Adaptive Uncertainty Resolution in Bayesian Combinatorial Optimization Problems
TL;DR: In this paper, the authors show that while probing yields significant improvement in the objective function, being adaptive about the probing is not beneficial beyond constant factors, whereas non-adaptive observations can be performed in parallel.
Abstract: In several applications such as databases, planning, and sensor networks, parameters such as selectivity, load, or sensed values are known only with some associated uncertainty. The performance of such a system (as captured by some objective function over the parameters) is significantly improved if some of these parameters can be probed or observed. In a resource constrained situation, deciding which parameters to observe in order to optimize system performance, itself becomes an interesting and important optimization problem. This general problem is the focus of this article.One of the most important considerations in this framework is whether adaptivity is required for the observations. Adaptive observations introduce blocking or sequential operations in the system whereas nonadaptive observations can be performed in parallel. One of the important questions in this regard is to characterize the benefit of adaptivity for probes and observation.We present general techniques for designing constant factor approximations to the optimal observation schemes for several widely used scheduling and metric objective functions. We show a unifying technique that relates this optimization problem to the outlier version of the corresponding deterministic optimization. By making this connection, our technique shows constant factor upper bounds for the benefit of adaptivity of the observation schemes. We show that while probing yields significant improvement in the objective function, being adaptive about the probing is not beneficial beyond constant factors.
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10 Jan 2016
TL;DR: The first approximation algorithms for a number of stochastic probing problems, which have applications, e.g., to path-planning and precedence-constrained scheduling, are obtained.
Abstract: A stochastic probing problem consists of a set of elements whose values are independent random variables. The algorithm knows the distributions of these variables, but not the actual outcomes. The only way to learn the actual outcomes is to probe these elements. However, there are constraints on which set of elements may be probed. (E.g., we may have to travel in some metric to probe elements but have limited time.) These constraints are called outer constraints. We want to develop an algorithm that picks some set of elements to maximize the (expected) value, subject to the picked subset of elements satisfying some other set of constraints, called the inner constraints. In the past, probing problems were studied for the case when both inner and outer constraints were intersections of matroids; these modeled kidney matching and Bayesian auctions applications. One limitation of past work was their reliance on linear-programming-like techniques, which made going beyond matroid-like structures difficult.In this work, we give a very general adaptivity gap result that holds for all prefix-closed outer constraints, as long as the inner constraints are intersections of matroids. The adaptivity gap is O(log n) for any constant number of inner matroid constraints. The prefix-closedness captures most "reasonable" outer constraints, like orienteering, connectivity, and precedence. Based on this we obtain the first approximation algorithms for a number of stochastic probing problems, which have applications, e.g., to path-planning and precedence-constrained scheduling.
59 citations
13 Dec 2010
TL;DR: This paper models this selective call out as an online recurrent Bayesian decision framework with bandwidth type constraints, and obtains natural algorithms with bounded performance guarantees for several natural optimization criteria.
Abstract: Ads on the Internet are increasingly sold via ad exchanges such as RightMedia, AdECN and Doubleclick Ad Exchange. These exchanges allow real-time bidding, that is, each time the publisher contacts the exchange, the exchange "calls out" to solicit bids from ad networks. This solicitation introduces a novel aspect, in contrast to existing literature. This suggests developing a joint optimization framework which optimizes over the allocation and well as solicitation.
We model this selective call out as an online recurrent Bayesian decision framework with bandwidth type constraints. We obtain natural algorithms with bounded performance guarantees for several natural optimization criteria. We show that these results hold under different call out constraint models, and different arrival processes. Interestingly, the paper shows that under MHR assumptions, the expected revenue of generalized second price auction with reserve is constant factor of the expected welfare. Also the analysis herein allow us prove adaptivity gap type results for the adwords problem.
51 citations
TL;DR: The two main results of the paper show that for α>273⋅n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that forα>129 all equibria in MaxGame are Trees and the price is constant.
Abstract: We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1,2,…,n}, each identified with a vertex of a graph (network), where the strategy of player i, i=1,…,n, is to build some edges adjacent to i. The cost of building an edge is α>0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SumGame variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MaxGame variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α>273⋅n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that for α>129 all equilibria in MaxGame are trees and the price of anarchy is constant. For SumGame this answers (almost completely) one of the fundamental open problems in the field—is price of anarchy of the network creation game constant for all values of α?—in an affirmative way, up to a tiny range of α.
51 citations
TL;DR: A rounding method based on random walks in polytopes is developed, which leads to improved approximation algorithms and integrality gaps for several assignment problems that arise in resource allocation and scheduling.
Abstract: We develop a rounding method based on random walks in polytopes, which leads to improved approximation algorithms and integrality gaps for several assignment problems that arise in resource allocation and scheduling. In particular, it generalizes the work of Shmoys & Tardos on the generalized assignment problem in two different directions, where the machines have hard capacities, and where some jobs can be dropped. We also outline possible applications and connections of this methodology to discrepancy theory and iterated rounding.
43 citations
TL;DR: In this paper, the authors developed a model to identify critical nodes in the supply chain of COVID-19 vaccine distribution in order to ensure a minimum service level (67%) under the possibility of lead time disruptions.
33 citations
References
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31 Aug 2004
TL;DR: This paper enrichs interactive sensor querying with statistical modeling techniques, and demonstrates that such models can help provide answers that are both more meaningful, and, by introducing approximations with probabilistic confidences, significantly more efficient to compute in both time and energy.
Abstract: Declarative queries are proving to be an attractive paradigm for ineracting with networks of wireless sensors. The metaphor that "the sensornet is a database" is problematic, however, because sensors do not exhaustively represent the data in the real world. In order to map the raw sensor readings onto physical reality, a model of that reality is required to complement the readings. In this paper, we enrich interactive sensor querying with statistical modeling techniques. We demonstrate that such models can help provide answers that are both more meaningful, and, by introducing approximations with probabilistic confidences, significantly more efficient to compute in both time and energy. Utilizing the combination of a model and live data acquisition raises the challenging optimization problem of selecting the best sensor readings to acquire, balancing the increase in the confidence of our answer against the communication and data acquisition costs in the network. We describe an exponential time algorithm for finding the optimal solution to this optimization problem, and a polynomial-time heuristic for identifying solutions that perform well in practice. We evaluate our approach on several real-world sensor-network data sets, taking into account the real measured data and communication quality, demonstrating that our model-based approach provides a high-fidelity representation of the real phenomena and leads to significant performance gains versus traditional data acquisition techniques.
1,218 citations
TL;DR: A new extension of the primal-dual schema and the use of Lagrangian relaxation to derive approximation algorithms for the metric uncapacitated facility location problem and the metric k-median problem achieving guarantees of 3 and 6 respectively.
Abstract: We present approximation algorithms for the metric uncapacitated facility location problem and the metric k-median problem achieving guarantees of 3 and 6 respectively. The distinguishing feature of our algorithms is their low running time: O(m logm) and O(m logm(L + log (n))) respectively, where n and m are the total number of vertices and edges in the underlying complete bipartite graph on cities and facilities. The main algorithmic ideas are a new extension of the primal-dual schema and the use of Lagrangian relaxation to derive approximation algorithms.
872 citations
TL;DR: The first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems are derived.
Abstract: We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems.
Our technique produces approximation algorithms that run in $O(n^2\log n)$ time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of $O(n^2\log n)$ time compares favorably with the best strongly polynomial exact algorithms running in $O(n^3)$ time for dense graphs. A similar result is obtained for the 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413--420].
809 citations
Proceedings Article•
01 Sep 1992
TL;DR: A 2-approximation algorithm for the Steiner tree problem was given in this paper with running time of O(n 2 log n) for the shortest path problem, where n is the number of vertices in a graph.
Abstract: We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems.Our technique produces approximation algorithms that run in O(n2 log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of O(n2 log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n3) time for dense graphs. A similar result is obtained for the 2-matching problem and its variants.We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [4].
773 citations
TL;DR: This work analyzes local search heuristics for the metric k-median and facility location problems and shows that local search with swaps has a locality gap of 5 and introduces a new local search operation which opens one or more copies of a facility and drops zero or more facilities.
Abstract: We analyze local search heuristics for the metric k-median and facility location problems. We define the locality gap of a local search procedure for a minimization problem as the maximum ratio of a locally optimum solution (obtained using this procedure) to the global optimum. For k-median, we show that local search with swaps has a locality gap of 5. Furthermore, if we permit up to p facilities to be swapped simultaneously, then the locality gap is 3+2/p. This is the first analysis of a local search for k-median that provides a bounded performance guarantee with only k medians. This also improves the previous known 4 approximation for this problem. For uncapacitated facility location, we show that local search, which permits adding, dropping, and swapping a facility, has a locality gap of 3. This improves the bound of 5 given by M. Korupolu, C. Plaxton, and R. Rajaraman [Analysis of a Local Search Heuristic for Facility Location Problems, Technical Report 98-30, DIMACS, 1998]. We also consider a capacitated facility location problem where each facility has a capacity and we are allowed to open multiple copies of a facility. For this problem we introduce a new local search operation which opens one or more copies of a facility and drops zero or more facilities. We prove that this local search has a locality gap between 3 and 4.
671 citations