Approximation Algorithms for Covering/Packing Integer Programs

Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min {c.x: x in Z^n_+, Ax > a, Bx < b, x < d}. We give a bicriteria-approximation algorithm that, given epsilon in (0, 1], finds a solution of cost O(ln(m)/epsilon^2) times optimal, meeting the covering constraints (Ax > a) and multiplicity constraints (x < d), and satisfying Bx < (1 + epsilon)b + beta, where beta is the vector of row sums beta_i = sum_j B_ij. Here m denotes the number of rows of A. This gives an O(ln m)-approximation algorithm for CIP -- minimum-cost covering integer programs with multiplicity constraints, i.e., the special case when there are no packing constraints Bx < b. The previous best approximation ratio has been O(ln(max_j sum_i A_ij)) since 1982. CIP contains the set cover problem as a special case, so O(ln m)-approximation is the best possible unless P=NP.Comment: Preliminary version appeared in IEEE Symposium on Foundations of Computer Science (2001). To appear in Journal of Computer and System Science

Symmetric Allocations for Distributed Storage

We consider the problem of optimally allocating a given total storage budget in a distributed storage system. A source has a data object which it can code and store over a set of storage nodes; it is allowed to store any amount of coded data in each node, as long as the total amount of storage used does not exceed the given budget. A data collector subsequently attempts to recover the original data object by accessing each of the nodes independently with some constant probability. By using an appropriate code, successful recovery occurs when the total amount of data in the accessed nodes is at least the size of the original data object. The goal is to find an optimal storage allocation that maximizes the probability of successful recovery. This optimization problem is challenging because of its discrete nature and nonconvexity, despite its simple formulation. Symmetric allocations (in which all nonempty nodes store the same amount of data), though intuitive, may be suboptimal; the problem is nontrivial even if we optimize over only symmetric allocations. Our main result shows that the symmetric allocation that spreads the budget maximally over all nodes is asymptotically optimal in a regime of interest. Specifically, we derive an upper bound for the suboptimality of this allocation and show that the performance gap vanishes asymptotically in the specified regime. Further, we explicitly find the optimal symmetric allocation for a variety of cases. Our results can be applied to distributed storage systems and other problems dealing with reliability under uncertainty, including delay tolerant networks (DTNs) and content delivery networks (CDNs).Comment: 7 pages, 3 figures, extended version of an IEEE GLOBECOM 2010 pape

Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels

In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erd\H{o}s on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum $d$-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also discuss an application to a problem of finding an optimal data allocation in a distributed storage system

Content-access QoS in peer-to-peer networks using a fast MDS erasure code

This paper describes an enhancement of content access Quality of Service in peer to peer (P2P) networks. The main idea is to use an erasure code to distribute the information over the peers. This distribution increases the users’ choice on disseminated encoded data and therefore statistically enhances the overall throughput of the transfer. A performance evaluation based on an original model using the results of a measurement campaign of sequential and parallel downloads in a real P2P network over Internet is presented. Based on a bandwidth distribution, statistical content-access QoS are guaranteed in function of both the content replication level in the network and the file dissemination strategies. A simple application in the context of media streaming is proposed. Finally, the constraints on the erasure code related to the proposed system are analysed and a new fast MDS erasure code is proposed, implemented and evaluated

Multicluster interleaving on paths and cycles

Interleaving codewords is an important method not only for combatting burst errors, but also for distributed data retrieval. This paper introduces the concept of multicluster interleaving (MCI), a generalization of traditional interleaving problems. MCI problems for paths and cycles are studied. The following problem is solved: how to interleave integers on a path or cycle such that any m (m/spl ges/2) nonoverlapping clusters of order 2 in the path or cycle have at least three distinct integers. We then present a scheme using a "hierarchical-chain structure" to solve the following more general problem for paths: how to interleave integers on a path such that any m (m/spl ges/2) nonoverlapping clusters of order L (L/spl ges/2) in the path have at least L+1 distinct integers. It is shown that the scheme solves the second interleaving problem for paths that are asymptotically as long as the longest path on which an MCI exists, and clearly, for shorter paths as well

Network File Storage With Graceful Performance Degradation

A file storage scheme is proposed for networks containing heterogeneous clients. In the scheme, the performance measured by file-retrieval delays degrades gracefully under increasingly serious faulty circumstances. The scheme combines coding with storage for better performance. The problem is NP-hard for general networks; and this paper focuses on tree networks with asymmetric edges between adjacent nodes. A polynomial-time memory-allocation algorithm is presented, which determines how much data to store on each node, with the objective of minimizing the total amount of data stored in the network. Then a polynomial-time data-interleaving algorithm is used to determine which data to store on each node for satisfying the quality-of-service requirements in the scheme. By combining the memory-allocation algorithm with the data-interleaving algorithm, an optimal solution to realize the file storage scheme in tree networks is established

Approximation algorithms for stochastic and risk-averse optimization

We present improved approximation algorithms in stochastic optimization. We prove that the multi-stage stochastic versions of covering integer programs (such as set cover and vertex cover) admit essentially the same approximation algorithms as their standard (non-stochastic) counterparts; this improves upon work of Swamy \& Shmoys which shows an approximability that depends multiplicatively on the number of stages. We also present approximation algorithms for facility location and some of its variants in the $2$-stage recourse model, improving on previous approximation guarantees. We give a $2.2975$-approximation algorithm in the standard polynomial-scenario model and an algorithm with an expected per-scenario $2.4957$-approximation guarantee, which is applicable to the more general black-box distribution model.Comment: Extension of a SODA'07 paper. To appear in SIAM J. Discrete Mat

Multi-Cluster interleaving in linear arrays and rings

Interleaving codewords is an important method not only for combatting burst-errors, but also for flexible data-retrieving. This paper defines the Multi-Cluster Interleaving (MCI) problem, an interleaving problem for parallel data-retrieving. The MCI problems on linear arrays and rings are studied. The following problem is completely solved: how to interleave integers on a linear array or ring such that any m (m greater than or equal to 2) non-overlapping segments of length 2 in the array or ring have at least 3 distinct integers. We then present a scheme using a 'hierarchical-chain structure' to solve the following more general problem for linear arrays: how to interleave integers on a linear array such that any m (m greater than or equal to 2) non-overlapping segments of length L (L greater than or equal to 2) in the array have at least L + 1 distinct integers. It is shown that the scheme using the 'hierarchical-chain structure' solves the second interleaving problem for arrays that are asymptotically as long as the longest array on which an MCI exists, and clearly, for shorter arrays as well