26 research outputs found

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    Mobile Ad Hoc Networks

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    Guiding readers through the basics of these rapidly emerging networks to more advanced concepts and future expectations, Mobile Ad hoc Networks: Current Status and Future Trends identifies and examines the most pressing research issues in Mobile Ad hoc Networks (MANETs). Containing the contributions of leading researchers, industry professionals, and academics, this forward-looking reference provides an authoritative perspective of the state of the art in MANETs. The book includes surveys of recent publications that investigate key areas of interest such as limited resources and the mobility of mobile nodes. It considers routing, multicast, energy, security, channel assignment, and ensuring quality of service. Also suitable as a text for graduate students, the book is organized into three sections: Fundamentals of MANET Modeling and Simulation—Describes how MANETs operate and perform through simulations and models Communication Protocols of MANETs—Presents cutting-edge research on key issues, including MAC layer issues and routing in high mobility Future Networks Inspired By MANETs—Tackles open research issues and emerging trends Illustrating the role MANETs are likely to play in future networks, this book supplies the foundation and insight you will need to make your own contributions to the field. It includes coverage of routing protocols, modeling and simulations tools, intelligent optimization techniques to multicriteria routing, security issues in FHAMIPv6, connecting moving smart objects to the Internet, underwater sensor networks, wireless mesh network architecture and protocols, adaptive routing provision using Bayesian inference, and adaptive flow control in transport layer using genetic algorithms

    FPT-approximation for FPT problems

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    Over the past decade, many results have focused on the design of parameterized approximation algorithms for W[1]-hard problems. However, there are fundamental problems within the class FPT for which the best known algorithms have seen no progress over the course of the decade. In this paper, we expand the study of FPT-approximation and initiate a systematic study of FPT-approximation for problems that are in FPT. We design FPT-approximation algorithms for problems that are in FPT, with running times that are significantly faster than the corresponding best known FPT-algorithm, and while achieving approximation ratios that are significantly better than what is possible in polynomial time

    Extremal problems on special graph colorings

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    In this thesis, we study several extremal problems on graph colorings. In particular, we study monochromatic connected matchings, paths, and cycles in 2-edge colored graphs, packing colorings of subcubic graphs, and directed intersection number of digraphs. In Chapter 2, we consider monochromatic structures in 2-edge colored graphs. A matching M in a graph G is connected if all the edges of M are in the same component of G. Following Łuczak, there are a number of results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in 2-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings, which is used to find necessary and sufficient conditions on the existence of monochromatic paths and cycles: for every fixed s and large n, we describe all values of n_1, ...,n_s such that for every 2-edge-coloring of the complete s-partite graph K_{n_1, ...,n_s} there exists a monochromatic (i) cycle C_{2n} with 2n vertices, (ii) cycle C_{at least 2n} with at least 2n vertices, (iii) path P_{2n} with 2n vertices, and (iv) path P_{2n+1} with 2n+1 vertices. Our results also imply for large n of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every 2-edge-coloring of the complete 3-partite graph K_{n,n,n} there is a monochromatic path P_{2n+1}. Moreover, we prove that for every sufficiently large n, if n = 3t+r where r in {0,1,2} and G is an n-vertex graph with minimum degree at least (3n-1)/4, then for every 2-edge-coloring of G, either there are cycles of every length {3, 4, 5, ..., 2t+r} of the same color, or there are cycles of every even length {4, 6, 8, ..., 2t+2} of the same color. This result is tight and implies the conjecture of Schelp that for every sufficiently large n, every (3n-1)-vertex graph G with minimum degree larger than 3|V(G)|/4, in each 2-edge-coloring of G there exists a monochromatic path P_{2n} with 2n vertices. It also implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for every positive integer n of the form n=3t+r where r in {0,1,2} and every n-vertex graph G with minimum degree at least 3n/4, in each 2-edge-coloring of G there exists a monochromatic cycle of length at least 2t+r. In Chapter 3, we consider a collection of special vertex colorings called packing colorings. For a sequence of non-decreasing positive integers S = (s_1, ..., s_k), a packing S-coloring is a partition of V(G) into sets V_1, ..., V_k such that for each integer i in {1, ..., k} the distance between any two distinct x,y in V_i is at least s_i+1. The smallest k such that G has a packing (1,2, ..., k)-coloring is called the packing chromatic number of G and is denoted by \chi_p(G). The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We show that for every fixed k and g at least 2k+2, almost every n-vertex cubic graph of girth at least g has the packing chromatic number greater than k, which answers the previous question in the negative. Moreover, we work towards the conjecture of Brešar, Klavžar, Rall and Wash that the packing chromatic number of 1-subdivision of subcubic graphs are bounded above by 5. In particular, we show that every subcubic graph is (1,1,2,2,3,3,k)-colorable for every integer k at least 4 via a coloring in which color k is used at most once, every 2-degenerate subcubic graph is (1,1,2,2,3,3)-colorable, and every subcubic graph with maximum average degree less than 30/11 is packing (1,1,2,2)-colorable. Furthermore, while proving the packing chromatic number of subcubic graphs is unbounded, we also consider improving upper bound on the independence ratio, alpha(G)/n, of cubic n-vertex graphs of large girth. We show that ``almost all" cubic labeled graphs of girth at least 16 have independence ratio at most 0.454. In Chapter 4, we introduce and study the directed intersection representation of digraphs. A directed intersection representation is an assignment of a color set to each vertex in a digraph such that two vertices form an edge if and only if their color sets share at least one color and the tail vertex has a strictly smaller color set than the head. The smallest possible size of the union of the color sets is defined to be the directed intersection number (DIN). We show that the directed intersection representation is well-defined for all directed acyclic graphs and the maximum DIN among all n vertex acyclic digraphs is at most 5n^2/8 + O(n) and at least 9n^2/16 + O(n)

    A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

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    Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

    Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm

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    Many of the most celebrated and influential results in graph coloring, such as Brooks' Theorem and Vizing's Theorem, relate a graph's chromatic number to its clique number or maximum degree. Currently, several of the most important and enticing open problems in coloring, such as Reed's ω,Δ,χ\omega, \Delta, \chi Conjecture, follow this theme. This thesis both broadens and deepens this classical paradigm. In Part~1, we tackle list-coloring problems in which the number of colors available to each vertex vv depends on its degree, denoted d(v)d(v), and the size of the largest clique containing it, denoted ω(v)\omega(v). We make extensive use of the probabilistic method in this part. We conjecture the ``list-local version'' of Reed's Conjecture, that is every graph is LL-colorable if LL is a list-assignment such that L(v)(1ε)(d(v)+1)+εω(v))|L(v)| \geq \lceil (1 - \varepsilon)(d(v) + 1) + \varepsilon\omega(v))\rceil for each vertex vv and ε1/2\varepsilon \leq 1/2, and we prove this for ε1/330\varepsilon \leq 1/330 under some mild additional assumptions. We also conjecture the ``mad\mathrm{mad} version'' of Reed's Conjecture, even for list-coloring. That is, for ε1/2\varepsilon \leq 1/2, every graph GG satisfies \chi_\ell(G) \leq \lceil (1 - \varepsilon)(\mad(G) + 1) + \varepsilon\omega(G)\rceil, where mad(G)\mathrm{mad}(G) is the maximum average degree of GG. We prove this conjecture for small values of ε\varepsilon, assuming ω(G)mad(G)log10mad(G)\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G). We actually prove a stronger result that improves bounds on the density of critical graphs without large cliques, a long-standing problem, answering a question of Kostochka and Yancey. In the proof, we use a novel application of the discharging method to find a set of vertices for which any precoloring can be extended to the remainder of the graph using the probabilistic method. Our result also makes progress towards Hadwiger's Conjecture: we improve the best known bound on the chromatic number of KtK_t-minor free graphs by a constant factor. We provide a unified treatment of coloring graphs with small clique number. We prove that for Δ\Delta sufficiently large, if GG is a graph of maximum degree at most Δ\Delta with list-assignment LL such that for each vertex vV(G)v\in V(G), L(v)72d(v)min{ln(ω(v))ln(d(v)),ω(v)ln(ln(d(v)))ln(d(v)),log2(χ(G[N(v)])+1)ln(d(v))}|L(v)| \geq 72\cdot d(v)\min\left\{\sqrt{\frac{\ln(\omega(v))}{\ln(d(v))}}, \frac{\omega(v)\ln(\ln(d(v)))}{\ln(d(v))}, \frac{\log_2(\chi(G[N(v)]) + 1)}{\ln(d(v))}\right\} and d(v)ln2Δd(v) \geq \ln^2\Delta, then GG is LL-colorable. This result simultaneously implies three famous results of Johansson from the 90s, as well as the following new bound on the chromatic number of any graph GG with ω(G)ω\omega(G)\leq \omega and Δ(G)Δ\Delta(G)\leq \Delta for Δ\Delta sufficiently large: χ(G)72ΔlnωlnΔ.\chi(G) \leq 72\Delta\sqrt{\frac{\ln\omega}{\ln\Delta}}. In Part~2, we introduce and develop the theory of fractional coloring with local demands. A fractional coloring of a graph is an assignment of measurable subsets of the [0,1][0, 1]-interval to each vertex such that adjacent vertices receive disjoint sets, and we think of vertices ``demanding'' to receive a set of color of comparatively large measure. We prove and conjecture ``local demands versions'' of various well-known coloring results in the ω,Δ,χ\omega, \Delta, \chi paradigm, including Vizing's Theorem and Molloy's recent breakthrough bound on the chromatic number of triangle-free graphs. The highlight of this part is the ``local demands version'' of Brooks' Theorem. Namely, we prove that if GG is a graph and f:V(G)[0,1]f : V(G) \rightarrow [0, 1] such that every clique KK in GG satisfies vKf(v)1\sum_{v\in K}f(v) \leq 1 and every vertex vV(G)v\in V(G) demands f(v)1/(d(v)+1/2)f(v) \leq 1/(d(v) + 1/2), then GG has a fractional coloring ϕ\phi in which the measure of ϕ(v)\phi(v) for each vertex vV(G)v\in V(G) is at least f(v)f(v). This result generalizes the Caro-Wei Theorem and improves its bound on the independence number, and it is tight for the 5-cycle

    36th International Symposium on Theoretical Aspects of Computer Science: STACS 2019, March 13-16, 2019, Berlin, Germany

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    Walking Through Waypoints

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    We initiate the study of a fundamental combinatorial problem: Given a capacitated graph G=(V,E)G=(V,E), find a shortest walk ("route") from a source sVs\in V to a destination tVt\in V that includes all vertices specified by a set WV\mathscr{W}\subseteq V: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable

    Multicoloring of graphs to secure a secret

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    Vertex coloring and multicoloring of graphs are a well known subject in graph theory, as well as their applications. In vertex multicoloring, each vertex is assigned some subset of a given set of colors. Here we propose a new kind of vertex multicoloring, motivated by the situation of sharing a secret and securing it from the actions of some number of attackers. We name the multicoloring a highly a-resistant vertex k-multicoloring, where a is the number of the attackers, and k the number of colors. For small values a we determine what is the minimal number of vertices a graph must have in order to allow such a coloring, and what is the minimal number of colors needed
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