Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time

The described multicoloring problem has direct applications in the context of wireless ad hoc and sensor networks. In order to coordinate the access to the shared wireless medium, the nodes of such a network need to employ some medium access control (MAC) protocol. Typical MAC protocols control the access to the shared channel by time (TDMA), frequency (FDMA), or code division multiple access (CDMA) schemes. Many channel access schemes assign a fixed set of time slots, frequencies, or (orthogonal) codes to the nodes of a network such that nodes that interfere with each other receive disjoint sets of time slots, frequencies, or code sets. Finding a valid assignment of time slots, frequencies, or codes hence directly corresponds to computing a multicoloring of a graph $G$. The scarcity of bandwidth, energy, and computing resources in ad hoc and sensor networks, as well as the often highly dynamic nature of these networks require that the multicoloring can be computed based on as little and as local information as possible

Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

The Capacity of Smartphone Peer-To-Peer Networks

We study three capacity problems in the mobile telephone model, a network abstraction that models the peer-to-peer communication capabilities implemented in most commodity smartphone operating systems. The capacity of a network expresses how much sustained throughput can be maintained for a set of communication demands, and is therefore a fundamental bound on the usefulness of a network. Because of this importance, wireless network capacity has been active area of research for the last two decades. The three capacity problems that we study differ in the structure of the communication demands. The first problem is pairwise capacity, where the demands are (source, destination) pairs. Pairwise capacity is one of the most classical definitions, as it was analyzed in the seminal paper of Gupta and Kumar on wireless network capacity. The second problem we study is broadcast capacity, in which a single source must deliver packets to all other nodes in the network. Finally, we turn our attention to all-to-all capacity, in which all nodes must deliver packets to all other nodes. In all three of these problems we characterize the optimal achievable throughput for any given network, and design algorithms which asymptotically match this performance. We also study these problems in networks generated randomly by a process introduced by Gupta and Kumar, and fully characterize their achievable throughput. Interestingly, the techniques that we develop for all-to-all capacity also allow us to design a one-shot gossip algorithm that runs within a polylogarithmic factor of optimal in every graph. This largely resolves an open question from previous work on the one-shot gossip problem in this model

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

Finer Tight Bounds for Coloring on Clique-Width

We revisit the complexity of the classical k-Coloring problem parameterized by clique-width. This is a very well-studied problem that becomes highly intractable when the number of colors k is large. However, much less is known on its complexity for small, concrete values of k. In this paper, we completely determine the complexity of k-Coloring parameterized by clique-width for any fixed k, under the SETH. Specifically, we show that for all k >= 3,epsilon>0, k-Coloring cannot be solved in time O^*((2^k-2-epsilon)^{cw}), and give an algorithm running in time O^*((2^k-2)^{cw}). Thus, if the SETH is true, 2^k-2 is the "correct" base of the exponent for every k. Along the way, we also consider the complexity of k-Coloring parameterized by the related parameter modular treewidth (mtw). In this case we show that the "correct" running time, under the SETH, is O^*({k choose floor[k/2]}^{mtw}). If we base our results on a weaker assumption (the ETH), they imply that k-Coloring cannot be solved in time n^{o(cw)}, even on instances with O(log n) colors

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

We consider the ILP Feasibility problem: given an integer linear program $\{Ax = b, x\geq 0\}$, where $A$ is an integer matrix with $k$ rows and $\ell$ columns and $b$ is a vector of $k$ integers, we ask whether there exists $x\in\mathbb{N}^\ell$ that satisfies $Ax = b$. Our goal is to study the complexity of ILP Feasibility when both $k$, the number of constraints (rows of $A$), and $\|A\|_\infty$, the largest absolute value in $A$, are small. Papadimitriou [J. ACM, 1981] was the first to give a fixed-parameter algorithm for ILP Feasibility in this setting, with running time $\left((\|A\mid b\|_\infty) \cdot k\right)^{O(k^2)}$. This was very recently improved by Eisenbrand and Weismantel [SODA 2018], who used the Steinitz lemma to design an algorithm with running time $(k\|A\|_\infty)^{O(k)}\cdot \|b\|_\infty^2$, and subsequently by Jansen and Rohwedder [2018] to $O(k\|A\|_\infty)^{k}\cdot \log \|b\|_\infty$. We prove that for $\{0,1\}$-matrices $A$, the dependency on $k$ is probably optimal: an algorithm with running time $2^{o(k\log k)}\cdot (\ell+\|b\|_\infty)^{o(k)}$ would contradict ETH. This improves previous non-tight lower bounds of Fomin et al. [ESA 2018]. We then consider ILPs with many constraints, but structured in a shallow way. Precisely, we consider the dual treedepth of the matrix $A$, which is the treedepth of the graph over the rows of $A$, with two rows adjacent if in some column they both contain a non-zero entry. It was recently shown by Kouteck\'{y} et al. [ICALP 2018] that ILP Feasibility can be solved in time $\|A\|_\infty^{2^{O(td(A))}}\cdot (k+\ell+\log \|b\|_\infty)^{O(1)}$. We present a streamlined proof of this fact and prove optimality: even assuming that all entries of $A$ and $b$ are in $\{-1,0,1\}$, the existence of an algorithm with running time $2^{2^{o(td(A))}}\cdot (k+\ell)^{O(1)}$ would contradict ETH.Comment: Added Corollary 2, extended Conclusion