SIAM Digital Library
 
 
 

SIAM J. on Computing

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

2004

Volume 33, Issue 6, pp. 1261-1531


Fast, Distributed Approximation Algorithms for Positive Linear Programming with Applications to Flow Control

Yair Bartal, John W. Byers, and Danny Raz

SIAM J. Comput. 33, pp. 1261-1279 (19 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
We study combinatorial optimization problems in which a set of distributed agents must achieve a global objective using only local information. Papadimitriou and Yannakakis [Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 121--129] initiated the study of such problems in a framework where distributed decision-makers must generate feasible solutions to positive linear programs with information only about local constraints. We extend their model by allowing these distributed decision-makers to perform local communication to acquire information over time and then explore the tradeoff between the amount of communication and the quality of the solution to the linear program that the decision-makers can obtain.
Our main result is a distributed algorithm that obtains a $(1 + \epsilon)$ approximation to the optimal linear programming solution while using only a polylogarithmic number of rounds of local communication. This algorithm offers a significant improvement over the logarithmic approximation ratio previously obtained by Awerbuch and Azar [Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, 1994, pp. 240--249] for this problem while providing a comparable running time. Our results apply directly to the application of network flow control, an application in which distributed routers must quickly choose how to allocate bandwidth to connections using only local information to achieve global objectives. The sequential version of our algorithm is faster and considerably simpler than the best known approximation algorithms capable of achieving a $(1 + \epsilon)$ approximation ratio for positive linear programming.

Preemptive Weighted Completion Time Scheduling of Parallel Jobs

Uwe Schwiegelshohn

SIAM J. Comput. 33, pp. 1280-1308 (29 pages)

Full Text: | Download PDF

Show Abstract
We present a new algorithm for the preemptive offline scheduling of independent jobs on a system consisting of m identical machines. The jobs can be parallel; that is, they may need the concurrent availability of several machines for their execution. To this end, we introduce a machine model which is based on existing multiprocessors and accounts for the penalty of preemption. After examining the relation between makespan and total weighted completion time costs for the scheduling of parallel jobs, we show that our new algorithm achieves an approximation factor of 2.37 for total weighted completion time scheduling if no preemption penalty is considered. This compares favorably to the thus far best approximation factor of 8.53 for the nonpreemptive case. To fine-tune the algorithm with respect to different preemption penalties, we use a fairly simple numerical optimization problem. Further, we present an algorithm to transform the preemptive schedule into a nonpreemptive one. This leads to an improved approximation factor of 7.11 for the nonpreemptive weighted completion time scheduling.

Algebraic Properties for Selector Functions

Lane A. Hemaspaandra, Harald Hempel, and Arfst Nickelsen

SIAM J. Comput. 33, pp. 1309-1337 (29 pages)

Full Text: | Download PDF

Show Abstract
The nondeterministic advice complexity of the P-selective sets is known to be exactly linear. Regarding the deterministic advice complexity of the P-selective sets---i.e., the amount of Karp--Lipton advice needed for polynomial-time machines to recognize them in general---the best current upper bound is quadratic [K. Ko, J. Comput. System Sci., 26 (1983), pp. 209--221] and the best current lower bound is linear [L. Hemaspaandra and L. Torenvliet, Theoret. Comput. Sci., 154 (1996), pp. 367--377].
We prove that every associatively P-selective set is commutatively, associatively P-selective. Using this, we establish an algebraic sufficient condition for the P-selective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all P-selective sets are associatively P-selective, then the deterministic advice complexity of the P-selective sets is linear. The weakest previously known sufficient condition was P = NP.
We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.

Graphs with Tiny Vector Chromatic Numbers and Huge Chromatic Numbers

Uriel Feige, Michael Langberg, and Gideon Schechtman

SIAM J. Comput. 33, pp. 1338-1368 (31 pages) | Cited 2 times

Full Text: | Download PDF

Show Abstract
Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246--265] introduced the notion of a vector coloring of a graph. In particular, they showed that every $k$-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly $\Delta^{1 - 2/k}$ colors. Here $\Delta$ is the maximum degree in the graph and is assumed to be of the order of $n^{\delta}$ for some $0 < \delta < 1$. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets.
We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than $n/\Delta^{1 - 2/k}$ (and hence cannot be colored with significantly fewer than $\Delta^{1 - 2/k}$ colors). For $k = O(\log n/\log\log n)$ we show vector k-colorable graphs that do not have independent sets of size (log n)c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylog n.
As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653--750] for this problem.

Disjoint NP-Pairs

Christian Glasser, Alan L. Selman, Samik Sengupta, and Liyu Zhang

SIAM J. Comput. 33, pp. 1369-1416 (48 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
We study the question of whether the class DisjNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provides additional evidence for the existence of P-inseparable disjoint NP-pairs.
We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair; an oracle relative to which optimal proof systems exist, and hence complete pairs exist, but no pair is NP-hard; and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.

Incremental Clustering and Dynamic Information Retrieval

Moses Charikar, Chandra Chekuri, Tomas Feder, and Rajeev Motwani

SIAM J. Comput. 33, pp. 1417-1440 (24 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
Motivated by applications such as document and image classification in information retrieval, we consider the problem of clustering dynamic point sets in a metric space. We propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retrieval application, and which should also be useful in other applications. The goal is to efficiently maintain clusters of small diameter as new points are inserted. We analyze several natural greedy algorithms and demonstrate that they perform poorly. We propose new deterministic and randomized incremental clustering algorithms which have a provably good performance, and which we believe should also perform well in practice. We complement our positive results with lower bounds on the performance of incremental algorithms. Finally, we consider the dual clustering problem where the clusters are of fixed diameter, and the goal is to minimize the number of clusters.

Tight Bounds for Testing Bipartiteness in General Graphs

Tali Kaufman, Michael Krivelevich, and Dana Ron

SIAM J. Comput. 33, pp. 1441-1483 (43 pages) | Cited 12 times

Full Text: | Download PDF

Show Abstract
In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs and one most suitable for bounded-degree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant complexity, while the complexity of testing bounded-degree graphs is $\tilde{\Theta}(\sqrt{n})$, where $n$ is the number of vertices in the graph (and $\tilde{\Theta}(f(n))$ means $\Theta(f(n)\cdot{\rm polylog}(f(n)))$). Thus there is a large gap between the complexity of testing in the two cases.
In this work we bridge the gap described above. In particular, we study the problem of testing bipartiteness in a model that is suitable for all densities. We present an algorithm whose complexity is $\tilde{O}(\min(\sqrt{n},n^2/m))$, where $m$ is the number of edges in the graph, and we match it with an almost tight lower bound.

Depth Optimal Sorting Networks Resistant to k Passive Faults

Marek Piotrów

SIAM J. Comput. 33, pp. 1484-1512 (29 pages)

Full Text: | Download PDF

Show Abstract
We study the problem of constructing a sorting network that is tolerant to faults and whose running time (i.e., depth) is as small as possible. We consider the scenario of worst-case comparator faults and follow the model of passive comparator failure proposed by Yao and Yao SIAM J. Comput., 14 (1985), pp. 120--128], in which a faulty comparator outputs its inputs directly without comparison. Our main result is the first construction of an N-input k-fault-tolerant sorting network with an asymptotically optimal depth $\theta$(log N + k). That improves over the result of Leighton and Ma [Proceedings of the 5th Annual ACM Symposium on Parallel Algorithms and Architectures, Velen, Germany, 1993, ACM, New York, pp. 30--41], whose network is of depth O(log N + klog\frac{log N}{log k})$.
Actually, we present a fault-tolerant correction network that can be added after any N-input sorting network to correct its output in the presence of at most k faulty comparators. Since the depth of the network is O(log N + k) and the constants hidden behind the "O" notation are small, the construction can be of practical use.
Developing the techniques necessary to show the main result, we construct a fault-tolerant network for the insertion problem. As a by-product, we get an N-input O(log N)-depth INSERT-network that is tolerant to random faults, thereby answering a question posed by Ma in his Ph.D. thesis [Fault-Tolerant Sorting Network, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1994].
The results are based on a new notion of constant delay comparator networks, that is, networks in which each register is used (compared) only in a period of time of a constant length. Copies of such networks can be pipelined with only a constant increase in the total depth per copy.

Arithmetic Circuits and Polynomial Replacement Systems

Pierre McKenzie, Heribert Vollmer, and Klaus W. Wagner

SIAM J. Comput. 33, pp. 1513-1531 (19 pages)

Full Text: | Download PDF

Show Abstract
This paper addresses the problems of counting proof-trees (as introduced by Venkateswaran and Tompa) and counting proof-circuits, a related but seemingly more natural question. These problems lead to a common generalization of straight-line programs which we call polynomial replacement systems {PRSs}. We contribute a classification of these systems and we investigate their complexity. Diverse problems falling within the scope of this study include, for example, counting proof-circuits and evaluating $\{\cup,+\}$-circuits over the natural numbers. A number of complexity results are obtained, including a proof that counting proof-circuits is $\numP$-complete.
Close

Your Recent History Recent History Help

Recently Viewed

  1. A Subquadratic Sequence Alignment Algorithm for Unrestricted Scoring Matrices
  2. Pseudotriangulations from Surfaces and a Novel Type of Edge Flip
  3. Complexity Results for First-Order Two-Variable Logic with Counting
  4. The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Dat...
  5. Asymptotically Tight Bounds for Performing BMMC Permutations on Parallel Disk Systems
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close