Symmetric Randomized Dependent Rounding

Various forms of dependent rounding are useful when handling a mixture of “hard” (e.g., matroid) constraints and “soft” (packing) constraints. We consider a few classes of such problems that arise in facility location, where one aims for small additive violations of the packing constraints, and where we require substantial “near-independence” properties among the variables being rounded. While the classical works in discrepancy theory (Beck-Fiala (1981) and Karp et al. (1987)) as well as more-modern works such as those of Bansal-Nagarajan (2016) – which have many further properties – yield such small additive violations, they do not yield the forms of near-independence that we require. We develop a new rounding technique, block-selection rounding, which generalizes dependent rounding to allow multiple linear constraints and multiple choices for each item. We show that it satisfies near-independence properties, and use this to develop approximation algorithms for knapsack median and knapsack center problems. We anticipate that this technique will have more-general applicability. ∗Research supported in part by NSF Awards CNS-1010789, CCF-1422569 and CCF-1749864, and by research awards from Adobe, Inc. †Department of Computer Science, University of Maryland, College Park, MD 20742. Email: [email protected] ‡Department of Computer Science, University of Maryland, College Park, MD 20742. Email: [email protected] §Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. Email: [email protected] ¶Department of Computer Science, University of Maryland, College Park, MD 20742. Email: [email protected] 0 ar X iv :1 70 9. 06 99 5v 2 [ cs .D S] 9 A pr 2 01 8