9 research outputs found

    Asynchronous Collective Tree Exploration by Tree-Mining

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    We investigate the problem of collaborative tree exploration with complete communication introduced by [FGKP06], in which a group of kk agents is assigned to collectively go through all edges of an unknown tree in an efficient manner and then return to the origin. The agents have unrestricted communication and computation capabilities. The algorithm's runtime is typically compared to the cost of offline traversal, which is at least max{2n/k,2D}\max\{2n/k,2D\} where nn is the number of nodes and DD is the tree depth. Since its introduction, two types of guarantee have emerged on the topic: the first is of the form r(k)(n/k+D)r(k)(n/k+D), where r(k)r(k) is called the competitive ratio, and the other is of the form 2n/k+f(k,D)2n/k+f(k,D), where f(k,D)f(k,D) is called the competitive overhead. In this paper, we present the first algorithm with linear-in-DD competitive overhead, thereby reconciling both approaches. Specifically, our bound is in 2n/k+O(klog2kD)2n/k + O(k^{\log_2 k} D) and thus leads to a competitive ratio in O(k/exp(0.8lnk))O(k/\exp(0.8\sqrt{\ln k})). This is the first improvement over the O(k/lnk)O(k/\ln k)-competitive algorithm known since the introduction of the problem in 2004. Our algorithm is obtained for an asynchronous generalization of collective tree exploration (ACTE). It is an instance of a general class of locally-greedy exploration algorithms that we define. We show that the additive overhead analysis of locally-greedy algorithms can be seen through the lens of a 2-player game that we call the tree-mining game and that could be of independent interest

    Computing Square Colorings on Bounded-Treewidth and Planar Graphs

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    A square coloring of a graph GG is a coloring of the square G2G^2 of GG, that is, a coloring of the vertices of GG such that any two vertices that are at distance at most 22 in GG receive different colors. We investigate the complexity of finding a square coloring with a given number of qq colors. We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time n2tw+4+O(1)n^{2^{\operatorname{tw} + 4}+O(1)} for graphs of treewidth at most tw\operatorname{tw}. The somewhat unusual exponent 2tw2^{\operatorname{tw}} in the running time is essentially optimal: we show that for any ϵ>0\epsilon>0, there is no algorithm with running time f(tw)n(2ϵ)twf(\operatorname{tw})n^{(2-\epsilon)^{\operatorname{tw}}} unless the Exponential-Time Hypothesis (ETH) fails. We also show that the square coloring problem is NP-hard on planar graphs for any fixed number q4q \ge 4 of colors. Our main algorithmic result is showing that the problem (when the number of colors qq is part of the input) can be solved in subexponential time 2O(n2/3logn)2^{O(n^{2/3}\log n)} on planar graphs. The result follows from the combination of two algorithms. If the number qq of colors is small (n1/3\le n^{1/3}), then we can exploit a treewidth bound on the square of the graph to solve the problem in time 2O(qnlogn)2^{O(\sqrt{qn}\log n)}. If the number of colors is large (n1/3\ge n^{1/3}), then an algorithm based on protrusion decompositions and building on our result for the bounded-treewidth case solves the problem in time 2O(nlogn/q)2^{O(n\log n/q)}.Comment: 72 pages, 15 figures, full version of a paper accepted at SODA 202

    Computing Square Colorings on Bounded-Treewidth and Planar Graphs

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    A {\em square coloring} of a graph GG is a coloring of the square G2G^2 of GG, that is, a coloring of the vertices of GG such that any two vertices that are at distance at most 22 in GG receive different colors. We investigate the complexity of finding a square coloring with a given number of qq colors. We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time n^{2^{\ttw + 4}+O(1)} for graphs of treewidth at most \ttw. The somewhat unusual exponent 2^\ttw in the running time is essentially optimal: we show that for any ϵ>0\epsilon>0, there is no algorithm with running time f(\ttw)n^{(2-\epsilon)^\ttw} unless the Exponential-Time Hypothesis (ETH) fails. We also show that the square coloring problem is NP-hard on planar graphs for any fixed number q4q \ge 4 of colors. Our main algorithmic result is showing that the problem (when the number of colors qq is part of the input) can be solved in subexponential time 2O(n2/3logn)2^{O(n^{2/3}\log n)} on planar graphs. The result follows from the combination of two algorithms. If the number qq of colors is small (n1/3\le n^{1/3}), then we can exploit a treewidth bound on the square of the graph to solve the problem in time 2O(qnlogn)2^{O(\sqrt{qn}\log n)}. If the number of colors is large (n1/3\ge n^{1/3}), then an algorithm based on protrusion decompositions and building on our result for the bounded-treewidth case solves the problem in time 2O(nlogn/q)2^{O(n\log n/q)}

    Eighth International Symposium “Monitoring of Mediterranean Coastal Areas. Problems and Measurement Techniques”

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    The 8th International Symposium "Monitoring of Mediterranean Coastal Areas. Problems and Measurements Techniques" was organized by CNR-IBE in collaboration with FCS Foundation, and Natural History Museum of the Mediterranean and under the patronage of University of Florence, Accademia dei Geogofili, Tuscany Region and Livorno Province. It is the occasion in which scholars can illustrate and exchange their activities and innovative proposals, with common aims to promote actions to preserve coastal marine environment. Considering Symposium interdisciplinary nature, the Scientific Committee, underlining this holistic view of Nature, decided to celebrate Alexander von Humboldt; a nature scholar that proposed the organic and inorganic nature’s aspects as a single system. It represents a sign of continuity considering that in-presence Symposium could not be carried out due to the COVID-19 pandemic restrictions. Subjects are related to coastal topics: morphology; flora and fauna; energy production; management and integrated protection; geography and landscape, cultural heritage and environmental assets, legal and economic aspects

    Proceedings of Eighth International Symposium “Monitoring of Mediterranean Coastal Areas. Problems and Measurement Techniques”

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    The 8th International Symposium "Monitoring of Mediterranean Coastal Areas. Problems and Measurements Techniques" was organized by CNR-IBE in collaboration with FCS Foundation, and Natural History Museum of the Mediterranean and under the patronage of University of Florence, Accademia dei Geogofili, Tuscany Region and Livorno Province. It is the occasion in which scholars can illustrate and exchange their activities and innovative proposals, with common aims to promote actions to preserve coastal marine environment. Considering Symposium interdisciplinary nature, the Scientific Committee, underlining this holistic view of Nature, decided to celebrate Alexander von Humboldt; a nature scholar that proposed the organic and inorganic nature’s aspects as a single system. It represents a sign of continuity considering that in-presence Symposium could not be carried out due to the COVID-19 pandemic restrictions. Subjects are related to coastal topics: morphology; flora and fauna; energy production; management and integrated protection; geography and landscape, cultural heritage and environmental assets, legal and economic aspects

    Partitioning into Isomorphic or Connected Subgraphs

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    This thesis deals mainly with the partitioning and connectedness of graphs. First, we show that the problem of partitioning the nodes of a graph into a specific number of subsets such that the induced subgraphs on these sets are isomorphic to one another is NP-complete. If the induced subgraphs have to be connected, the problem remains NP-complete. Then we inspect some special graph classes for which the problem is solvable in polynomial time. Afterwards, we deal with the problem of defining a polytope by incidence vectors of nodes, which induce a connected graph. We inspect some facet-defining inequalities and their general structure. For some graph classes we state the full description. We then proceed to the problem of partitioning the nodes of a graph into a given number of parts such that the induced graphs are connected. For the corresponding polytope we show the dimension and some facet defining inequalities. This theoretical inspection is advanced by the problem of partitioning a graph into different parts such that the parts induce a connected graph in order to maximize the induced cut. We introduce different ideas for solving this problem in SCIP and show the numerical results. This leads to interesting problems on MIPs in general. As the problem in literature generally deals with the feasible region, we focus on the objective function. To do that, we inspect the problem of finding MIPs for problems with nonlinear objective functions. We discuss properties and requirements showing the existence or non-existence of particular formulations. Lastly, we inspect the problem of partitioning the nodes of a graph such that all but one class are isomorphic. This problem becomes interesting if the part not inducing the isomorphism is minimized. For this purpose we also introduce a technique, which generates the parts by brute-force. Instead of partitioning the graph into isomorphic parts, we proceed to the problem of similar graphs. In this case we inspect different similarities and show algorithms which implement these

    Optimisation problems in wireless sensor networks : Local algorithms and local graphs

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    This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating–dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating–dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs – these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating–dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs
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