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Algorithms that Access the Input via Queries
Abstract
Problems where an algorithm cannot simply access the whole input but needs to obtain information about it using queries arise naturally in many settings. We discuss different aspects of models where an algorithm needs to query the input, and of how the performance of algorithms for such models can be measured. After that, we give some concrete examples of algorithmic settings and results for scenarios where algorithms access the input via queries. Finally, we discuss recent results for the setting of computing with explorable uncertainty with parallel queries and with untrusted predictions.
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In computing with explorable uncertainty, one considers problems where the values of some input elements are uncertain, typically represented as intervals, but can be obtained using queries. Previous work has considered query minimization in the settings where queries are asked sequentially (adaptive model) or all at once (non-adaptive model). We introduce a new model where k queries can be made in parallel in each round, and the goal is to minimize the number of query rounds. Using competitive analysis, we present upper and lower bounds on the number of query rounds required by any algorithm in comparison with the optimal number of query rounds for the given instance. Given a set of uncertain elements and a family of m subsets of that set, we study the problems of sorting all m subsets and of determining the minimum value (or the minimum element(s)) of each subset. We also study the selection problem, i.e., the problem of determining the i -th smallest value and identifying all elements with that value in a given set of uncertain elements. Our results include 2-round-competitive algorithms for sorting and selection and an algorithm for the minimum value problem that uses at most ( 2 + ε ) · opt k + O 1 ε · lg m query rounds for every 0 < ε < 1 , where opt k is the optimal number of query rounds.
Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of \emph{adaptive} testing algorithms, wherein each query may be determined by the answers received to prior queries, and their \emph{non-adaptive} counterparts, in which all queries are independent of answers obtained from previous queries. In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of "rounds of adaptivity" it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in k+1 rounds, where the queries in the i'th round may depend on the answers obtained in the previous rounds. Then, we ask the following question: Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity? We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every and there exists a property of functions for which (1) there exists a k-adaptive tester for with query complexity , yet (2) any -adaptive tester for must make queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs.
We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but
we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to
find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates
an optimal set-cover of size OPT within O(logN) factor with high probability using O( OPT ·log2
N ) queries where N is the number of elements in the universal set.
We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown
n-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem
we present an O(log2
n)-competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known
algorithm has a competitive ratio of W( Ö{nlogn} )\Omega ( \sqrt{n\log n} ) and therefore our result achieves an exponential improvement.
We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is uncertain. Initially, for each edge e of the graph only a set Ae, called an uncertainty area, that contains the actual edge weight we is known. The algorithm can ‘update ’ e to obtain the edge weight we ∈ Ae. The task is to output the edge set of a minimum spanning tree after a minimum number of updates. An algorithm is k-update competitive if it makes at most k times as many updates as the optimum. We present a 2-update competitive algorithm if all areas Ae are open or trivial, which is the best possible among deterministic algorithms. The condition on the areas Ae is to exclude degenerate inputs for which no constant update competitive algorithm can exist. Next, we consider a setting where the vertices of the graph correspond to points in Euclidean space and the weight of an edge is equal to the distance of its endpoints. The location of each point is initially given as an uncertainty area, and an update reveals the exact location of the point. We give a general relation between the edge uncertainty and the vertex uncertainty versions of a problem and use it to derive a 4-update competitive
Due to its fast, dynamic, and distributed growth process, it is hard to obtain an accurate map of the Internet. In many cases, such a map-representing the structure of the Internet as a graph with nodes and links-is a prerequisite when investigating properties of the Internet. A common way to obtain such maps is to make certain local measurements at a small subset of the nodes, and then to combine these in order to "discover" (an approximation of) the actual graph. Each of these measurements is potentially quite costly. It is thus a natural objective to minimize the number of measurements which still discover the whole graph. We formalize this problem as a combinatorial optimization problem and consider it for two different models characterized by different types of measurements. We give several upper and lower bounds on the competitive ratio (for the online network discovery problem) and the approximation ratio (for the offline network verification problem) in both models. Furthermore, for one of the two models, we compare four simple greedy strategies in an experimental analysis
How efficiently can we find an unknown graph using distance or shortest path queries between its vertices? We assume that the unknown graph G is connected, unweighted, and has bounded degree. In the reconstruction problem, the goal is to find the graph G. In the verification problem, we are given a hypothetical graph Ĝ and want to check whether G is equal to Ĝ.
We provide a randomized algorithm for reconstruction using Õ(n&frac;32) distance queries, based on Voronoi cell decomposition. Next, we analyze natural greedy algorithms for reconstruction using a shortest path oracle and also for verification using either oracle, and show that their query complexity is n1+o(1). We further improve the query complexity when the graph is chordal or outerplanar. Finally, we show some lower bounds, and consider an approximate version of the reconstruction problem.
We study the necessity of interaction between individuals for obtaining approximately efficient economic allocations. We view this as a formalization of Hayek's classic point of view that focuses on the information transfer advantages that markets have relative to centralized planning. We study two settings: combinatorial auctions with unit demand bidders (bipartite matching) and combinatorial auctions with subadditive bidders. In both settings we prove that non-interactive protocols require exponentially larger communication costs than do interactive ones, even ones that only use a modest amount of interaction.
Historically, there is a close connection between geometry and optImization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well. Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics. The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization. The second algorithm we discuss finds its roots in the classical "geometry of numbers", developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation.
Property testing is concerned with the design of super-fast algorithms for the structural analysis of large quantities of data. The aim is to unveil global features of the data, such as determining whether the data has a particular property or estimating global parameters. Remarkably, it is possible for decisions to be made by accessing only a small portion of the data. Property testing focuses on properties and parameters that go beyond simple statistics. This book provides an extensive and authoritative introduction to property testing. It provides a wide range of algorithmic techniques for the design and analysis of tests for algebraic properties, properties of Boolean functions, graph properties, and properties of distributions.
We study the problem of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n-vertex graph: one that uses only bipartite independent set queries, and the other one with independent set queries.
We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem: on query , the oracle returns the size of the cut between S and . We provide algorithms computing an exact minimum s-t cut in G with queries, and computing an exact global minimum cut of G with only queries (while learning the graph requires queries).
Given a graph with "uncertainty intervals" on the edges, we want to identify a minimum spanning tree by querying some edges for their exact edge weights which lie in the given uncertainty intervals. Our objective is to minimize the number of edge queries. It is known that there is a deterministic algorithm with best possible competitive ratio 2 [T. Erlebach, et al., in Proceedings of STACS, Schloss Dagstuhl, Dagstuhl, Germany, 2008, pp. 277-288]. Our main result is a randomized algorithm with expected competitive ratio 1 + 1/√2 ≈ 1.707, solving the long-standing open problem of whether an expected competitive ratio strictly less than 2 can be achieved [T. Erlebach and M. Hoffmann, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 116 (2015)]. We also present novel results for various extensions, including arbitrary matroids and more general querying models.
Blackbox optimization typically arises when the functions defining the objective and constraints of an optimization problem are computed through a computer simulation. The blackbox is expensive to compute, can have limited precision and can be contaminated with numerical noise. It may also fail to return a valid output, even when the input appears acceptable. Launching twice the simulation from the same input may produce different outputs. These unreliable properties are frequently encountered when dealing with real optimization problems. The term blackbox is used to indicate that the internal structure of the target problem, such as derivatives or their approximations, cannot be exploited as it may be unknown, hidden, unreliable, or inexistent. There are situations where some structure such as bounds or linear constraints may be exploited and in some cases a surrogate of the problem is supplied or a model may be constructed and trusted. This chapter surveys algorithms for this class of problems, including a supporting convergence analysis based on the nonsmooth calculus. The chapter also lists numerous published applications of these methods to real optimization problems. © 2014 Springer Science+Business Media New York. All rights are reserved.
In this paper, we consider the problem of reconstructing a hidden graph with m edges using additive queries. Given a graph G = (V;E) and a set of vertices S V , an additive query, Q(S), asks for the number of edges in the subgraph induced by S. The information theoretic lower bound for the query complexity of reconstructing a graph with n vertices and m edges is m log n 2 m logm ! : In this paper we give the rst polynomial time algorithm with query complexity that matches this lower bound1. This solves the open problem by (S. Choi and J. Han Kim. Optimal Query Complexity Bounds for Finding Graphs. STOC, 749{758, 2008). In the paper, we actually show an algorithm for the generalized problem of reconstructing weighted graphs. In the weighted case, an additive query, Q(S), asks for the sum of weights of edges in the subgraph induces by S. The complexity of the algorithm also matches the information theoretic lower bound.
Untrusted predictions improve trustable query policies
- T Erlebach
- M Hoffmann
- M S De Lima
- N Megow
- J Schlöter
Query-competitive algorithms for computing with uncertainty
- T Erlebach
- M Hoffmann
New query lower bounds for submodular function minimization
- A Graur
- T Pollner
- V Ramaswamy
- S M Weinberg
Query-based learning
- S Jain
- F Stephan
Uncertainty Exploration: Algorithms, Competitive Analysis, and Computational Experiments
- J Meißner
Query-competitive sorting with uncertainty
- M M Halldórsson
- M S De Lima
Improving online algorithms via ML predictions
- M Purohit
- Z Svitkina
- R Kumar
- S Bengio
- H M Wallach
- H Larochelle
- K Grauman
- N Cesa-Bianchi