107 research outputs found
Simpler and Better Algorithms for Minimum-Norm Load Balancing
Recently, Chakrabarty and Swamy (STOC 2019) introduced the minimum-norm load-balancing problem on unrelated machines, wherein we are given a set J of jobs that need to be scheduled on a set of m unrelated machines, and a monotone, symmetric norm; We seek an assignment sigma: J -> [m] that minimizes the norm of the resulting load vector load_{sigma} in R_+^m, where load_{sigma}(i) is the load on machine i under the assignment sigma. Besides capturing all l_p norms, symmetric norms also capture other norms of interest including top-l norms, and ordered norms. Chakrabarty and Swamy (STOC 2019) give a (38+epsilon)-approximation algorithm for this problem via a general framework they develop for minimum-norm optimization that proceeds by first carefully reducing this problem (in a series of steps) to a problem called min-max ordered load balancing, and then devising a so-called deterministic oblivious LP-rounding algorithm for ordered load balancing.
We give a direct, and simple 4+epsilon-approximation algorithm for the minimum-norm load balancing based on rounding a (near-optimal) solution to a novel convex-programming relaxation for the problem. Whereas the natural convex program encoding minimum-norm load balancing problem has a large non-constant integrality gap, we show that this issue can be remedied by including a key constraint that bounds the "norm of the job-cost vector." Our techniques also yield a (essentially) 4-approximation for: (a) multi-norm load balancing, wherein we are given multiple monotone symmetric norms, and we seek an assignment respecting a given budget for each norm; (b) the best simultaneous approximation factor achievable for all symmetric norms for a given instance
Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions
Two-stage stochastic optimization is a framework for modeling uncertainty,
where we have a probability distribution over possible realizations of the
data, called scenarios, and decisions are taken in two stages: we make
first-stage decisions knowing only the underlying distribution and before a
scenario is realized, and may take additional second-stage recourse actions
after a scenario is realized. The goal is typically to minimize the total
expected cost. A criticism of this model is that the underlying probability
distribution is itself often imprecise! To address this, a versatile approach
that has been proposed is the {\em distributionally robust 2-stage model}:
given a collection of probability distributions, our goal now is to minimize
the maximum expected total cost with respect to a distribution in this
collection.
We provide a framework for designing approximation algorithms in such
settings when the collection is a ball around a central distribution and the
central distribution is accessed {\em only via a sampling black box}.
We first show that one can utilize the {\em sample average approximation}
(SAA) method to reduce the problem to the case where the central distribution
has {\em polynomial-size} support. We then show how to approximately solve a
fractional relaxation of the SAA (i.e., polynomial-scenario
central-distribution) problem. By complementing this via LP-rounding algorithms
that provide {\em local} (i.e., per-scenario) approximation guarantees, we
obtain the {\em first} approximation algorithms for the distributionally robust
versions of a variety of discrete-optimization problems including set cover,
vertex cover, edge cover, facility location, and Steiner tree, with guarantees
that are, except for set cover, within -factors of the guarantees known
for the deterministic version of the problem
Near-Optimal and Robust Mechanism Design for Covering Problems with Correlated Players
We consider the problem of designing incentive-compatible, ex-post
individually rational (IR) mechanisms for covering problems in the Bayesian
setting, where players' types are drawn from an underlying distribution and may
be correlated, and the goal is to minimize the expected total payment made by
the mechanism. We formulate a notion of incentive compatibility (IC) that we
call {\em support-based IC} that is substantially more robust than Bayesian IC,
and develop black-box reductions from support-based-IC mechanism design to
algorithm design. For single-dimensional settings, this black-box reduction
applies even when we only have an LP-relative {\em approximation algorithm} for
the algorithmic problem. Thus, we obtain near-optimal mechanisms for various
covering settings including single-dimensional covering problems, multi-item
procurement auctions, and multidimensional facility location.Comment: Major changes compared to the previous version. Please consult this
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Welfare Maximization and Truthfulness in Mechanism Design with Ordinal Preferences
We study mechanism design problems in the {\em ordinal setting} wherein the
preferences of agents are described by orderings over outcomes, as opposed to
specific numerical values associated with them. This setting is relevant when
agents can compare outcomes, but aren't able to evaluate precise utilities for
them. Such a situation arises in diverse contexts including voting and matching
markets.
Our paper addresses two issues that arise in ordinal mechanism design. To
design social welfare maximizing mechanisms, one needs to be able to
quantitatively measure the welfare of an outcome which is not clear in the
ordinal setting. Second, since the impossibility results of Gibbard and
Satterthwaite~\cite{Gibbard73,Satterthwaite75} force one to move to randomized
mechanisms, one needs a more nuanced notion of truthfulness.
We propose {\em rank approximation} as a metric for measuring the quality of
an outcome, which allows us to evaluate mechanisms based on worst-case
performance, and {\em lex-truthfulness} as a notion of truthfulness for
randomized ordinal mechanisms. Lex-truthfulness is stronger than notions
studied in the literature, and yet flexible enough to admit a rich class of
mechanisms {\em circumventing classical impossibility results}. We demonstrate
the usefulness of the above notions by devising lex-truthful mechanisms
achieving good rank-approximation factors, both in the general ordinal setting,
as well as structured settings such as {\em (one-sided) matching markets}, and
its generalizations, {\em matroid} and {\em scheduling} markets.Comment: Some typos correcte
Improved Algorithms for MST and Metric-TSP Interdiction
We consider the MST-interdiction problem: given a multigraph G = (V, E), edge weights {w_e >= 0}_{e in E}, interdiction costs {c_e >= 0}_{e in E}, and an interdiction budget B >= 0, the goal is to remove a subset R of edges of total interdiction cost at most B so as to maximize the w-weight of an MST of G-R:=(V,E-R).
Our main result is a 4-approximation algorithm for this problem. This improves upon the previous-best 14-approximation [Zenklusen, FOCS 2015]. Notably, our analysis is also significantly simpler and cleaner than the one in [Zenklusen, FOCS 2015]. Whereas Zenklusen uses a greedy algorithm with an involved analysis to extract a good interdiction set from an over-budget set, we utilize a generalization of knapsack called the tree knapsack problem that nicely captures the key combinatorial aspects of this "extraction problem." We prove a simple, yet strong, LP-relative approximation bound for tree knapsack, which leads to our improved guarantees for MST interdiction. Our algorithm and analysis are nearly tight, as we show that one cannot achieve an approximation ratio better than 3 relative to the upper bound used in our analysis (and the one in [Zenklusen, FOCS 2015]).
Our guarantee for MST-interdiction yields an 8-approximation for metric-TSP interdiction (improving over the 28-approximation in [Zenklusen, FOCS 2015]). We also show that maximum-spanning-tree interdiction is at least as hard to approximate as the minimization version of densest-k-subgraph
Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications
We consider the matroid median problem, wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation by Krishnaswamy et al. We illustrate the power and versatility of our techniques by deriving: (a) an 8-approximation for the two-matroid median problem, a generalization of matroid median that we introduce involving two matroids; and (b) a 24-approximation algorithm for matroid median with penalties, which is a vast improvement over the 360-approximation obtained by Krishnaswamy et al. We show that a variety of seemingly disparate facility-location problems considered in the literature -- data placement problem, mobile facility location, k-median forest, metric uniform minimum-latency UFL -- in fact reduce to the matroid median or two-matroid median problems, and thus obtain improved approximation guarantees for all these problems. Our techniques also yield an improvement for the knapsack median problem
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