way home, which is located in a grid cell chosen by an adversary. Initially, the agent is provided with an infinite sequence of instructions, that dictate the movements performed by the agent. Each instruction corresponds to a movement to an adjacent grid cell and the set of instructions can be a function of the initial locations of the agent and home. The challenge of our problem stems from faults in the movements made by the agent. In every step, with some constant probability 0≤p≤1 , the agent performs a random movement instead of following the current instruction. This paper provides two results on this problem. First, we show that for some values of p , there does not exist any set of instructions that guide the agent home in finite expected time. Second, we complement this impossibility result with an algorithm that, for sufficiently small values of p , yields a finite expected hitting time for home. In particular, we show that for any p<1 , our approach gives a hitting rate that decays polynomially as a function of time. In that sense, our approach is far superior to a standard random walk in terms of hitting time. The main contribution and take-home message of this paper is to show that, for some value of 0.01139⋯<p<0.6554… , there exists a phase transition on the solvability of the problem.

A circle graph is a graph in which the adjacency of vertices can be represented as the intersection of chords of a circle. The problem of calculating the chromatic number is known to be NP-complete, even on circle graphs. In this paper, we propose a new integer linear programming formulation for a coloring problem on circle graphs. We also show that the linear relaxation problem of our formulation finds the fractional chromatic number of a given circle graph. As a byproduct, our formulation gives a polynomial-sized linear programming formulation for calculating the fractional chromatic number of a circle graph. We also extend our result to a formulation for a capacitated stowage stack minimization problem.

The concept of graph burning and burning number ( bn(G) ) of a graph G was introduced recently [1]. Graph burning models the spread of contagion (fire) in a graph in discrete time steps. bn(G) is the minimum time needed to burn a graph G .The problem is NP-complete. In this paper, we develop first heuristics to solve the problem in general (connected) graphs. In order to test the performance of our algorithms, we applied them on some graph classes with known burning number such as theta graphs, we tested our algorithms on DIMACS and BHOSLIB that are known benchmarks for NP-hard problems in graph theory. We also improved the upper bound for burning number on general graphs in terms of their distance to cluster. Then we generated a data set of 2000 random graphs with known distance to cluster and tested our heuristics on them.