Query-competitive algorithms for cheapest set problems under uncertainty

Considering the model of computing under uncertainty where element weights are uncertain but can be obtained at a cost by query operations, we study the problem of identifying a cheapest (minimum-weight) set among a given collection of feasible sets using a minimum number of queries of element weights. For the general case we present an algorithm that makes at most d ? OPT + d queries, where d is the maximum cardinality of any given set and OPT is the optimal number of queries needed to identify a cheapest set. For the minimum multi-cut problem in trees with d terminal pairs, we give an algorithm that makes at most d ? OPT + 1 queries. For the problem of computing a minimum-weight base of a given matroid, we give an algorithm that makes at most 2 ? OPT queries, generalizing a known result for the minimum spanning tree problem. For each of the above algorithms we give matching lower bounds. We also settle the complexity of the verification version of the general cheapest set problem and the minimum multi-cut problem in trees under uncertainty.

[1]  Thomas Erlebach,et al.  Query-Competitive Algorithms for Computing with Uncertainty , 2015, Bull. EATCS.

[2]  Daniel Vanderpooten,et al.  Min-max and min-max regret versions of combinatorial optimization problems: A survey , 2009, Eur. J. Oper. Res..

[3]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[4]  Sanjeev Khanna,et al.  On computing functions with uncertainty , 2001, PODS '01.

[5]  Thomas Erlebach,et al.  Query-Competitive Algorithms for Cheapest Set Problems under Uncertainty , 2014, MFCS.

[6]  Jennifer Widom,et al.  Offering a Precision-Performance Tradeoff for Aggregation Queries over Replicated Data , 2000, VLDB.

[7]  Maarten Löffler,et al.  Largest and Smallest Tours and Convex Hulls for Imprecise Points , 2006, SWAT.

[8]  Mihalis Yannakakis,et al.  Primal-dual approximation algorithms for integral flow and multicut in trees , 1997, Algorithmica.

[9]  Anders Yeo,et al.  A Polynomial Kernel for Multicut in Trees , 2009, STACS.

[10]  Thomas Erlebach,et al.  Computing Minimum Spanning Trees with Uncertainty , 2008, STACS.

[11]  David G. Kirkpatrick,et al.  Input-Thrifty Extrema Testing , 2011, ISAAC.

[12]  Dexter Kozen,et al.  The Design and Analysis of Algorithms , 1991, Texts and Monographs in Computer Science.

[13]  Alejandro López-Ortiz,et al.  On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods , 2014, Theory of Computing Systems.

[14]  Michael Hoffmann,et al.  Verification Problem of Maximal Points under Uncertainty , 2013, IWOCA.

[15]  Yogish Sabharwal,et al.  The Update Complexity of Selection and Related Problems , 2015, Theory of Computing Systems.

[16]  Rajeev Raman,et al.  Efficient Update Strategies for Geometric Computing with Uncertainty , 2003, CIAC.

[17]  Jennifer Widom,et al.  Computing the median with uncertainty , 2000, STOC '00.

[18]  Bernt Lindström.  A Theorem on Families of Sets , 1972, J. Comb. Theory, Ser. A.

[19]  Rajeev Motwani,et al.  Computing shortest paths with uncertainty , 2003, J. Algorithms.

[20]  Simon Kahan,et al.  A model for data in motion , 1991, STOC '91.

[21]  William J. Cook,et al.  Combinatorial Optimization: Cook/Combinatorial , 1997 .

[22]  Jiong Guo,et al.  Fixed-parameter tractability and data reduction for multicut in trees , 2005 .