75 research outputs found

    Approximation Algorithms for Multicoloring Planar Graphs and Powers of Square and Triangular Meshes

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    International audienceA multicoloring of a weighted graph G is an assignment of sets of colors to the vertices of G so that two adjacent vertices receive two disjoint sets of colors. A multicoloring problem on G is to find a multicoloring of G. In particular, we are interested in a minimum multicoloring that uses the least total number of colors. The main focus of this work is to obtain upper bounds on the weighted chromatic number of some classes of graphs in terms of the weighted clique number. We first propose an 11/6-approximation algorithm for multicoloring any weighted planar graph. We then study the multicoloring problem on powers of square and triangular meshes. Among other results, we show that the infinite triangular mesh is an induced subgraph of the fourth power of the infinite square mesh and we present 2-approximation algorithms for multicoloring a power square mesh and the second power of a triangular mesh, 3-approximation algorithms for multicoloring powers of semi-toroidal meshes and of triangular meshes and 4-approximation algorithm for multicoloring the power of a toroidal mesh. We also give similar algorithms for the Cartesian product of powers of paths and of cycles

    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 aa-resistant vertex kk-multicoloring, where aa is the number of the attackers, and kk the number of colors. For small values aa 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.Comment: 19 pages, 5 figure

    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

    Capacity Enhancement in 60 GHz Based D2D Networks by Relay Selection and Scheduling

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    Millimeter-wave or 60 GHz communication is a promising technology that enables data rates in multigigabits. However, its tremendous propagation loss and signal blockage may severely affect the network throughput. In current data-centric device-to-device (D2D) communication networks, the devices with intended data communications usually lay in close proximity, unlike the case in voice-centric networks. So the network can be visualized as a naturally formed groups of devices. In this paper, we jointly consider resource scheduling and relay selection to improve network capacity in 60 GHz based D2D networks. Two types of transmission scenarios are considered in wireless personal area networks (WPANs), intra and intergroup. A distributed receiver based relay selection scheme is proposed for intragroup transmission, while a distance based relay selection scheme is proposed for intergroup transmission. The outage analysis of our proposed relay selection scheme is provided along with the numerical results. We then propose a concurrent transmission scheduling algorithm based on vertex coloring technique. The proposed scheduling algorithm employs time and space division in mmWave WPANs. Using vertex multicoloring, we allow transmitter-receiver (Tx-Rx) communication pairs to span over more colors, enabling better time slot utilization. We evaluate our scheduling algorithm in single-hop and multihop scenarios and discover that it outperforms other schemes by significantly improving network throughput

    Fast algorithms for two scheduling problems

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    The thesis deals with problems from two distint areas of scheduling theory. In the first part we consider the preemptive Sum Multicoloring (pSMC) problem. In an instance of pSMC, pairwise conflicting jobs are represented by a conflict graph, and the time demands of jobs are given by integer weights on the nodes. The goal is to schedule the jobs in such a way that the sum of their finish times is minimized. We give the first polynomial algorithm for pSMC on paths and cycles, running in time O(min(n², n log p)), where n is the number of nodes and p is the largest time demand. This answers a question raised by Halldórsson et al. [51] about the hardness of this problem. Our result identifies a gap between binary-tree conflict graphs - where the question is NP-hard - and paths. In the second part of the thesis we consider the problem of scheduling n jobs on m machines of different speeds s.t. the makespan is minimized (Q||C_max). We provide a fast and simple, deterministic monotone 2.8-approximation algorithm for Q||C_max. Monotonicity is relevant in the context of truthful mechanisms: when each machine speed is only known to the machine itself, we need to motivate that machines "declare" their true speeds to the scheduling mechanism. So far the best deterministic truthful mechanism that is polynomial in n and m; was a 5-approximation by Andelman et al. [3]. A randomized 2-approximation method, satisfying a weaker definition of truthfulness, was given by Archer and Tardos [4, 5]. As a core result, we prove the conjecture of Auletta et al. [8], that the greedy list scheduling algorithm Lpt is monotone if machine speeds are all integer powers of two (2-divisible machines). Proving the worst case bound of 2.8 involves studying the approximation ratio of Lpt on 2-divisible machines. As a side result, we obtain a tight bound of (sqrt(3) + 1)/2 ~= 1.3660 for the "one fast machine" case, i.e., when m - 1 machine speeds are equal, and there is only one faster machine. In this special case the best previous lower and upper bounds were 4/3 - epsilon < Lpt/Opt <= 3/2 - 1/(2m), shown in a classic paper by Gonzalez et al. [42]. Moreover, the authors of [42] conjectured the bound 4/3 to be tight. Thus, the results of the thesis answer three open questions in scheduling theory.In dieser Arbeit befassen wir uns mit Problemen aus zwei verschiedenen Teilgebieten der Scheduling-Theorie. Im ersten Teil betrachten wir das sog. preemptive Sum Multicoloring (pSMC) Problem. In einer Eingabe für pSMC werden paarweise Konflikte zwischen Jobs durch einen Konfliktgraphen repräsentiert; der Zeitbedarf eines Jobs ist durch ein ganzzahliges, positives Gewicht in seinem jeweiligen Knoten gegeben. Die Aufgabe besteht darin, die Jobs so den Maschinen zuzuweisen, dass die Summe ihrer Maschinenlaufzeiten minimiert wird. Wir liefern den ersten Algorithmus für pSMC auf Pfaden und Kreisen mit polynomieller Laufzeit; er benötigt O(min(n², n log p)) Zeit, wobei n die Anzahl der Jobs und p die maximale Zeitanforderung darstellen. Dies liefert eine Antwort auf die von Halldórsson et al. [51] aufgeworfene Frage der Komplexitätsklasse von pSMC. Unser Resultat identifiziert eine Diskrepanz zwischen der Komplexität auf binären Bäumen - für diese ist das Problem NP-schwer - und Pfaden. Im zweiten Teil dieser Arbeit betrachten wir das Problem, n Jobs auf m Maschinen mit unterschiedlichen Geschwindigkeiten so zu verteilen, dass der Makespan minimiert wird (Q||C_max). Wir präsentieren einen einfachen deterministischen monotonen Algorithmus mit Approximationsgüte 2.8 für Q||C_max. Monotonie ist relevant im Zusammenhang mit truthful Mechanismen: wenn die Geschwindigkeiten der Maschinen nur diesen selbst bekannt sind, müssen sie motiviert werden, dem Scheduling Mechanismus ihre tatsächlichen Geschwindigkeiten offenzulegen. Der beste bisherige deterministische truthful Mechanismus mit polynomieller Laufzeit in n und m von Andelman et al. [3] erreicht Approximationsgüte fünf. Eine randomisierte Methode mit ApproximationsgÄute zwei, die jedoch nur eine schwächere Definition von truthful Mechanismen unterstützt, wurde von Archer und Tardos [4, 5] entwickelt. Als ein zentrales Ergebnis beweisen wir die Vermutung von Auletta et al. [8], dass der greedy list-scheduling Algorithmus Lpt monoton ist, falls alle Maschinengeschwindigkeiten ganze Potenzen von zwei sind (2-divisible Maschinen). Der Beweis der obigen Approximationsschranke von 2.8 benutzt die Approximationsgüte von Lpt auf 2-divisible Maschinen. Als Nebenresultat erhalten wir eine scharfe Schranke von (sqrt(3) + 1)/2 ~= 1.3660 für den Fall "einer schnellen Maschine", d.h. m - 1 Maschinen haben identische Geschwindigkeiten und es gibt nur eine schnellere Maschine. Die bisherigen besten unteren und oberen Schranken für diesen Spezialfall waren 4/3 - epsilon < Lpt/Opt <= 3/2 - 1/(2m). Letztere wurden 1977 von Gonzalez, Ibara und Sahni [42] bewiesen, die mutmaßten, dass die tatächliche obere Schranke bei 4=3 läge. Alles in allem, liefert diese Arbeit Antworten auf drei offene Fragen im Bereich der Scheduling-Theorie

    Parameterized Approximation Schemes using Graph Widths

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    Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard to solve exactly when parameterized by treewidth or clique-width. Our main contribution is to present a natural randomized rounding technique that extends well-known ideas and can be used for both of these widths. Applying this very generic technique we obtain approximation schemes for a number of problems, evading both polynomial-time inapproximability and parameterized intractability bounds

    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
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