Inapproximability of Maximum Biclique Problems, Minimum kk-Cut and Densest At-Least-kk-Subgraph 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 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 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. - Maximum Balanced Biclique (MBB): given a bipartite graph $G$, find a balanced complete bipartite subgraph of $G$ with maximum number of vertices. - Minimum $k$-Cut: given a weighted graph $G$, find a set of edges with minimum total weight whose removal partitions $G$ into $k$ connected components. - Densest At-Least-$k$-Subgraph (DAL$k$S): given a weighted graph $G$, find a set $S$ of at least $k$ vertices such that the induced subgraph on $S$ has maximum density (the ratio between the total weight of edges and the number of vertices). We show that, assuming SSEH and NP $\nsubseteq$ BPP, no polynomial time algorithm gives $n^{1 - \varepsilon}$-approximation for MEB or MBB for every constant $\varepsilon > 0$. Moreover, assuming SSEH, we show that it is NP-hard to approximate Minimum $k$-Cut and DAL$k$S to within $(2 - \varepsilon)$ factor of the optimum for every constant $\varepsilon > 0$. The ratios in our results are essentially tight since trivial algorithms give $n$-approximation to both MEB and MBB and efficient $2$-approximation algorithms are known for Minimum $k$-Cut [SV95] and DAL$k$S [And07, KS09]. Our first result is proved by combining a technique developed by Raghavendra et al. [RST12] to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test [BK09] whereas our second result is shown via elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

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

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., [Marx08, 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 $\text{OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t(\text{OPT})\text{poly}(N)$ time and outputs a solution of size $f(\text{OPT})$, for any functions $t$ and $f$ that are independent of $N$ (for Clique, we want $f(\text{OPT})=\omega(1)$)? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no $o(\text{OPT})$-FPT-approximation algorithm for Clique and no $f(\text{OPT})$-FPT-approximation algorithm for DomSet, for any function $f$ (e.g., this holds even if $f$ is 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) [Dinur16, MR16], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even $(1 - \epsilon)$-satisfiable for some constant $\epsilon > 0$. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. 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.Comment: 43 pages. To appear in FOCS'1

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 Khot\u27s long code test [Bansal and Khot, FOCS, 2009] whereas our last result is shown via an elementary reduction

Fractional coverings, greedy coverings, and rectifier networks

A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix M that has a 1 in the entry (i,j) iff there is a path from the j-th source to the i-th sink. The smallest number of edges in a rectifier network that computes M is a classic complexity measure on matrices, which has been studied for more than half a century. We explore two techniques that have hitherto found little to no applications in this theory. They build upon a basic fact that depth-2 rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. Using fractional and greedy coverings (defined in the standard way), we obtain new results in this area. First, we show that all fractional coverings of the so-called full triangular matrix have cost at least n log n. This provides (a fortiori) a new proof of the tight lower bound on its depth-2 complexity (the exact value has been known since 1965, but previous proofs are based on different arguments). Second, we show that the greedy heuristic is instrumental in tightening the upper bound on the depth-2 complexity of the Kneser-Sierpinski (disjointness) matrix. The previous upper bound is O(n^{1.28}), and we improve it to O(n^{1.17}), while the best known lower bound is Omega(n^{1.16}). Third, using fractional coverings, we obtain a form of direct product theorem that gives a lower bound on unbounded-depth complexity of Kronecker (tensor) products of matrices. In this case, the greedy heuristic shows (by an argument due to Lovász) that our result is only a logarithmic factor away from the "full" direct product theorem. Our second and third results constitute progress on open problem 7.3 and resolve, up to a logarithmic factor, open problem 7.5 from a recent book by Jukna and Sergeev (in Foundations and Trends in Theoretical Computer Science (2013)

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

We study the 2-ary constraint satisfaction problems (2-CSPs), which can be stated as follows: given a constraint graph $G=(V,E)$, an alphabet set $\Sigma$ and, for each $\{u, v\}\in E$, a constraint $C_{uv} \subseteq \Sigma\times\Sigma$, the goal is to find an assignment $\sigma: V \to \Sigma$ that satisfies as many constraints as possible, where a constraint $C_{uv}$ is satisfied if $(\sigma(u),\sigma(v))\in C_{uv}$. While the approximability of 2-CSPs is quite well understood when $|\Sigma|$ is constant, many problems are still open when $|\Sigma|$ becomes super constant. One such problem is whether it is hard to approximate 2-CSPs to within a polynomial factor of $|\Sigma| |V|$. Bellare et al. (1993) suggested that the answer to this question might be positive. Alas, despite efforts to resolve this conjecture, it remains open to this day. In this work, we separate $|V|$ and $|\Sigma|$ and ask a related but weaker question: is it hard to approximate 2-CSPs to within a polynomial factor of $|V|$ (while $|\Sigma|$ may be super-polynomial in $|V|$)? Assuming the exponential time hypothesis (ETH), we answer this question positively by showing that no polynomial time algorithm can approximate 2-CSPs to within a factor of $|V|^{1 - o(1)}$. Note that our ratio is almost linear, which is almost optimal as a trivial algorithm gives a $|V|$-approximation for 2-CSPs. Thanks to a known reduction, our result implies an ETH-hardness of approximating Directed Steiner Network with ratio $k^{1/4 - o(1)}$ where $k$ is the number of demand pairs. The ratio is roughly the square root of the best known ratio achieved by polynomial time algorithms (Chekuri et al., 2011; Feldman et al., 2012). Additionally, under Gap-ETH, our reduction for 2-CSPs not only rules out polynomial time algorithms, but also FPT algorithms parameterized by $|V|$. Similar statement applies for DSN parameterized by $k$.Comment: 36 pages. A preliminary version appeared in ITCS'1