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Claire Kenyon

Publications and source records attributed to Claire Kenyon.

6 recordsLinked to original sources
arXiv2020

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.

cs.DS↗
arXiv2005

The reverse greedy algorithm for the metric k-median problem

The Reverse Greedy algorithm (RGreedy) for the k-median problem works as follows. It starts by placing facilities on all nodes. At each step, it removes a facility to minimize the resulting total distance from the customers to the remaining facilities. It stops when k facilities remain. We prove that, if the distance function is metric, then the approximation ratio of RGreedy is between ?(log n/ log log n) and O(log n).

cs.DS↗
arXiv2004

Glauber Dynamics on Trees and Hyperbolic Graphs

We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with $n$ vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap $|λ_1-λ_2|$) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in $n$. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time $τ_2$ satisfies $τ_2=O(1)$, then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp.

math.PR↗
arXiv2002

On the Sum-of-Squares Algorithm for Bin Packing

In this paper we present a theoretical analysis of the deterministic on-line {\em Sum of Squares} algorithm ($SS$) for bin packing introduced and studied experimentally in \cite{CJK99}, along with several new variants. $SS$ is applicable to any instance of bin packing in which the bin capacity $B$ and item sizes $s(a)$ are integral (or can be scaled to be so), and runs in time $O(nB)$. It performs remarkably well from an average case point of view: For any discrete distribution in which the optimal expected waste is sublinear, $SS$ also has sublinear expected waste. For any discrete distribution where the optimal expected waste is bounded, $SS$ has expected waste at most $O(\log n)$. In addition, we discuss several interesting variants on $SS$, including a randomized $O(nB\log B)$-time on-line algorithm $SS^*$, based on $SS$, whose expected behavior is essentially optimal for all discrete distributions. Algorithm $SS^*$ also depends on a new linear-programming-based pseudopolynomial-time algorithm for solving the NP-hard problem of determining, given a discrete distribution $F$, just what is the growth rate for the optimal expected waste. This article is a greatly expanded version of the conference paper \cite{sumsq2000}.

cs.DS↗
arXiv2002

Polynomial-Time Approximation Scheme for Data Broadcast

The data broadcast problem is to find a schedule for broadcasting a given set of messages over multiple channels. The goal is to minimize the cost of the broadcast plus the expected response time to clients who periodically and probabilistically tune in to wait for particular messages. The problem models disseminating data to clients in asymmetric communication environments, where there is a much larger capacity from the information source to the clients than in the reverse direction. Examples include satellites, cable TV, internet broadcast, and mobile phones. Such environments favor the ``push-based'' model where the server broadcasts (pushes) its information on the communication medium and multiple clients simultaneously retrieve the specific information of individual interest. This paper presents the first polynomial-time approximation scheme (PTAS) for data broadcast with O(1) channels and when each message has arbitrary probability, unit length and bounded cost. The best previous polynomial-time approximation algorithm for this case has a performance ratio of 9/8.

cs.DS↗