Min-max graph partitioning and small set expansion

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal-size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an $O(\sqrt{\log n\log k})$-approximation algorithm. This improves over an $O(\log^2 n)$ approximation for the second version, and roughly $O(k\log n)$ approximation for the first version that follows from other previous work. We also give an improved O(1)-approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the Small-Set Expansion problem. In this problem, we are given a graph G and the goal is to find a non-empty set $S\subseteq V$ of size $|S| \leq \rho n$ with minimum edge-expansion. We give an $O(\sqrt{\log{n}\log{(1/\rho)}})$ bicriteria approximation algorithm for the general case of Small-Set Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor

New Approximation Bounds for Small-Set Vertex Expansion

The vertex expansion of the graph is a fundamental graph parameter. Given a graph $G=(V,E)$ and a parameter $\delta \in (0,1/2]$, its $\delta$-Small-Set Vertex Expansion (SSVE) is defined as $$\min_{S : |S| = \delta |V|} \frac{|{\partial^V(S)}|}{ \min \{ |S|, |S^c| \} }$$ where $\partial^V(S)$ is the vertex boundary of a set $S$. The SSVE~problem, in addition to being of independent interest as a natural graph partitioning problem, is also of interest due to its connections to the Strong Unique Games problem. We give a randomized algorithm running in time $n^{{\sf poly}(1/\delta)}$, which outputs a set $S$ of size $\Theta(\delta n)$, having vertex expansion at most $$\max\left(O(\sqrt{\phi^* \log d \log (1/\delta)}) , \tilde{O}(d\log^2(1/\delta)) \cdot \phi^* \right),$$ where $d$ is the largest vertex degree of the graph, and $\phi^*$ is the optimal $\delta$-SSVE. The previous best-known guarantees for this were the bi-criteria bounds of $\tilde{O}(1/\delta)\sqrt{\phi^* \log d}$ and $\tilde{O}(1/\delta)\phi^* \sqrt{\log n}$ due to Louis-Makarychev [TOC'16]. Our algorithm uses the basic SDP relaxation of the problem augmented with ${\rm poly}(1/\delta)$ rounds of the Lasserre/SoS hierarchy. Our rounding algorithm is a combination of the rounding algorithms of Raghavendra-Tan [SODA'12] and Austrin-Benabbas-Georgiou [SODA'13]. A key component of our analysis is novel Gaussian rounding lemma for hyperedges which might be of independent interest.Comment: 55 Page

Many Sparse Cuts via Higher Eigenvalues

Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq 2\sqrt{\lambda_2} where $w$ is the total edge weight of a subset or a cut and $\lambda_2$ is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer $k \in [n]$, there exist $ck$ disjoint subsets $S_1, ..., S_{ck}$, such that $$\max_i \phi(S_i) \leq C \sqrt{\lambda_{k} \log k}$$ where $\lambda_i$ is the $i^{th}$ smallest eigenvalue of the normalized Laplacian and $c0$ are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any $k$, there is a subset $S$ whose weight is at most a \bigO(1/k) fraction of the total weight and $\phi(S) \le C \sqrt{\lambda_k \log k}$. Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding $k$ subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right

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

How to Round Subspaces: A New Spectral Clustering Algorithm

A basic problem in spectral clustering is the following. If a solution obtained from the spectral relaxation is close to an integral solution, is it possible to find this integral solution even though they might be in completely different basis? In this paper, we propose a new spectral clustering algorithm. It can recover a $k$-partition such that the subspace corresponding to the span of its indicator vectors is $O(\sqrt{opt})$ close to the original subspace in spectral norm with $opt$ being the minimum possible ($opt \le 1$ always). Moreover our algorithm does not impose any restriction on the cluster sizes. Previously, no algorithm was known which could find a $k$-partition closer than $o(k \cdot opt)$. We present two applications for our algorithm. First one finds a disjoint union of bounded degree expanders which approximate a given graph in spectral norm. The second one is for approximating the sparsest $k$-partition in a graph where each cluster have expansion at most $\phi_k$ provided $\phi_k \le O(\lambda_{k+1})$ where $\lambda_{k+1}$ is the $(k+1)^{st}$ eigenvalue of Laplacian matrix. This significantly improves upon the previous algorithms, which required $\phi_k \le O(\lambda_{k+1}/k)$.Comment: Appeared in SODA 201

Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if \lambda_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut