Pasin Manurangsi

Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph

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(n1/4 + 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 2O(log2/3 n) (Alon et al.) respectively. In this work, we show, assuming the exponential time hypothesis (ETH), that there is no polynomial-time algorithm that approximates Densest k-Subgraph to within n1/(loglogn)c factor of the optimum, where c > 0 is a universal constant independent of n. In addition, our result has perfect completeness, meaning that we prove that it is ETH-hard to even distinguish between the case in which G contains a k-clique and the case in which every induced k-subgraph of G has density at most 1/n-1/(loglogn)c in polynomial time. Moreover, if we make a stronger assumption that there is some constant ε > 0 such that no subexponential-time algorithm can distinguish between a satisfiable 3SAT formula and one which is only (1 - ε)-satisfiable (also known as Gap-ETH), then the ratio above can be improved to nf(n) for any function f whose limit is zero as n goes to infinity (i.e. f ϵ o(1)).

Computing an Approximately Optimal Agreeable Set of Items

We study the problem of finding a small subset of items that is agreeable to all agents, meaning that all agents value the subset at least as much as its complement. Previous work has shown worst-case bounds, over all instances with a given number of agents and items, on the number of items that may need to be included in such a subset. Our goal in this paper is to efficiently compute an agreeable subset whose size approximates the size of the smallest agreeable subset for a given instance. We consider three well-known models for representing the preferences of the agents: ordinal preferences on single items, the value oracle model, and additive utilities. In each of these models, we establish virtually tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time.

On the parameterized complexity of approximating dominating set

We study the parameterized complexity of approximating the k-Dominating Set (domset) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F(k) · k whenever the graph G has a dominating set of size k. When such an algorithm runs in time T(k)poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for k-domset. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the ”most infamous” open problems in Parameterized Complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1]≠FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T, F and every constant ε > 0: (i) Assuming W[1]≠FPT, there is no F(k)-FPT-approximation algorithm for k-domset, (ii) Assuming the Exponential Time Hypothesis (ETH), there is no F(k)-approximation algorithm for k-domset that runs in T(k)no(k) time, (iii) Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F(k)-approximation algorithm for k-domset that runs in T(k)nk − ε time, (iv) Assuming the k-sum Hypothesis, for every integer k ≥ 3, there is no F(k)-approximation algorithm for k-domset that runs in T(k) n⌈ k/2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under W[1]≠FPT and (log1/4 − ε k)-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F(k)-FPT-approximation algorithm for any function F was shown under gapETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form nδ k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well studied problem or a variant of one; this allows us to easily apply known techniques to solve them.

Improved Approximation Algorithms for Projection Games

The projection games (aka Label-Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label-Cover. In this paper we design several approximation algorithms for projection games:

From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More

We consider questions that arise from the intersection between theareas of approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx, 2008; Fellow et al., 2012; Downey & Fellow 2013]) 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 int(opt) poly(N) time and outputs a solution of size f(opt), forany 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) [Dinur, 2016, Manurangsi & Raghavendra 2016], which states that no 2^{o(n)}-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 - c)-satisfiable for some constant c ≈ 0.Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced 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[1]-hard [Lin, 2015; Khot & Raman, 2000; Moser & Sikdar, 2009]. Additionally, we rule out k^{o(1)}-FPT-approximation algorithm for Densest k-Subgraph although this ratio does not yet match the trivial O(k)-approximation algorithm.To the best of our knowledge, prior results only rule out constantfactor approximation for Clique [Hajiaghayi et al., 2013; KK13, Bonnet et al., 2015] and log^{1/4+c}(opt) approximation for DomSet for any constant c ≈ 0 [Chen & Lin, 2016]. Our result on Clique significantly improves on [Hajiaghayi et al., 2013; Bonnet et al., 2015]. However, our result on DomSet is incomparable to [Chen & Lin, 2016] since their results hold under ETH while our results hold under Gap-ETH, which is a stronger assumption.

Asymptotic Existence of Fair Divisions for Groups

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn randomly from probability distributions, we show that when all groups contain an equal number of players, a welfare-maximizing allocation is likely to be envy-free if the number of items exceeds the total number of players by a logarithmic factor. On the other hand, an envy-free allocation is unlikely to exist if the number of items is less than the total number of players. In addition, we show that a simple truthful mechanism, namely the random assignment mechanism, yields an allocation that satisfies the weaker notion of approximate envy-freeness with high probability.

Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small set of vertices whose expansion is almost zero and one in which all small sets of vertices have expansion almost one. In this work, we prove conditional inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph G, find a complete bipartite subgraph of G with maximum number of edges. We show that, assuming SSEH and that NP != BPP, no polynomial time algorithm gives n^{1 - epsilon}-approximation for MEB for every constant epsilon > 0. - Maximum Balanced Biclique (MBB): given a bipartite graph G, find a balanced complete bipartite subgraph of G with maximum number of vertices. Similar to MEB, we prove n^{1 - epsilon} ratio inapproximability for MBB for every epsilon > 0, assuming SSEH and that NP != BPP. - Minimum k-Cut: given a weighted graph G, find a set of edges with minimum total weight whose removal splits the graph into k components. We prove that this problem is NP-hard to approximate to within (2 - epsilon) factor of the optimum for every epsilon > 0, assuming SSEH. The ratios in our results are essentially tight since trivial algorithms give n-approximation to both MEB and MBB and 2-approximation algorithms are known for Minimum k-Cut [Saran and Vazirani, SIAM J. Comput., 1995]. Our first two results are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani [Raghavendra et al., CCC, 2012] to avoid locality of gadget reductions with a generalization of Bansal and Khots long code test [Bansal and Khot, FOCS, 2009] whereas our last result is shown via an elementary reduction.

Approximating Dense Max 2-CSPs

In this paper, we present a polynomial-time algorithm that approximates sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within

An Improved Integrality Gap for the Călinescu-Karloff-Rabani Relaxation for Multiway Cut

O(N^{\varepsilon})

ETH-Hardness of Approximating 2-CSPs and Directed Steiner Network

approximation ratio for any constant