Johnson Coverage Hypothesis: Inapproximability of k-means and k-median in L_p metrics

K-median and k-means are the two most popular objectives for clustering algorithms. Despite intensive effort, a good understanding of the approximability of these objectives, particularly in $\ell_p$-metrics, remains a major open problem. In this paper, we significantly improve upon the hardness of approximation factors known in literature for these objectives in $\ell_p$-metrics. We introduce a new hypothesis called the Johnson Coverage Hypothesis (JCH), which roughly asserts that the well-studied max k-coverage problem on set systems is hard to approximate to a factor greater than 1-1/e, even when the membership graph of the set system is a subgraph of the Johnson graph. We then show that together with generalizations of the embedding techniques introduced by Cohen-Addad and Karthik (FOCS '19), JCH implies hardness of approximation results for k-median and k-means in $\ell_p$-metrics for factors which are close to the ones obtained for general metrics. In particular, assuming JCH we show that it is hard to approximate the k-means objective: $\bullet$ Discrete case: To a factor of 3.94 in the $\ell_1$-metric and to a factor of 1.73 in the $\ell_2$-metric; this improves upon the previous factor of 1.56 and 1.17 respectively, obtained under UGC. $\bullet$ Continuous case: To a factor of 2.10 in the $\ell_1$-metric and to a factor of 1.36 in the $\ell_2$-metric; this improves upon the previous factor of 1.07 in the $\ell_2$-metric obtained under UGC. We also obtain similar improvements under JCH for the k-median objective. Additionally, we prove a weak version of JCH using the work of Dinur et al. (SICOMP '05) on Hypergraph Vertex Cover, and recover all the results stated above of Cohen-Addad and Karthik (FOCS '19) to (nearly) the same inapproximability factors but now under the standard NP$\neq$P assumption (instead of UGC).

On Approximability of Clustering Problems Without Candidate Centers

The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the approximability of k-means, and other classic clustering problems such as k-median and k-minsum, our knowledge of the hardness of approximation factors of these problems remains quite poor. In this paper, we significantly improve upon the hardness of approximation factors known in the literature for these objectives. We show that if the input lies in a general metric space, it is NP-hard to approximate: $\bullet$ Continuous k-median to a factor of $2-o(1)$; this improves upon the previous inapproximability factor of 1.36 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ Continuous k-means to a factor of $4- o(1)$; this improves upon the previous inapproximability factor of 2.10 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ k-minsum to a factor of $1.415$; this improves upon the APX-hardness shown by Guruswami and Indyk (SODA '03). Our results shed new and perhaps counter-intuitive light on the differences between clustering problems in the continuous setting versus the discrete setting (where the candidate centers are given as part of the input).

A PTAS for $\ell_p$-Low Rank Approximation

A number of recent works have studied algorithms for entrywise $\ell_p$-low rank approximation, namely algorithms which given an $n \times d$ matrix $A$ (with $n \geq d$), output a rank-$k$ matrix $B$ minimizing $\|A-B\|_p^p=\sum_{i,j} |A_{i,j} - B_{i,j}|^p$. We show the following: On the algorithmic side, for $p \in (0,2)$, we give the first $n^{\text{poly}(k/\epsilon)}$ time $(1+\epsilon)$-approximation algorithm. For $p = 0$, there are various problem formulations, a common one being the binary setting for which $A\in\{0,1\}^{n\times d}$ and $B = U \cdot V$, where $U\in\{0,1\}^{n \times k}$ and $V\in\{0,1\}^{k \times d}$. There are also various notions of multiplication $U \cdot V$, such as a matrix product over the reals, over a finite field, or over a Boolean semiring. We give the first PTAS for what we call the Generalized Binary $\ell_0$-Rank-$k$ Approximation problem, for which these variants are special cases. Our algorithm runs in time $(1/\epsilon)^{2^{O(k)}/\epsilon^{2}} \cdot nd \cdot \log^{2^k} d$. For the specific case of finite fields of constant size, we obtain an alternate algorithm with time $n \cdot d^{\text{poly}(k/\epsilon)}$. On the hardness front, for $p \in (1,2)$, we show under the Small Set Expansion Hypothesis and Exponential Time Hypothesis (ETH), there is no constant factor approximation algorithm running in time $2^{k^{\delta}}$ for a constant $\delta > 0$, showing an exponential dependence on $k$ is necessary. For $p = 0$, we observe that there is no approximation algorithm for the Generalized Binary $\ell_0$-Rank-$k$ Approximation problem running in time $2^{2^{\delta k}}$ for a constant $\delta > 0$. We also show for finite fields of constant size, under the ETH, that any fixed constant factor approximation algorithm requires $2^{k^{\delta}}$ time for a constant $\delta > 0$.

Why You Should Charge Your Friends for Borrowing Your Stuff

We consider goods that can be shared with k-hop neighbors (i.e., the set of nodes within k hops from an owner) on a social network. We examine incentives to buy such a good by devising game-theoretic models where each node decides whether to buy the good or free ride. First, we find that social inefficiency, specifically excessive purchase of the good, occurs in Nash equilibria. Second, the social inefficiency decreases as k increases and thus a good can be shared with more nodes. Third, and most importantly, the social inefficiency can also be significantly reduced by charging free riders an access cost and paying it to owners, leading to the conclusion that organizations and system designers should impose such a cost. These findings are supported by our theoretical analysis in terms of the price of anarchy and the price of stability; and by simulations based on synthetic and real social networks.