Wireless sensor networks (WSNs) and other monitoring systems often use sensors of multiple strengths within the same network. This information is lost when the network is modeled by a typical non-weighted graph. We introduce the mixed-weight open locating-dominating set (mixed-weight OLD-set), an extension of the open locating-dominating set, that allows vertices in the graph to have different weights. We show that finding the minimum mixed-weight OLD-set is NP-complete, establish probabilistic bounds for the size of mixed-weight OLD-sets in random graphs, and provide simulation results supporting those bounds.

We study finite and infinite Sidon sets in N^d. The additive energy of two sets is used to obtain new upper bounds for the cardinalities of finite Sidon subsets of some sets as well as to provide short proofs of already known results. We also disprove a conjecture of Lindstrom on the largest cardinality of a Sidon set in [1,N]x[1,N] and relate it to a known conjecture of Vinogradov concerning the size of the smallest quadratic residue modulo a prime p. For infinite Sidon sets [email protected]?N^d, we prove that [email protected]?inf"n"->"~|[email protected]?[1,n]^d|n^d/logn>0. Finally, we show how to map infinite Sidon sets in N^d to N^d^^^' in an effective way. As an application, we find an explicit Sidon set of positive integers A such that |[email protected]?[1,n]|>=n^1^/^3^+^o^(^1^).

We solve an arithmetic problem due to Erdös and Freud (1986) investigated also by Freiman, Nathanson and Sárközy: How many elements from a given set of integers one must take to represent a power of 2 by their sum?

We prove that any $r$-coloring of the edges of $K_m$ contains a monochromatic even cycle, where $m = 3r + 1$ if $r$ is odd and $m =3r$ if $r$ is even. We also prove that $K_{m−1}$ has an $r$-coloring without monochromatic even cycles.

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger’s theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension. We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc. As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually ‘thick’ families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension. Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.

We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various areas, such as number theory, combinatorics and mathematical physics.

We consider the problem of generating a connected robust identifying code of a graph, by which we mean a subgraph with two properties: (i) it is connected, (ii) it is robust identifying, in the sense that the (subgraph-) induced neighborhoods of any two vertices differ by at least 2r + 1 vertices, where r is the robustness parameter. This particular formulation builds upon a rich literature on the identifying code problem but adds a property that is important for some practical networking applications. We concretely show that this modified problem is NP-complete and provide an otherwise efficient algorithm for computing it for an arbitrary graph. We demonstrate a connection between the the sizes of certain connected identifying codes and error-correcting code of a given distance. One consequence of this is that robustness leads to connectivity of identifying codes.

Un code identifiant est un ensemble de sommets d'un graphe tel que, d'une part, chaque sommet hors du code a un voisin dans le code (propriete de domination) et, d'autre part, tous les sommets ont un voisinage distinct a l'interieur du code (propriete de separation). Dans cette these, nous nous interessons a des aspects combinatoires et algorithmiques relatifs aux codes identifiants.Pour la partie combinatoire, nous etudions tout d'abord des questions extremales en donnant une caracterisation complete des graphes non-orientes finis ayant comme taille minimum de code identifiant leur ordre moins un. Nous caracterisons egalement les graphes diriges finis, les graphes non-orientes infinis et les graphes orientes infinis ayant pour seul code identifiant leur ensemble de sommets. Ces resultats repondent a des questions ouvertes precedemment etudiees dans la litterature.Puis, nous etudions la relation entre la taille minimum d'un code identifiant et le degre maximum d'un graphe, en particulier en donnant divers majorants pour ce parametre en fonction de l'ordre et du degre maximum. Ces majorants sont obtenus via deux techniques. L'une est basee sur la construction d'ensembles independants satisfaisant certaines proprietes, et l'autre utilise la combinaison de deux outils de la methode probabiliste : le lemme local de Lovasz et une borne de Chernoff. Nous donnons egalement des constructions de familles de graphes en relation avec ce type de majorants, et nous conjecturons que ces constructions sont optimales a une constante additive pres.Nous presentons egalement de nouveaux minorants et majorants pour la cardinalite minimum d'un code identifiant dans des classes de graphes particulieres. Nous etudions les graphes de maille au moins 5 et de degre minimum donne en montrant que la combinaison de ces deux parametres influe fortement sur la taille minimum d'un code identifiant. Nous appliquons ensuite ces resultats aux graphes reguliers aleatoires. Puis, nous donnons des minorants pour la taille d'un code identifiant des graphes d'intervalles et des graphes d'intervalles unitaires. Enfin, nous donnons divers minorants et majorants pour cette quantite lorsque l'on se restreint aux graphes adjoints. Cette derniere question est abordee via la notion nouvelle de codes arete-identifiants.Pour la partie algorithmique, il est connu que le probleme de decision associes a la notion de code identifiant est NP-complet meme pour des classes de graphes restreintes. Nous etendons ces resultats a d'autres classes de graphes telles que celles des graphes split, des co-bipartis, des adjoints ou d'intervalles. Pour cela nous proposons des reductions polynomiales depuis divers problemes algorithmiques classiques. Ces resultats montrent que dans beaucoup de classes de graphes, le probleme des codes identifiants est algorithmiquement plus difficile que des problems lies (tel que le probleme des ensembles dominants).Par ailleurs, nous completons les connaissances relatives a l'approximabilite du probleme d'optimisation associe aux codes identifiants. Nous etendons le resultat connu de NP-difficulte pour l'approximation de ce probleme avec un facteur sous-logarithmique (en fonction de la taille du graphe instance) aux graphes bipartis, split et co-bipartis, respectivement. Nous etendons egalement le resultat connu d'APX-completude pour les graphes de degre maximum donne a une sous-classe des graphes split, aux graphes bipartis de degre maximum 4 et aux graphes adjoints. Enfin, nous montrons l'existence d'un algorithme de type PTAS pour les graphes d'intervalles unitaires.

This paper presents a multi-phase algorithm to solve the global illumination problem. In the first phase dependent tests are applied, i.e. the random walks of different pixels are built from the same random numbers. The result of the first phase is used to identify homogeneous pixel groups in the image. The criterion of the formation of such groups is that averaging the color inside these groups should result in less error than handling the pixels independently. The second phase of the algorithm is a conventional random walk method that uses independent random samples in different pixels. The final result is calculated as the average of the results of the dependent tests and the lowpass filtered version of the independent tests. This low-pass filter averages the pixel values inside the homogenous groups. The algorithm takes advantage of the fact that the image can contain larger homogeneous regions that can be calculated from much less number of samples. Thus we can focus on those pixels where significant changes happen.

The problem of detection of a sparse number of damages in a structure is considered. The idea relies on the newly developed framework for compressed change detection [1], which leverages the unique covering property of identifying codes to detect statistical changes in stochastic phenomena. Since only a small number of damage scenarios can occur simultaneously, change detection is applied to responses of pairs of sensors that form an identifying code over a learned damage-sensing graph. An asymptotic analysis of the detection delay and the probability of detection of the proposed approach is provided when the number of damage scenarios is large.

The paper shows that almost every $n$-vertex graph is, uniquely, determined by its subgraphs with $3 \log_2{n}$ vertices. Therefore, for checking the isomorphism of almost every pair of $n$-vertex graphs, it is sufficient to compare their $3 \log_2{n}$-vertex subgraphs.

The metric dimension of a graph G is the minimum size of a subset S of vertices of G such that all other vertices are uniquely determined by their distances to the vertices in S . In this paper we investigate the metric dimension for two different models of random forests, in each case obtaining normal limit distributions for this parameter.

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$.

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.

The advent of frequency-agile radios holds the potential for improving the utilization of spectrum by allowing wireless systems to dynamically adapt their spectral footprint based on the local conditions. Whether this is done using market mechanisms or opportunistic approaches, the gains result from shifting some responsibility for avoiding harmful interference from the static "regulatory layer" to layers that can adapt at runtime. However, this leaves open the major problem of how to enforce/incentivize compliance and what the structure of "light-handed" regulation should be. This paper examines this and focuses on two specific technical problems: (a) determining whether harmful interference is occurring and (b) assigning liability by detecting the culprits. "Light-handed regulation" is interpreted as making unambiguous (and easily certified) requirements on the behavior of individual devices themselves while still preserving significant freedom to innovate at both the device and the system level. The basic idea explored here is to require the PHY/MAC layers of a cognitive radio to guarantee silence during certain time- slots where the exact sequence of required silences is given by a device/system -specific code. Thus, if a system is a source of harmful interference, the interference pattern itself contains the signature of the culprit. Nevertheless, identifying the unique interference pattern becomes challenging as both the number of cognitive radios and the number of harmful interferers increases. The key tradeoffs are explored in terms of the "regulatory overhead" (amount of enforced silence) needed to make guarantees. The quality of regulatory guarantees is expressed by the time required to convict the guilty, the number of potential cognitive systems that can be supported, and the number of simultaneously guilty parties that can be resolved. We show that the time to conviction need only scale logarithmically in the potential number of cognitive users. The base of the logarithm is determined by the amount of overhead that we will tolerate and how many guilty parties we want to be able to resolve.

Suppose E >O and k > I. We show that if II > n,,(k. a) and .4 L Z,, satisfies IAl > (( l/k) + E)n then there is a subset B L A such that 0 < 1 BI <I, and xhi B h = 0 (in 2,). The case k = 3 solves a problem of Stalley and another problem of Erdos and Graham. For an integer HI > 0, let snd(nt) denote the smallest integer that does not divide PI. We prove that for every I-: > 0 there is a constant c = ~(8:) z I, such that for every n > 0 and every rn, n ’ +’ 6 WI < n’llog’n every set A E j I, Z,..., II ) of cardinality IAl > c.n/snd(m) contains a subset Bcl.4 so that ChcB h =m. This is best possible, up to the constant C. In particular it implies that for every II there is an m such that every set A c (l,..., II jof cardinality IAl > cx/log II contains a subset BG A so that xhtS h = ,n, thus settling a problem of Erdds and Graham. ’ IYX7

Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every n-vertex distance-hereditary graph, equivalently a graph of rank-width at most 1, can be computed in time $${\mathcal {O}}(n^2\cdot \log _2 n)$$O(n2·log2n), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. As a corollary, we show that the path-width of every n-element matroid of branch-width at most 2 can be computed in time $${\mathcal {O}}(n^2\cdot \log _2 n)$$O(n2·log2n), provided that the matroid is given by its binary representation. To establish this result, we present a characterization of the linear rank-width of distance-hereditary graphs in terms of their canonical split decompositions. This characterization is similar to the known characterization of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex separation and search number of a graph. Inf. Comput., 113(1):50–79, 1994]. However, different from forests, it is non-trivial to relate substructures of the canonical split decomposition of a graph with some substructures of the given graph. We introduce a notion of ‘limbs’ of canonical split decompositions, which correspond to certain vertex-minors of the original graph, for the right characterization.

Let k,m,n⩾2 be integers. Let A be a subset of {0,1,…,n} with 0∈A and the greatest common divisor of all elements of A is 1. Suppose that |A|>1l+12-klmn+2l, where l=⌈k/m⌉. We prove that if m⩾3, or m=2 and k even, then there exists a power of m which can be represented as a sum of k elements (not necessarily distinct) of A.

Let @?, n and r be positive integers. Define F^n={0,1}^n. The Hamming distance between words x and y of F^n is denoted by d(x,y). The ball of radius r is defined as B"r(X)={y@?F^n|@?x@?X:d(x,y)@?r}, where X is a subset of F^n. A code C@?F^n is called (r,@?@?)-identifying if for all X,Y@?F^n such that |X|@?@?, |Y|@?@? and X Y, the sets B"r(X)@?C and B"r(Y)@?C are different. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. In this paper, we present various results concerning (r,@?@?)-identifying codes in the Hamming space F^n. First we concentrate on improving the lower bounds on (r,@?1)-identifying codes for r>1. Then we proceed by introducing new lower bounds on (r,@?@?)-identifying codes with @?>=2. We also prove that (r,@?@?)-identifying codes can be constructed from known ones using a suitable direct sum when @?>=2. Constructions for (r,@?2)-identifying codes with the best known cardinalities are also given.