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A mobile agent navigating along edges of a simple connected graph, either finite or countably infinite, has to find an inert target (treasure) hidden in one of the nodes. This task is known as treasure hunt. The agent has no a priori knowledge of the graph, of the location of the treasure or of the initial distance to it. The cost of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until finding the treasure. Awerbuch, Betke, Rivest and Singh [3] considered graph exploration and treasure hunt for finite graphs in a restricted model where the agent has a fuel tank that can be replenished only at the starting node $s$. The size of the tank is $B=2(1+\alpha)r$, for some positive real constant $\alpha$, where $r$, called the radius of the graph, is the maximum distance from $s$ to any other node. The tank of size $B$ allows the agent to make at most $\lfloor B\rfloor$ edge traversals between two consecutive visits at node $s$. Let $e(d)$ be the number of edges whose at least one extremity is at distance less than $d$ from $s$. Awerbuch, Betke, Rivest and Singh [3] conjectured that it is impossible to find a treasure hidden in a node at distance at most $d$ at cost nearly linear in $e(d)$. We first design a deterministic treasure hunt algorithm working in the model without any restrictions on the moves of the agent at cost $\mathcal{O}(e(d) \log d)$, and then show how to modify this algorithm to work in the model from [3] with the same complexity. Thus we refute the above twenty-year-old conjecture. We observe that no treasure hunt algorithm can beat cost $\Theta(e(d))$ for all graphs and thus our algorithms are also almost optimal.

$k$-Approximate distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the $k$-approximation of the distance between any two vertices $u$ and $v$ can be determined efficiently by merely inspecting the labels of $u$ and $v$, without using any other information. One of the important problems is finding natural classes of graphs admitting exact or approximate distance labeling schemes with labels of polylogarithmic size. In this paper, we describe a $4$-approximate distance labeling scheme forthe class of $K_4$-free bridged graphs. This scheme uses labels of poly-logarithmic length $O(\log n^3)$ allowing a constant decoding time. Given the labels of two vertices $u$ and $v$, the decoding function returnsa value between the exact distance $d_G(u,v)$ and its quadruple $4d_G(u,v)$., Comment: 29 pages, 21 figures

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices $u$ and $v$ can be determined efficiently by merely inspecting the labels of $u$ and $v$, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such a way that given the labels of a source node and a destination node, it is possible to compute efficiently the port number of the edge from the source that heads in the direction of the destination. One of important problems is finding natural classes of graphs admitting distance and/or routing labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on $n$ nodes enjoys distance and routing labeling schemes with labels of $O(\log^3 n)$ bits., Comment: 34 pages, 10 figures

Let $\mathcal S$ be a family of subsets of a set $X$ of cardinality $m$ and $\text{VC-dim}(\mathcal S)$ be the Vapnik-Chervonenkis dimension of $\mathcal S$. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if $G_1(\mathcal S)=(V,E)$ is the subgraph of the hypercube $Q_m$ induced by $\mathcal S$ (called the 1-inclusion graph of $\mathcal S$), then $\frac{|E|}{|V|}\le \text{VC-dim}({\mathcal S})$. Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation. In this note, we adapt the shifting technique to prove that if $\mathcal S$ is an arbitrary set family and $G_{1,2}(\mathcal S)=(V,E)$ is the 1,2-inclusion graph of $\mathcal S$ (i.e., the subgraph of the square $Q^2_m$ of the hypercube $Q_m$ induced by $\mathcal S$), then $\frac{|E|}{|V|}\le \binom{d}{2}$, where $d:=\text{cVC-dim}^*(\mathcal S)$ is the clique-VC-dimension of $\mathcal S$ (which we introduce in this paper). The 1,2-inclusion graphs are exactly the subgraphs of halved cubes and comprise subgraphs of Johnson graphs as a subclass., Comment: 15 pages, 4 figures

In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs $G$ of Cartesian products $G_1\times\cdots\times G_m$ of arbitrary connected graphs. Namely, we show that $\frac{|E(G)|}{|V(G)|}\le \lceil 2\max\{ \text{dens}(G_1),\ldots,\text{dens}(G_m)\} \rceil\log|V(G)|$, where $\text{dens}(H)$ is the maximum ratio $\frac{|E(H')|}{|V(H')|}$ over all subgraphs $H'$ of $H$. We introduce the notions of VC-dimension $\text{VC-dim}(G)$ and VC-density $\text{VC-dens}(G)$ of a subgraph $G$ of a Cartesian product $G_1\times\cdots\times G_m$, generalizing the classical Vapnik-Chervonenkis dimension of set-families (viewed as subgraphs of hypercubes). We prove that if $G_1,\ldots,G_m$ belong to the class ${\mathcal G}(H)$ of all finite connected graphs not containing a given graph $H$ as a minor, then for any subgraph $G$ of $G_1\times\cdots\times G_m$ a sharper inequality $\frac{|E(G)|}{|V(G)|}\le \text{VC-dim}(G)\alpha(H)$ holds, where $\alpha(H)$ is the density of the graphs from ${\mathcal G}(H)$. We refine and sharpen those two results to several specific graph classes. We also derive upper bounds (some of them polylogarithmic) for the size of adjacency labeling schemes of subgraphs of Cartesian products., Comment: 20 pages, 17 figures

A graph environment must be explored by a collection of mobile robots. Some of the robots, a priori unknown, may turn out to be unreliable. The graph is weighted and each node is assigned a deadline. The exploration is successful if each node of the graph is visited before its deadline by a reliable robot. The edge weight corresponds to the time needed by a robot to traverse the edge. Given the number of robots which may crash, is it possible to design an algorithm, which will always guarantee the exploration, independently of the choice of the subset of unreliable robots by the adversary? We find the optimal time, during which the graph may be explored. Our approach permits to find the maximal number of robots, which may turn out to be unreliable, and the graph is still guaranteed to be explored. We concentrate on line graphs and rings, for which we give positive results. We start with the case of the collections involving only reliable robots. We give algorithms finding optimal times needed for exploration when the robots are assigned to fixed initial positions as well as when such starting positions may be determined by the algorithm. We extend our consideration to the case when some number of robots may be unreliable. Our most surprising result is that solving the line exploration problem with robots at given positions, which may involve crash-faulty ones, is NP-hard. The same problem has polynomial solutions for a ring and for the case when the initial robots' positions on the line are arbitrary. The exploration problem is shown to be NP-hard for star graphs, even when the team consists of only two reliable robots.

We consider the problem of collectively delivering some message from a specified source to a designated target location in a graph, using multiple mobile agents. Each agent has a limited energy which constrains the distance it can move. Hence multiple agents need to collaborate to move the message, each agent handing over the message to the next agent to carry it forward. Given the positions of the agents in the graph and their respective budgets, the problem of finding a feasible movement schedule for the agents can be challenging. We consider two variants of the problem: in non-returning delivery, the agents can stop anywhere; whereas in returning delivery, each agent needs to return to its starting location, a variant which has not been studied before. We first provide a polynomial-time algorithm for returning delivery on trees, which is in contrast to the known (weak) NP-hardness of the non-returning version. In addition, we give resource-augmented algorithms for returning delivery in general graphs. Finally, we give tight lower bounds on the required resource augmentation for both variants of the problem. In this sense, our results close the gap left by previous research., Comment: 19 pages. An extended abstract of this paper was published at the 23rd International Colloquium on Structural Information and Communication Complexity 2016, SIROCCO'16

A set of identical, mobile agents is deployed in a weighted network. Each agent has a battery -- a power source allowing it to move along network edges. An agent uses its battery proportionally to the distance traveled. We consider two tasks : convergecast, in which at the beginning, each agent has some initial piece of information, and information of all agents has to be collected by some agent; and broadcast in which information of one specified agent has to be made available to all other agents. In both tasks, the agents exchange the currently possessed information when they meet. The objective of this paper is to investigate what is the minimal value of power, initially available to all agents, so that convergecast or broadcast can be achieved. We study this question in the centralized and the distributed settings. In the centralized setting, there is a central monitor that schedules the moves of all agents. In the distributed setting every agent has to perform an algorithm being unaware of the network. In the centralized setting, we give a linear-time algorithm to compute the optimal battery power and the strategy using it, both for convergecast and for broadcast, when agents are on the line. We also show that finding the optimal battery power for convergecast or for broadcast is NP-hard for the class of trees. On the other hand, we give a polynomial algorithm that finds a 2-approximation for convergecast and a 4-approximation for broadcast, for arbitrary graphs. In the distributed setting, we give a 2-competitive algorithm for convergecast in trees and a 4-competitive algorithm for broadcast in trees. The competitive ratio of 2 is proved to be the best for the problem of convergecast, even if we only consider line networks. Indeed, we show that there is no (2 -- $\epsilon$)-competitive algorithm for convergecast or for broadcast in the class of lines, for any $\epsilon$ \textgreater{} 0.

For a family of geometric objects in the plane $\mathcal{F}=\{S_1,\ldots,S_n\}$, define $\chi(\mathcal{F})$ as the least integer $\ell$ such that the elements of $\mathcal{F}$ can be colored with $\ell$ colors, in such a way that any two intersecting objects have distinct colors. When $\mathcal{F}$ is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most $k$ pseudo-disks, it can be proven that $\chi(\mathcal{F})\le 3k/2 + o(k)$ since the problem is equivalent to cyclic coloring of plane graphs. In this paper, we study the same problem when pseudo-disks are replaced by a family $\mathcal{F}$ of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of $\mathcal{F}$ are only allowed to "touch" each other. Such a family is said to be $k$-touching if no point of the plane is contained in more than $k$ elements of $\mathcal{F}$. We give bounds on $\chi(\mathcal{F})$ as a function of $k$, and in particular we show that $k$-touching segments can be colored with $k+5$ colors. This partially answers a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems of strings., Comment: 19 pages. A preliminary version of this work appeared in the proceedings of EuroComb'09 under the title "Coloring a set of touching strings"