Improved distributed algorithms for coloring interval graphs with application to multicoloring trees

Post-print (lokagerð höfundar)We give a distributed (1+eps)-approximation algorithm for the minimum vertex coloring problem on interval graphs, which runs in the LOCAL model and operates in O((1/eps) log* n) rounds. If nodes are aware of their interval representations, then the algorithm can be adapted to the CONGEST model using the same number of rounds. Prior to this work, only constant factor approximations using O(log* n) rounds were known. Linial's ring coloring lower bound implies that the dependency on log* n cannot be improved. We further prove that the dependency on 1/eps is also optimal. To obtain our CONGEST model algorithm, we develop a color rotation technique that may be of independent interest. We demonstrate that color rotations can also be applied to obtain a (1+eps)-approximate multicoloring of directed trees in O((1/eps)log* n) rounds.Magnus M. Halldorsson is supported by grants 152679-05 and 174484-05 from the Icelandic Research Fund. Christian Konrad is supported by the Centre for Discrete Mathematics and its Applications (DIMAP) at Warwick University and by EPSRC award EP/N011163/1."Peer Reviewed

Parameterized Approximation Schemes using Graph Widths

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

Fast algorithms for two scheduling problems

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

Recoloring Interval Graphs with Limited Recourse Budget

We consider the problem of coloring an interval graph dynamically. Intervals arrive one after the other and have to be colored immediately such that no two intervals of the same color overlap. In each step only a limited number of intervals may be recolored to maintain a proper coloring (thus interpolating between the well-studied online and offline settings). The number of allowed recolorings per step is the so-called recourse budget. Our main aim is to prove both upper and lower bounds on the required recourse budget for interval graphs, given a bound on the allowed number of colors. For general interval graphs with n vertices and chromatic number k it is known that some recoloring is needed even if we have 2k colors available. We give an algorithm that maintains a 2k-coloring with an amortized recourse budget of $\u1d4aa(log n)$. For maintaining a k-coloring with k ≤ n, we give an amortized upper bound of \u1d4aa(k⋅ k! ⋅ √n), and a lower bound of $Ω(k) for k ∈ \u1d4aa(√n)$, which can be as large as $Ω(√n$). For unit interval graphs it is known that some recoloring is needed even if we have k+1 colors available. We give an algorithm that maintains a (k+1)-coloring with at most $\u1d4aa(k²)$ recolorings per step in the worst case. We also give a lower bound of $Ω(log n)$ on the amortized recourse budget needed to maintain a k-coloring. Additionally, for general interval graphs we show that if one does not insist on maintaining an explicit coloring, one can have a k-coloring algorithm which does not incur a factor of $\u1d4aa(k ⋅ k! ⋅ √n)$ in the running time. For this we provide a data structure, which allows for adding intervals in $\u1d4aa(k² log³ n)$ amortized time per update and querying for the color of a particular interval in $\u1d4aa(log n) time$. Between any two updates, the data structure answers consistently with some optimal coloring. The data structure maintains the coloring implicitly, so the notion of recourse budget does not apply to it

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

Local Certification of Some Geometric Intersection Graph Classes

In the context of distributed certification, the recognition of graph classes has started to be intensively studied. For instance, different results related to the recognition of planar, bounded tree-width and $H$-minor free graphs have been recently obtained. The goal of the present work is to design compact certificates for the local recognition of relevant geometric intersection graph classes, namely interval, chordal, circular arc, trapezoid and permutation. More precisely, we give proof labeling schemes recognizing each of these classes with logarithmic-sized certificates. We also provide tight logarithmic lower bounds on the size of the certificates on the proof labeling schemes for the recognition of any of the aforementioned geometric intersection graph classes