Showing papers by "Magnús M. Halldórsson published in 2011"

Wireless capacity with oblivious power in general metrics

Abstract: The capacity of a wireless network is the maximum possible amount of simultaneous communication, taking interference into account. Formally, we treat the following problem. Given is a set of links, each a sender-receiver pair located in a metric space, and an assignment of power to the senders. We seek a maximum subset of links that are feasible in the SINR model: namely, the signal received on each link should be larger than the sum of the interferences from the other links. We give a constant-factor approximation that holds for any length-monotone, sub-linear power assignment and any distance metric.We use this to give essentially tight characterizations of capacity maximization under power control using oblivious power assignments. Specifically, we show that the mean power assignment is optimal for capacity maximization of bi-directional links, and give a tight θ(log n)-approximation of scheduling bi-directional links with power control using oblivious power. For uni-directional links we give a nearly optimal O(log n + log log Δ)-approximation to the power control problem using mean power, where Δ is the ratio of longest and shortest links. Combined, these results clarify significantly the centralized complexity of wireless communication problems.

Nearly optimal bounds for distributed wireless scheduling in the SINR model

Abstract: We study the wireless scheduling problem in the physically realistic SINR model. More specifically: we are given a set of n links, each a sender-receiver pair. We would like to schedule the links using the minimum number of slots, using the SINR model of interference among simultaneously transmitting links. In the basic problem, all senders transmit with the same uniform power. In this work, we provide a distributed O(log n)-approximation for the scheduling problem, matching the best ratio known for centralized algorithms. This is based on an algorithm studied by Kesselheim and Vocking, improving their analysis by a logarithmic factor. We show this to be best possible for any such distributed algorithm. Our analysis extends also to linear power assignments, and as well as for more general assignments, modulo assumptions about message acknowledgement mechanisms.

Wireless Capacity and Admission Control in Cognitive Radio

Abstract: We give algorithms with constant-factor performance guarantees for several capacity and throughput problems in the SINR model. The algorithms are all based on a novel LP formulation for capacity problems. First, we give a new constant-factor approximation algorithm for selecting the maximum subset of links that can be scheduled simultaneously, under any non-decreasing and sublinear power assignment. For the case of uniform power, we extend this to the case of variable QoS requirements and link-dependent noise terms. Second, we approximate a problem related to cognitive radio: find a maximum set of links that can be simultaneously scheduled without affecting a given set of previously assigned links. Finally, we obtain constant-factor approximation of weighted capacity under linear power assignment.

Alternation graphs

Abstract: A graph G=(V,E) is an alternation graph if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x,y)∈E for each x≠y. In this paper we give an effective characterization of alternation graphs in terms of orientations. Namely, we show that a graph is an alternation graph if and only if it admits a semi-transitive orientation defined in the paper. This allows us to prove a number of results about alternation graphs, in particular showing that the recognition problem is in NP, and that alternation graphs include all 3-colorable graphs. We also explore bounds on the size of the word representation of the graph. A graph G is a k-alternation graph if it is represented by a word in which each letter occurs exactly k times; the alternation number of G is the minimum k for which G is a k-alternation graph. We show that the alternation number is always at most n, while there exist graphs for which it is n/2.

Wireless Capacity With Arbitrary Gain Matrix

Abstract: Given a set of wireless links, a fundamental problem is to find the largest subset that can transmit simultaneously, within the SINR model of interference. Significant progress on this problem has been made in recent years. In this note, we study the problem in the setting where we are given a fixed set of arbitrary powers each sender must use, and an arbitrary gain matrix defining how signals fade. This variation of the problem appears immune to most algorithmic approaches studied in the literature. Indeed it is very hard to approximate since it generalizes the max independent set problem. Here, we propose a simple semi-definite programming approach to the problem that yields constant factor approximation, if the optimal solution is strictly larger than half of the input size.

Online Scheduling with Interval Conflicts

Abstract: In the problem of Scheduling with Interval Conflicts, there is a ground set of items indexed by integers, and the input is a collection of conflicts, each containing all the items whose index lies within some interval on the real line. Conflicts arrive in an online fashion. A scheduling algorithm must select, from each conflict, at most one survivor item, and the goal is to maximize the number (or weight) of items that survive all the conflicts they are involved in. We present a centralized deterministic online algorithm whose competitive ratio is O(log sigma), where sigma is the size of the largest conflict. For the distributed setting, we present another deterministic algorithm whose competitive ratio is 2 log sigma, in the special contiguous case, in which the item indices constitute a contiguous interval of integers. Our upper bounds are complemented by two lower bounds: one that shows that even in the contiguous case, all deterministic algorithms (centralized or distributed) have competitive ratio Omega(log sigma), and that in the non-contiguous case, no deterministic oblivious algorithm (i.e., a distributed algorithm that does not use communication) can have a bounded competitive ratio.

Sum edge coloring of multigraphs via configuration LP

Abstract: We consider the scheduling of biprocessor jobs under sum objective (BPSMSM). Given a collection of unit-length jobs where each job requires the use of two processors, find a schedule such that no two jobs involving the same processor run concurrently. The objective is to minimize the sum of the completion times of the jobs. Equivalently, we would like to find a sum edge coloring of a given multigraph, that is, a partition of its edge set into matchings M1,…,Mt minimizing Σi=1ti|Mi|. This problem is APX-hard, even in the case of bipartite graphs [Marx 2009]. This special case is closely related to the classic open shop scheduling problem. We give a 1.8298-approximation algorithm for BPSMSM improving the previously best ratio known of 2 [Bar-Noy et al. 1998]. The algorithm combines a configuration LP with greedy methods, using nonstandard randomized rounding on the LP fractions. We also give an efficient combinatorial 1.8886-approximation algorithm for the case of simple graphs, which gives an improved 1.79568 + O(log d¯/d¯)-approximation in graphs of large average degree d¯.

Nearly Optimal Bounds for Distributed Wireless Scheduling in the SINR Model

Abstract: We study the wireless scheduling problem in the SINR model. More specifically, given a set of $n$ links, each a sender-receiver pair, we wish to partition (or \emph{schedule}) the links into the minimum number of slots, each satisfying interference constraints allowing simultaneous transmission. In the basic problem, all senders transmit with the same uniform power. We give a distributed $O(\log n)$-approximation algorithm for the scheduling problem, matching the best ratio known for centralized algorithms. It holds in arbitrary metric space and for every length-monotone and sublinear power assignment. It is based on an algorithm of Kesselheim and V\"ocking, whose analysis we improve by a logarithmic factor. We show that every distributed algorithm uses $\Omega(\log n)$ slots to schedule certain instances that require only two slots, which implies that the best possible absolute performance guarantee is logarithmic.

Algorithms - ESA 2011 : 19th annual European symposium, Saarbrücken, Germany, September 5-9, 2011 : proceedings

Abstract: This book constitutes the refereed proceedings of the 19th Annual European Symposium on Algorithms, ESA 2011, held in Saarbrucken, Germany, in September 2011 in the context of the combined conference ALGO 2011. The 67 revised full papers presented were carefully reviewed and selected from 255 initial submissions: 55 out of 209 in track design and analysis and 12 out of 46 in track engineering and applications. The papers are organized in topical sections on approximation algorithms, computational geometry, game theory, graph algorithms, stable matchings and auctions, optimization, online algorithms, exponential-time algorithms, parameterized algorithms, scheduling, data structures, graphs and games, distributed computing and networking, strings and sorting, as well as local search and set systems.

Wireless capacity with arbitrary gain matrix

Abstract: Given a set of wireless links, a fundamental problem is to find the largest subset that can transmit simultaneously, within the SINR model of interference. Significant progress on this problem has been made in recent years. In this note, we study the problem in the setting where we are given a fixed set of arbitrary powers each sender must use, and an arbitrary gain matrix defining how signals fade. This variation of the problem appears immune to most algorithmic approaches studied in the literature. Indeed it is very hard to approximate since it generalizes the max independent set problem. Here, we propose a simple semi-definite programming approach to the problem that yields constant factor approximation, if the optimal solution is strictly larger than half of the input size.