6 research outputs found
Do Capacity Constraints Constrain Coalitions?
We study strong equilibria in symmetric capacitated cost-sharing games. In
these games, a graph with designated source and sink is given, and each
edge is associated with some cost. Each agent chooses strategically an -
path, knowing that the cost of each edge is shared equally between all agents
using it. Two variants of cost-sharing games have been previously studied: (i)
games where coalitions can form, and (ii) games where edges are associated with
capacities; both variants are inspired by real-life scenarios. In this work we
combine these variants and analyze strong equilibria (profiles where no
coalition can deviate) in capacitated games. This combination gives rise to new
phenomena that do not occur in the previous variants. Our contribution is
two-fold. First, we provide a topological characterization of networks that
always admit a strong equilibrium. Second, we establish tight bounds on the
efficiency loss that may be incurred due to strategic behavior, as quantified
by the strong price of anarchy (and stability) measures. Interestingly, our
results are qualitatively different than those obtained in the analysis of each
variant alone, and the combination of coalitions and capacities entails the
introduction of more refined topology classes than previously studied
Composable Sketches for Functions of Frequencies: Beyond the Worst Case
Recently there has been increased interest in using machine learning
techniques to improve classical algorithms. In this paper we study when it is
possible to construct compact, composable sketches for weighted sampling and
statistics estimation according to functions of data frequencies. Such
structures are now central components of large-scale data analytics and machine
learning pipelines. However, many common functions, such as thresholds and p-th
frequency moments with p > 2, are known to require polynomial-size sketches in
the worst case. We explore performance beyond the worst case under two
different types of assumptions. The first is having access to noisy advice on
item frequencies. This continues the line of work of Hsu et al. (ICLR 2019),
who assume predictions are provided by a machine learning model. The second is
providing guaranteed performance on a restricted class of input frequency
distributions that are better aligned with what is observed in practice. This
extends the work on heavy hitters under Zipfian distributions in a seminal
paper of Charikar et al. (ICALP 2002). Surprisingly, we show analytically and
empirically that "in practice" small polylogarithmic-size sketches provide
accuracy for "hard" functions.Comment: Full version of a paper from ICML 2020. Python implementation
available as part of the supplemental material accompanying the ICML
publicatio
On Estimating Edit Distance: Alignment, Dimension Reduction, and Embeddings
Edit distance is a fundamental measure of distance between strings and has been widely studied in computer science. While the problem of estimating edit distance has been studied extensively, the equally important question of actually producing an alignment (i.e., the sequence of edits) has received far less attention. Somewhat surprisingly, we show that any algorithm to estimate edit distance can be used in a black-box fashion to produce an approximate alignment of strings, with modest loss in approximation factor and small loss in run time. Plugging in the result of Andoni, Krauthgamer, and Onak, we obtain an alignment that is a (log n)^{O(1/epsilon^2)} approximation in time O~(n^{1 + epsilon}).
Closely related to the study of approximation algorithms is the study of metric embeddings for edit distance. We show that min-hash techniques can be useful in designing edit distance embeddings through three results: (1) An embedding from Ulam distance (edit distance over permutations) to Hamming space that matches the best known distortion of O(log n) and also implicitly encodes a sequence of edits between the strings; (2) In the case where the edit distance between the input strings is known to have an upper bound K, we show that embeddings of edit distance into Hamming space with distortion f(n) can be modified in a black-box fashion to give distortion O(f(poly(K))) for a class of periodic-free strings; (3) A randomized dimension-reduction map with contraction c and asymptotically optimal expected distortion O(c), improving on the previous O~(c^{1 + 2 / log log log n}) distortion result of Batu, Ergun, and Sahinalp
Min-Cost Bipartite Perfect Matching with Delays
In the min-cost bipartite perfect matching with delays (MBPMD) problem, requests arrive online at points of a finite metric space. Each request is either positive or negative and has to be matched to a request of opposite polarity. As opposed to traditional online matching problems, the algorithm does not have to serve requests as they arrive, and may choose to match them later at a cost. Our objective is to minimize the sum of the distances between matched pairs of requests (the connection cost) and the sum of the waiting times of the requests (the delay cost). This objective exhibits a natural tradeoff between minimizing the distances and the cost of waiting for better matches. This tradeoff appears in many real-life scenarios, notably, ride-sharing platforms. MBPMD is related to its non-bipartite variant, min-cost perfect matching with delays (MPMD), in which each request can be matched to any other request. MPMD was introduced by Emek et al. (STOC\u2716), who showed an O(log^2(n)+log(Delta))-competitive randomized algorithm on n-point metric spaces with aspect ratio Delta.
Our contribution is threefold. First, we present a new lower bound construction for MPMD and MBPMD. We get a lower bound of Omega(sqrt(log(n)/log(log(n)))) on the competitive ratio of any randomized algorithm for MBPMD. For MPMD, we improve the lower bound from Omega(sqrt(log(n))) (shown by Azar et al., SODA\u2717) to Omega(log(n)/log(log(n))), thus, almost matching their upper bound of O(log(n)). Second, we adapt the algorithm of Emek et al. to the bipartite case, and provide a simplified analysis that improves the competitive ratio to O(log(n)). The key ingredient of the algorithm is an O(h)-competitive randomized algorithm for MBPMD on weighted trees of height h. Third, we provide an O(h)-competitive deterministic algorithm for MBPMD on weighted trees of height h. This algorithm is obtained by adapting the algorithm for MPMD by Azar et al. to the apparently more complicated bipartite setting