Simple, deterministic, constant-round coloring in the congested clique

We settle the complexity of the (\Delta+1)-coloring and (\Delta+1)-list coloring problems in the CONGESTED CLIQUE model by presenting a simple deterministic algorithm for both problems running in a constant number of rounds. This matches the complexity of the recent breakthrough randomized constant-round (\Delta+1)-list coloring algorithm due to Chang et al. (PODC’19), and significantly improves upon the state-of-the-art O(log(\Delta))-round deterministic (\Delta+1)-coloring bound of Parter (ICALP’18). A remarkable property of our algorithm is its simplicity. Whereas the state-of-the-art randomized algorithms for this problem are based on the quite involved local coloring algorithm of Chang et al. (STOC’18), our algorithm can be described in just a few lines. At a high level, it applies a careful derandomization of a recursive procedure which partitions the nodes and their respective palettes into separate bins. We show that after O(1) recursion steps, the remaining uncolored subgraph within each bin has linear size, and thus can be solved locally by collecting it to a single node. This algorithm can also be implemented in the Massively Parallel Computation (MPC) model provided that each machine has linear (in n) the number of nodes in the input graph) space. We also show an extension of our algorithm to the MPC regime in which machines have sublinear space: we present the first deterministic (\Delta+1)-list coloring algorithm designed for sublinear-space MPC, which runs in O(log(\Delta + loglog(n)) rounds

Randomized approximation algorithms : facility location, phylogenetic networks, Nash equilibria

Despite a great effort, researchers are unable to find efficient algorithms for a number of natural computational problems. Typically, it is possible to emphasize the hardness of such problems by proving that they are at least as hard as a number of other problems. In the language of computational complexity it means proving that the problem is complete for a certain class of problems. For optimization problems, we may consider to relax the requirement of the outcome to be optimal and accept an approximate (i.e., close to optimal) solution. For many of the problems that are hard to solve optimally, it is actually possible to efficiently find close to optimal solutions. In this thesis, we study algorithms for computing such approximate solutions

Distributed MIS in O(log log n) Awake Complexity

Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best known (randomized) MIS algorithms have O(log n) round complexity on general graphs [Luby, STOC 1986] (where n is the number of nodes), while the best known lower bound is [EQUATION] [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the O(log n) round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Energy is a premium resource in various settings such as battery-powered wireless networks and sensor networks. The bulk of the energy is used by nodes when they are awake, i.e., when they are sending, receiving, and even just listening for messages. On the other hand, when a node is sleeping, it does not perform any communication and thus spends very little energy. Several recent works have addressed the problem of designing energy-efficient distributed algorithms for various fundamental problems. These algorithms operate by minimizing the number of rounds in which any node is awake, also called the (worst-case) awake complexity. An intriguing open question is whether one can design a distributed MIS algorithm that has significantly smaller awake complexity compared to existing algorithms. In particular, the question of obtaining a distributed MIS algorithm with o(log n) awake complexity was left open in [Chatterjee, Gmyr, Pandurangan, PODC 2020]. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of O(log n) and also bypassing its fundamental [EQUATION] round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in O(log log n) awake complexity with high probability.1 However, this algorithm has a round complexity that is O(poly(n)). We then show how to drastically improve the round complexity at the cost of a slight increase in awake complexity by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability computes an MIS in O((log log n) log* n) awake complexity and O((log3 n)(log log n) log* n) round complexity. Our algorithms work in the CONGEST model where messages of size O(log n) bits can be sent per edge per round

Distributed MIS in O(log log n) Awake Complexity

Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best known (randomized) MIS algorithms have O(log n) round complexity on general graphs [Luby, STOC 1986] (where n is the number of nodes), while the best known lower bound is [EQUATION] [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the O(log n) round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Energy is a premium resource in various settings such as battery-powered wireless networks and sensor networks. The bulk of the energy is used by nodes when they are awake, i.e., when they are sending, receiving, and even just listening for messages. On the other hand, when a node is sleeping, it does not perform any communication and thus spends very little energy. Several recent works have addressed the problem of designing energy-efficient distributed algorithms for various fundamental problems. These algorithms operate by minimizing the number of rounds in which any node is awake, also called the (worst-case) awake complexity. An intriguing open question is whether one can design a distributed MIS algorithm that has significantly smaller awake complexity compared to existing algorithms. In particular, the question of obtaining a distributed MIS algorithm with o(log n) awake complexity was left open in [Chatterjee, Gmyr, Pandurangan, PODC 2020]. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of O(log n) and also bypassing its fundamental [EQUATION] round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in O(log log n) awake complexity with high probability.1 However, this algorithm has a round complexity that is O(poly(n)). We then show how to drastically improve the round complexity at the cost of a slight increase in awake complexity by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability computes an MIS in O((log log n) log* n) awake complexity and O((log3 n)(log log n) log* n) round complexity. Our algorithms work in the CONGEST model where messages of size O(log n) bits can be sent per edge per round

Dagstuhl News January - December 2011

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic