In the Densest k-Subgraph (DkS) problem, given an undirected graph G and an integer k, the goal is to find a subgraph of G on k vertices that contains maximum number of edges. Even though Bhaskara et al.'s state-of-the-art algorithm for the problem achieves only O(n 1/4+ε ) approximation ratio, previous attempts at proving hardness of approximation, including those under average case assumptions, fail to achieve a polynomial ratio; the best ratios ruled out under any worst case assumption and any average case assumption are only any constant (Raghavendra and Steurer) and 2 O(log 2/3 n) (Alon et al.) respectively.In this work, we show, assuming the exponential time hypothesis (ETH), that there is no polynomialtime algorithm that approximates DkS to within n

Abstract:The Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove conditional inapproximability results with essentially optimal ratios for the following graph problems based on this hypothesis: Maximum Edge Biclique, Maximum Balanced Biclique, Minimum k-Cut and Densest At-Least-k-Subgraph. Our hardness results for the two biclique problems are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test whereas our results for Minimum k-Cut and Densest At-Least-k-Subgraph are shown via elementary reductions.

We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envyfreeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions.

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Mar08; FGMS12; DF13]), are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting OPT be the optimum and N be the size of the input, is there an algorithm that runs in t(OPT) poly(N ) time and outputs a solution of size f (OPT), for any functions t and f that are independent of N (for Clique, we want f (OPT) = ω(1))?In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no o(OPT)-FPT-approximation algorithm for Clique and no f (OPT)-FPT-approximation algorithm for DomSet, for any function f (e.g., this holds even if f is an exponential or the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Din16; MR16], which states that no 2 o(n) -time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 − ε)-satisfiable for some constant ε > 0.Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs. Previously only exact versions of these problems were known to be W

Consider the setup where n parties are each given an element in the finite field and the goal is to compute the sum in a secure fashion and with as little communication as possible. We study this problem in the anonymized model of Ishai et al. (FOCS 2006) where each party may broadcast anonymous messages on an insecure channel. We present a new analysis of the one-round “split and mix” protocol of Ishai et al. In order to achieve the same security parameter, our analysis reduces the required number of messages by a multiplicative factor. We also prove lower bounds showing that the dependence of the number of messages on the domain size, the number of parties, and the security parameter is essentially tight. Using a reduction of Balle et al. (2019), our improved analysis of the protocol of Ishai et al. yields, in the same model, an -differentially private protocol for aggregation that, for any constant and any , incurs only a constant error and requires only a constant number of messages per party. Previously, such a protocol was known only for messages per party.