Incremental Medians via Online Bidding

In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance, and the algorithm produces a nested sequence of facility sets where the kth set has size k. The algorithm is c-cost-competitive if the cost of each set is at most c times the cost of the optimum set of size k. We give improved incremental algorithms for the metric version: an 8-cost-competitive deterministic algorithm, a 2e ~ 5.44-cost-competitive randomized algorithm, a (24+epsilon)-cost-competitive, poly-time deterministic algorithm, and a (6e+epsilon ~ .31)-cost-competitive, poly-time randomized algorithm. The algorithm is s-size-competitive if the cost of the kth set is at most the minimum cost of any set of size k, and has size at most s k. The optimal size-competitive ratios for this problem are 4 (deterministic) and e (randomized). We present the first poly-time O(log m)-size-approximation algorithm for the offline problem and first poly-time O(log m)-size-competitive algorithm for the incremental problem. Our proofs reduce incremental medians to the following online bidding problem: faced with an unknown threshold T, an algorithm submits "bids" until it submits a bid that is at least the threshold. It pays the sum of all its bids. We prove that folklore algorithms for online bidding are optimally competitive.Comment: conference version appeared in LATIN 2006 as "Oblivious Medians via Online Bidding

Sequential and Parallel Algorithms for Mixed Packing and Covering

Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients. Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set cover, bin-packing, knapsack, scheduling problems, minimum-weight triangulation, etc. This paper gives approximation algorithms for the general class of problems. The sequential algorithm is a simple greedy algorithm that can be implemented to find an epsilon-approximate solution in O(epsilon^-2 log m) linear-time iterations. The parallel algorithm does comparable work but finishes in polylogarithmic time. The results generalize previous work on pure packing and covering (the special case when the constraints are all "less-than" or all "greater-than") by Michael Luby and Noam Nisan (1993) and Naveen Garg and Jochen Konemann (1998)

Lotsize optimization leading to a pp-median problem with cardinalities

We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation and a primal heuristic for this problem we also consider a more abstract version which we relate to several other classical optimization problems like the p-median problem, the facility location problem or the matching problem.Comment: 14 page

Nearly Linear-Work Algorithms for Mixed Packing/Covering and Facility-Location Linear Programs

We describe the first nearly linear-time approximation algorithms for explicitly given mixed packing/covering linear programs, and for (non-metric) fractional facility location. We also describe the first parallel algorithms requiring only near-linear total work and finishing in polylog time. The algorithms compute $(1+\epsilon)$-approximate solutions in time (and work) $O^*(N/\epsilon^2)$, where $N$ is the number of non-zeros in the constraint matrix. For facility location, $N$ is the number of eligible client/facility pairs

A Nearly Linear-Time PTAS for Explicit Fractional Packing and Covering Linear Programs

We give an approximation algorithm for packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm computes feasible primal and dual solutions whose costs are within a factor of 1+eps of the optimal cost in time O((r+c)log(n)/eps^2 + n).Comment: corrected version of FOCS 2007 paper: 10.1109/FOCS.2007.62. Accepted to Algorithmica, 201

K-Medians, Facility Location, and the Chernoff-Wald Bound

We study the general (non-metric) facility-location and weighted k-medians problems, as well as the fractional facility-location and unweighted k-medians problems. We describe a natural randomized rounding scheme and use it to derive approximation algorithms for all of these problems. For facility location and weighted k-medians, the respective algorithms are polynomial-time [Hk + d]- and [(1 + )d; ln(n + n=)k]-approximation algorithms. These performance guarantees improve on the best previous performance guarantees, due respectively to Hochbaum (1982) and Lin and Vitter (1992). For fractional k-medians, the algorithm is a new, Lagrangian-relaxation, [(1 + )d; (1 + )k]