A number of recent works have studied algorithms for entrywise p-low rank approximation, namely algorithms which given an n×d matrix A (with n ≥ d), output a rank-k matrix B minimizing A − B p p = i,j |Ai,j − Bi,j| p when p > 0; and A − B 0 = i,j [Ai,j = Bi,j] for p = 0, where [·] is the Iverson bracket, that is, A − B 0 denotes the number of entries (i, j) for which Ai,j = Bi,j. For p = 1, this is often considered more robust than the SVD, while for p = 0 this corresponds to minimizing the number of disagreements, or robust PCA. This problem is known to be NP-hard for p ∈ {0, 1}, already for k = 1, and while there are polynomial time approximation algorithms, their approximation factor is at best poly(k). It was left open if there was a polynomial-time approximation scheme (PTAS) for p-approximation for any p ≥ 0. We show the following:1. On the algorithmic side, for p ∈ (0, 2), we give the first n poly(k/ε) time (1 + ε)approximation algorithm. For p = 0, there are various problem formulations, a common one being the binary setting in which A ∈ {0, 1} n×d and B = U · V , where U ∈ {0, 1} n×k and V ∈ {0, 1} k×d . There are also various notions of multiplication U · V , such as a matrix product over the reals, over a finite field, or over a Boolean semiring. We give the first almost-linear time approximation scheme for what we call the Generalized Binary 0-Rank-k problem, for which these variants are special cases. Our algorithm computes (1 + ε)-approximation in time (1/ε) 2 O(k) /ε 2 · nd 1+o(1) , where o(1) hides a factor (log log d) 1.1 / log d. In addition, for the case of finite fields of constant size, we obtain an alternate PTAS running in time n · d poly(k/ε) . Definition 2. (Generalized Binary 0 -Rank-k) Given a matrix A ∈ {0, 1} n×d with n ≥ d, an integer k, and an inner product function ., . :Our first result for p = 0 is as follows.Theorem 2 (PTAS for p = 0). For any ε ∈ (0, 1 2 ), there is a (1+ε)-approximation algorithm for the Generalized Binary 0 -Rank-k problem running in time (1/ε) 2 O(k) /ε 2 · nd 1+o(1) and succeeds with constant probability 1 , where o(1) hides a factor (log log d)Hence, we obtain the first almost-linear time approximation scheme for the Generalized Binary 0 -Rank-k problem, for any constant k. In particular, this yields the first polynomial time (1+ε)-approximation for constant k for 0 -low rank approximation of binary matrices when the underlying field is F 2 or the Boolean semiring. Even for k = 1, no PTAS was known before.Theorem 2 is doubly-exponential in k, and we show below that this is necessary for any approximation algorithm for Generalized Binary 0 -Rank-k. However, in the special case when the base field is F 2 , or more generally F q and A, U, and V have entries belonging to F q , it is possible to obtain an algorithm running in time n·d poly(k/ε) , which is an improvement for certain super-constant values of k and ε. We formally define the problem and state our result next. Definition 3. (Entrywise 0 -Rank-k Approximation over F q ) Given an n × d matrix A with e...

The k-means problem consists of finding k centers in R d that minimize the sum of the squared distances of all points in an input set P from R d to their closest respective center. Awasthi et. al. recently showed that there exists a constant ε ′ > 1 such that it is NP-hard to approximate the k-means objective within a factor of c. We establish that the constant ε ′ is at least 1.0013.For a given set of points P ⊂ R d , the k-means problem consists of finding a partition of P into k clusters (C 1 , . . . , C k ) with corresponding centers (c 1 , . . . , c k ) that minimize the sum of the squared distances of all points in P to their corresponding center, i.e. the quantity arg minwhere || · || denotes the Euclidean distance. The k-means problem has been well-known since the fifties, when Lloyd [Llo57] developed the famous local search heuristic also known as the k-means algorithm. Various exact, approximate, and heuristic algorithms have been developed since then. For a constant number of clusters k and a constant dimension d, the problem can be solved by enumerating weighted Voronoi diagrams [IKI94]. If the dimension is arbitrary but the number of centers is constant, many polynomialtime approximation schemes are known. For example, [FL11] gives an algorithm with running time O(nd + 2 poly(1/ε,k) ). In the general case, only constant-factor approximation algorithms are known [JV01, KMN + 04], but no algorithm with an approximation ratio smaller than 9 has yet been found. Surprisingly, no hardness results for the k-means problem were known even as recently as ten years ago. Today, it is known that the k-means problem is NP-hard, even for constant k and arbitrary dimension d [ADHP09, Das08] and also for arbitrary k and

We study the problem of allocating a set of indivisible items to agents with additive utilities to maximize the Nash social welfare. Cole and Gkatzelis [3] recently proved that this problem admits a constant factor approximation. We complement their result by showing that this problem is APX-hard.

In the k-Cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. The current best algorithms are an O(n (2−o(1))k ) randomized algorithm due to Karger and Stein, and anÕ(n 2k ) deterministic algorithm due to Thorup. Moreover, several 2-approximation algorithms are known for the problem (due to Saran and Vazirani, Naor and Rabani, and Ravi and Sinha).It has remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this paper we show an O(k O(k) n (2ω/3+o(1))k )-time algorithm for k-Cut. Moreover, we show an (1 + ε)-approximation algorithm that runs in time O((k/ε) O(k) n k+O(1) ), and a 1.81-approximation in fixed-parameter time O(2 OIn this paper we consider the k-Cut problem: given an edge-weighted graph G = (V, E, w) and an integer k, delete a minimum-weight set of edges so that G has at least k connected components. This problem is a natural generalization of the global min-cut problem, where the goal is to break the graph into k = 2 pieces. This problem has been actively studied in theory of both exact and approximation algorithms, where each result brought new insights and tools on graph cuts.It is not a priori clear how to obtain poly-time algorithms for any constant k, since guessing one vertex from each part only reduces the problem to the NP-hard Multiway Cut problem. Indeed, the first result along these lines was the work of Goldschmidt and Hochbaum [GH94] who gave an O(n (1/2−o(1))k 2 )-time exact algorithm for k-Cut. Since then, the exact exponent in terms of k has been actively studied. The current best runtime is achieved by an O(n 2(k−1) ) randomized algorithm due to Karger and Stein [KS96], which performs random edge contractions until the remaining graph has k nodes, and shows that the resulting cut is optimal with probability at least Ω(n −2(k−1) ). The asymptotic runtime ofÕ(n 2(k−1) ) was later matched by a deterministic algorithm of Thorup [Tho08]. His algorithm was based on tree-packing theorems; it showed how to efficiently find a tree for which the optimal k-cut crosses it 2k − 2 times. Enumerating over all possible 2k − 2 edges of this tree gives the algorithm.These elegant O(n 2k )-time algorithms are the state-of-the-art, and it has remained an open question to improve on them. An easy observation is that the problem is closely related to k-Clique, so we may not expect the exponent of n to go below (ω/3)k. Given the interest in fine-grained analysis of algorithms, where in the range [(ω/3)k, 2k − 2] does the correct answer lie?Our main results give faster deterministic and randomized algorithms for the problem.Theorem 1.1 (Faster Randomized Algorithm). Let W be a positive integer. There is a randomized algorithm for k-Cut on graphs with edge weights in [W ] with runtimethat succeeds with probability 1 − 1/poly(n).Theorem 1.2 (Even Faster Deterministic Algorithm). Let W be a positive integer. For any ε >...

Given a graph G = (V, E) and an integer k ∈ N, we study k-Vertex Separator (resp. k-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has at most k vertices. Our primary focus is on the case where k is either a constant or a slowly growing function of n (e.g. O(log n) or n o(1) ). Our problems can be interpreted as a special case of three general classes of problems that have been studied separately (balanced graph partitioning, Hypergraph Vertex Cover (HVC), and fixed parameter tractability (FPT)).Our main result is an O(log k)-approximation algorithm for k-Vertex Separator that runs in time 2, and an O(log k)-approximation algorithm for k-Edge Separator that runs in time n O(1) . Our result on k-Edge Separator improves the best previous graph partitioning algorithm [KNS09] for small k. Our result on k-Vertex Separator improves the simple (k + 1)-approximation from HVC [BAMSN15]. When OPT > k, the running time 2 O(k) n O(1) is faster than the lower bound k Ω(OPT) n Ω(1) for exact algorithms assuming the Exponential Time Hypothesis [DDvH14]. While the running time of 2 O(k) n O(1) for k-Vertex Separator seems unsatisfactory, we show that the superpolynomial dependence on k may be needed to achieve a polylogarithmic approximation ratio, based on hardness of Densest k-Subgraph.We also study k-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. With additional ideas from FPT algorithms and graph theory, we present an O(log k)-approximation algorithm for k-Path Transversal that runs in time 2 O(k 3 log k) n O(1) . Previously, the existence of even (1−δ)k-approximation algorithm for fixed δ > 0 was open [Cam15].