In this paper, we present a new iterative rounding framework for many clustering problems. Using this, we obtain an (α 1 + ≤ 7.081 + )-approximation algorithm for k-median with outliers, greatly improving upon the large implicit constant approximation ratio of Chen [16]. For k-means with outliers, we give an (α 2 + ≤ 53.002 + )-approximation, which is the first O(1)-approximation for this problem. The iterative algorithm framework is very versatile; we show how it can be used to give α 1 -and (α 1 + )approximation algorithms for matroid and knapsack median problems respectively, improving upon the previous best approximations ratios of 8 [42] and 17.46 [9].The natural LP relaxation for the k-median/k-means with outliers problem has an unbounded integrality gap. In spite of this negative result, our iterative rounding framework shows that we can round an LP solution to an almost-integral solution of small cost, in which we have at most two fractionally open facilities. Thus, the LP integrality gap arises due to the gap between almost-integral and fully-integral solutions. Then, using a pre-processing procedure, we show how to convert an almost-integral solution to a fully-integral solution losing only a constant-factor in the approximation ratio. By further using a sparsification technique, the additive factor loss incurred by the conversion can be reduced to any > 0. * Microsoft Research India. for general and Euclidean metrics respectively. Both problems admit PTASs [2, 21, 19] on fixed-dimensional Euclidean metrics.Despite their simplicity and elegance, a significant shortcoming these formulations face in real-world data sets is that they are not robust to noisy points, i.e., a few outliers can completely change the cost as well as structure of solutions. To overcome this shortcoming, Charikar et al. [12] introduced the robust k-median (RkMed) problem (also called k-median with outliers), which we now define.Definition 1.1 (The Robust k-Median and k-Means Problems). The input to the Robust k-Median (RkMed) problem is a set C of clients, a set F of facility locations, a metric space (C ∪ F, d), integers k and m. The objective is to choose a subset S ⊆ F of cardinality at most k, and a subset C * ⊆ C of cardinality at least m such that the total cost j∈C * d(j, S) is minimized. In the Robust k-Means (RkMeans) problem, we have the same input and the goal is to minimize j∈C * d 2 (j, S).The problem is not just interesting from the clustering point of view. In fact, such a joint view of clustering and removing outliers has been observed to be more effective [15,39] for even the sole task of outlier detection, a very important problem in the real world. Due to these use cases, there has been much recent work [15,23,40] in the applied community on these problems. However, their inherent complexity from the theoretical side is much less understood. For RkMed, Charikar et al. [12] give an algorithm that violates the number of outliers by a factor of (1 + ), and has cost at most 4(1 + 1/ ) times the optimal cost. Chen [1...

Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be d-dimensional vectors. While the approximability of the scalar problems is well understood, there has been a significant gap between the approximation algorithms and the hardness results for the multidimensional variants. In this paper, we close this gap by giving almost tight hardness results for these problems.

A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k ≥ 3 and r ≥ 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r = k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r = k − 1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := k−2 2 colors. In particular, for k ≤ 6, it is NP-hard to 2-color (k − 1)-rainbow colorable k-uniform hypergraphs.Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r] n . We prove that any such polymorphism f : [r] n → [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a ∈ [q] S such that fixing x |S = a determines the value of f (x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover.

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