The Complexity of Approximating Vertex Expansion

We study the complexity of approximating the vertex expansion of graphs $G = (V,E)$, defined as $$\Phi^V := \min_{S \subset V} n \cdot \frac{|N(S)|}{|S| |V \backslash S|}.$$ We give a simple polynomial-time algorithm for finding a subset with vertex expansion $O(\sqrt{OPT \log d})$ where $d$ is the maximum degree of the graph. Our main result is an asymptotically matching lower bound: under the Small Set Expansion (SSE) hypothesis, it is hard to find a subset with expansion less than $C\sqrt{OPT \log d}$ for an absolute constant $C$. In particular, this implies for all constant $\epsilon > 0$, it is SSE-hard to distinguish whether the vertex expansion $< \epsilon$ or at least an absolute constant. The analogous threshold for edge expansion is $\sqrt{OPT}$ with no dependence on the degree; thus our results suggest that vertex expansion is harder to approximate than edge expansion. In particular, while Cheeger's algorithm can certify constant edge expansion, it is SSE-hard to certify constant vertex expansion in graphs. Our proof is via a reduction from the {\it Unique Games} instance obtained from the \SSE hypothesis to the vertex expansion problem. It involves the definition of a smoother intermediate problem we call {\sf Analytic Vertex Expansion} which is representative of both the vertex expansion and the conductance of the graph. Both reductions (from the UGC instance to this problem and from this problem to vertex expansion) use novel proof ideas

The Densest k-Subhypergraph Problem

The Densest $k$-Subgraph (D$k$S) problem, and its corresponding minimization problem Smallest $p$-Edge Subgraph (S$p$ES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both D$k$S and S$p$ES from graphs to hypergraphs. We consider the Densest $k$-Subhypergraph problem (given a hypergraph $(V, E)$, find a subset $W\subseteq V$ of $k$ vertices so as to maximize the number of hyperedges contained in $W$) and define the Minimum $p$-Union problem (given a hypergraph, choose $p$ of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an $O(n^{4(4-\sqrt{3})/13 + \epsilon}) \leq O(n^{0.697831+\epsilon})$-approximation (for arbitrary constant $\epsilon > 0$) for Densest $k$-Subhypergraph and an $\tilde O(n^{2/5})$-approximation for Minimum $p$-Union. We also give an $O(\sqrt{m})$-approximation for Minimum $p$-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.Comment: 21 page

Approximation Algorithms for Hypergraph Small Set Expansion and Small Set Vertex Expansion

The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which, for a parameter $\delta \in (0,1/2]$, asks to compute the cut having the least expansion while having at most $\delta$ fraction of the vertices on the smaller side of the cut. We present two algorithms. Our first algorithm gives an $\tilde O(\delta^{-1} \sqrt{\log n})$ approximation. The second algorithm finds a set with expansion $\tilde O(\delta^{-1}(\sqrt{d_{\text{max}}r^{-1}\log r\, \phi^*} + \phi^*))$ in a $r$--uniform hypergraph with maximum degree $d_{\text{max}}$ (where $\phi^*$ is the expansion of the optimal solution). Using these results, we also obtain algorithms for the Small Set Vertex Expansion problem: we get an $\tilde O(\delta^{-1} \sqrt{\log n})$ approximation algorithm and an algorithm that finds a set with vertex expansion $O\left(\delta^{-1}\sqrt{\phi^V \log d_{\text{max}} } + \delta^{-1} \phi^V\right)$ (where $\phi^V$ is the vertex expansion of the optimal solution). For $\delta=1/2$, Hypergraph Small Set Expansion is equivalent to the hypergraph expansion problem. In this case, our approximation factor of $O(\sqrt{\log n})$ for expansion in hypergraphs matches the corresponding approximation factor for expansion in graphs due to ARV

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

Expansion Testing using Quantum Fast-Forwarding and Seed Sets

Expansion testing aims to decide whether an $n$-node graph has expansion at least $\Phi$, or is far from any such graph. We propose a quantum expansion tester with complexity $\widetilde{O}(n^{1/3}\Phi^{-1})$. This accelerates the $\widetilde{O}(n^{1/2}\Phi^{-2})$ classical tester by Goldreich and Ron [Algorithmica '02], and combines the $\widetilde{O}(n^{1/3}\Phi^{-2})$ and $\widetilde{O}(n^{1/2}\Phi^{-1})$ quantum speedups by Ambainis, Childs and Liu [RANDOM '11] and Apers and Sarlette [QIC '19], respectively. The latter approach builds on a quantum fast-forwarding scheme, which we improve upon by initially growing a seed set in the graph. To grow this seed set we use a so-called evolving set process from the graph clustering literature, which allows to grow an appropriately local seed set.Comment: v3: final version to appear in Quantu

Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain

Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$ and vertex conductance $\Psi^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap $\gamma^*(P)$ satisfies $$\Psi^*(G)^{2}/\log\Delta \lesssim \gamma^*(P) \lesssim \Psi^*(G).$$ Our bound is optimal under the Small Set Expansion Hypothesis, and answers a question of Olesker-Taylor and Zanetti, who obtained such a result with $\log\Delta$ replaced by $\log|V|$. In order to obtain our result, we show how to embed a negative-type semi-metric $d$ defined on $V$ into a negative-type semi-metric $d'$ supported in $\mathbb{R}^{O(\log\Delta)}$, such that the (fractional) matching number of the weighted graph $(V,E,d)$ is approximately equal to that of $(V,E,d')$.Comment: 6 page