1,720,979 research outputs found
Learning Partitions Using Rank Queries
Full text linkWe consider the problem of learning an unknown partition of an n element universe using rank queries. Such queries take as input a subset of the universe and return the number of parts of the partition it intersects. We give a simple O(n)-query, efficient, deterministic algorithm for this problem. We also generalize to give an O(n + klog r)-rank query algorithm for a general partition matroid where k is the number of parts and r is the rank of the matroid
Adaptive Boolean Monotonicity Testing in Total Influence Time
Full text linkTesting monotonicity of a Boolean function f:{0,1}^n -> {0,1} is an important problem in the field of property testing. It has led to connections with many interesting combinatorial questions on the directed hypercube: routing, random walks, and new isoperimetric theorems. Denoting the proximity parameter by epsilon, the best tester is the non-adaptive O~(epsilon^{-2}sqrt{n}) tester of Khot-Minzer-Safra (FOCS 2015). A series of recent results by Belovs-Blais (STOC 2016) and Chen-Waingarten-Xie (STOC 2017) have led to Omega~(n^{1/3}) lower bounds for adaptive testers. Reducing this gap is a significant question, that touches on the role of adaptivity in monotonicity testing of Boolean functions.
We approach this question from the perspective of parametrized property testing, a concept recently introduced by Pallavoor-Raskhodnikova-Varma (ACM TOCT 2017), where one seeks to understand performance of testers with respect to parameters other than just the size. Our result is an adaptive monotonicity tester with one-sided error whose query complexity is O(epsilon^{-2}I(f)log^5 n), where I(f) is the total influence of the function. Therefore, adaptivity provably helps monotonicity testing for low influence functions
Learning Spanning Forests Optimally using CUT Queries in Weighted Undirected Graphs
Full text linkIn this paper we describe a randomized algorithm which returns a maximal
spanning forest of an unknown {\em weighted} undirected graph making
queries in expectation. For weighted graphs, this is optimal due
to a result in [Auza and Lee, 2021] which shows an lower bound for
zero-error randomized algorithms. %To our knowledge, it is the only regime of
this problem where we have upper and lower bounds tight up to constants. These
questions have been extensively studied in the past few years, especially due
to the problem's connections to symmetric submodular function minimization. We
also describe a simple polynomial time deterministic algorithm that makes
queries on undirected unweighted graphs and
returns a maximal spanning forest, thereby (slightly) improving upon the
state-of-the-art
A Primal-Dual Analysis of Monotone Submodular Maximization
Full text linkIn this paper we design a new primal-dual algorithm for the classic discrete
optimization problem of maximizing a monotone submodular function subject to a
cardinality constraint achieving the optimal approximation of . This
problem and its special case, the maximum -coverage problem, have a wide
range of applications in various fields including operations research, machine
learning, and economics. While greedy algorithms have been known to achieve
this approximation factor, our algorithms also provide a dual certificate which
upper bounds the optimum value of any instance. This certificate may be used in
practice to certify much stronger guarantees than the worst-case
approximation factor
Learning Partitions using Rank Queries
Full text linkWe consider the problem of learning an unknown partition of an element universe using rank queries. Such queries take as input a subset of the universe and return the number of parts of the partition it intersects. We give a simple -query, efficient, deterministic algorithm for this problem. We also generalize to give an -rank query algorithm for a general partition matroid where is the number of parts and is the rank of the matroid
Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling
Full text linkGiven a non-negative real matrix A, the matrix scaling problem is to determine if it is possible to scale the rows and columns so that each row and each column sums to a specified target value for it.
The matrix scaling problem arises in many algorithmic applications, perhaps most notably as a preconditioning step in solving linear system of equations. One of the most natural and by now classical approach to matrix scaling is the Sinkhorn-Knopp algorithm (also known as the RAS method) where one alternately scales either all rows or all columns to meet the target values. In addition to being extremely simple and natural, another appeal of this procedure is that it easily lends itself to parallelization. A central question is to understand the rate of convergence of the Sinkhorn-Knopp algorithm.
Specifically, given a suitable error metric to measure deviations from target values, and an error bound epsilon, how quickly does the Sinkhorn-Knopp algorithm converge to an error below epsilon? While there are several non-trivial convergence results known about the Sinkhorn-Knopp algorithm, perhaps somewhat surprisingly, even for natural error metrics such as ell_1-error or ell_2-error, this is not entirely understood.
In this paper, we present an elementary convergence analysis for the Sinkhorn-Knopp algorithm that improves upon the previous best bound. In a nutshell, our approach is to show (i) a simple bound on the number of iterations needed so that the KL-divergence between the current row-sums and the target row-sums drops below a specified threshold delta, and (ii) then show that for a suitable choice of delta, whenever KL-divergence is below delta, then the ell_1-error or the ell_2-error is below epsilon. The well-known Pinsker's inequality immediately allows us to translate a bound on the KL divergence to a bound on ell_1-error. To bound the ell_2-error in terms of the KL-divergence, we establish a new inequality, referred to as (KL vs ell_1/ell_2) inequality in the paper. This new inequality is a strengthening of the Pinsker's inequality that we believe is of independent interest. Our analysis of ell_2-error significantly improves upon the best previous convergence bound for ell_2-error.
The idea of studying Sinkhorn-Knopp convergence via KL-divergence is not new and has indeed been previously explored. Our contribution is an elementary, self-contained presentation of this approach and an interesting new inequality that yields a significantly stronger convergence guarantee for the extensively studied ell_2-error
Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median
Full text linkWe consider a generalization of k-median and k-center, called the ordered k-median problem. In this problem, we are given a metric space (D,{c_{ij}}) with n=|D| points, and a non-increasing weight vector w in R_+^n, and the goal is to open k centers and assign each point j in D to a center so as to minimize w_1 *(largest assignment cost)+w_2 *(second-largest assignment cost)+...+w_n *(n-th largest assignment cost). We give an (18+epsilon)-approximation algorithm for this problem. Our algorithms utilize Lagrangian relaxation and the primal-dual schema, combined with an enumeration procedure of Aouad and Segev. For the special case of {0,1}-weights, which models the problem of minimizing the l largest assignment costs that is interesting in and of by itself, we provide a novel reduction to the (standard) k-median problem, showing that LP-relative guarantees for k-median translate to guarantees for the ordered k-median problem; this yields a nice and clean (8.5+epsilon)-approximation algorithm for {0,1} weights
Generalized Center Problems with Outliers
Full text linkWe study the F-center problem with outliers: given a metric space (X,d), a general down-closed family F of subsets of X, and a parameter m, we need to locate a subset S in F of centers such that the maximum distance among the closest m points in X to S is minimized.
Our main result is a dichotomy theorem. Colloquially, we prove that there is an efficient 3-approximation for the F-center problem with outliers if and only if we can efficiently optimize a poly-bounded linear function over F subject to a partition constraint. One concrete upshot of our result is a polynomial time 3-approximation for the knapsack center problem with outliers for which no (true) approximation algorithm was known
Directed Hypercube Routing, a Generalized Lehman-Ron Theorem, and Monotonicity Testing
Full text linkMotivated by applications to monotonicity testing, Lehman and Ron (JCTA, 2001) proved the existence of a collection of vertex disjoint paths between comparable sub-level sets in the directed hypercube. The main technical contribution of this paper is a new proof method that yields a generalization of their theorem: we prove the existence of two edge-disjoint collections of vertex disjoint paths. Our main conceptual contributions are conjectures on directed hypercube flows with simultaneous vertex and edge capacities of which our generalized Lehman-Ron theorem is a special case. We show that these conjectures imply directed isoperimetric theorems, and in particular, the robust directed Talagrand inequality due to Khot, Minzer, and Safra (SIAM J. on Comp, 2018). These isoperimetric inequalities, that relate the directed surface area (of a set in the hypercube) to its distance to monotonicity, have been crucial in obtaining the best monotonicity testers for Boolean functions. We believe our conjectures pave the way towards combinatorial proofs of these directed isoperimetry theorems
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