We consider column-sparse covering integer programs, a generalization of set cover, which have attracted a long line of research developing (randomized) approximation algorithms. We develop a new rounding scheme based on the Partial Resampling variant of the Lovász Local Lemma developed by Harris & Srinivasan (2013).This achieves an approximation ratio of 1 +, where a min is the minimum covering constraint and ∆ 1 is the maximum ℓ 1 -norm of any column of the covering matrix (whose entries are scaled to lie in [0, 1]). When there are additional constraints on the sizes of the variables, we show an approximation ratio of 1 + O log(∆1+1) aminǫ + log(∆1+1) amin to satisfy these size constraints up to multiplicative factor 1 + ǫ, or an approximation ratio of ln ∆ 0 + O( √ log ∆ 0 ) to satisfy the size constraints exactly (where ∆ 0 is the maximum number of non-zero entries in any column of the covering matrix). We also show nearly-matching inapproximability and integrality-gap lower bounds. These results improve asymptotically, in several different ways, over results shown by Srinivasan (2006) and Kolliopoulos & Young (2005).We show also that the rounding process leads to negative correlation among the variables. This allows us to automatically handle multi-criteria programs, efficiently achieving approximation ratios which are essentially equivalent to the single-criterion case and apply even when the number of criteria is large.

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In the past 20 years, increasing complexity in real world data has lead to the study of higher-order data models based on partitioning hypergraphs. However, hypergraph partitioning admits multiple formulations as hyperedges can be cut in multiple ways. Building upon a class of hypergraph partitioning problems introduced by Li & Milenkovic [36], we study the problem of minimizing ratio-cut objectives over hypergraphs given by a new class of cut functions, monotone submodular cut functions (mscfs), which captures hypergraph expansion and conductance as special cases.We first define the ratio-cut improvement problem, a family of local relaxations of the minimum ratiocut problem. This problem is a natural extension of the Andersen & Lang cut improvement problem [5] to the hypergraph setting. We demonstrate the existence of efficient algorithms for approximately solving this problem. These algorithms run in almost-linear time for the case of hypergraph expansion, and when the hypergraph rank is at most O(1).Next, we provide an efficient O(log n)-approximation algorithm for finding the minimum ratio-cut of G. We generalize the cut-matching game framework of Khandekar et al. [31] to allow for the cut player to play unbalanced cuts, and matching player to route approximate single-commodity flows. Using this framework, we bootstrap our algorithms for the ratio-cut improvement problem to obtain approximation algorithms for minimum ratio-cut problem for all mscfs. This also yields the first almost-linear time O(log n)-approximation algorithms for the case of hypergraph expansion, and constant hypergraph rank.Finally, we extend a result of Louis & Makarychev [41] to a broader set of objective functions by giving a polynomial time O √ log n -approximation algorithm for the minimum ratio-cut problem based on rounding ℓ 2 2 -metric embeddings.

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