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: \[ ϕ(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq 2\sqrt{λ_2} \] where $w$ is the total edge weight of a subset or a cut and $λ_2$ 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 \[ \max_i ϕ(S_i) \leq C \sqrt{λ_{k} \log k} \] where $λ_i$ is the $i^{th}$ smallest eigenvalue of the normalized Laplacian and $c<1,C>0$ 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 $ϕ(S) \le C \sqrt{λ_k \log k}$. 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.
We present a very simple and intuitive algorithm to find balanced sparse cuts in a graph via shortest-paths. Our algorithm combines a new multiplicative-weights framework for solving unit-weight multi-commodity flows with standard ball growing arguments. Using Dijkstra's algorithm for computing the shortest paths afresh every time gives a very simple algorithm that runs in time $\widetilde{O}(m^2/ϕ)$ and finds an $\widetilde{O}(ϕ)$-sparse balanced cut, when the given graph has a $ϕ$-sparse balanced cut. Combining our algorithm with known deterministic data-structures for answering approximate …
The problem of computing the vertex expansion of a graph is an NP-hard problem. The current best worst-case approximation guarantees for computing the vertex expansion of a graph are a $O(\sqrt{\log n})$-approximation algorithm due to Feige, Hajiaghayi and Lee [SIAM J. Comp., 2008], and $O(\sqrt{OPT \log d})$ bound in graphs having vertex degrees at most $d$, due to Louis, Raghavendra and Vempala [FOCS 2013]. We study a natural semi-random model of graphs with sparse vertex cuts. For certain ranges of parameters, we give an algorithm to recover the planted sparse vertex cut exactly. For a larg…
We present the first work-optimal polylogarithmic-depth parallel algorithm for the minimum cut problem on non-sparse graphs. For $m\geq n^{1+ε}$ for any constant $ε>0$, our algorithm requires $O(m \log n)$ work and $O(\log^3 n)$ depth and succeeds with high probability. Its work matches the best $O(m \log n)$ runtime for sequential algorithms [MN STOC 2020, GMW SOSA 2021]. This improves the previous best work by Geissmann and Gianinazzi [SPAA 2018] by $O(\log^3 n)$ factor, while matching the depth of their algorithm. To do this, we design a work-efficient approximation algorithm and paralleliz…
In this article, we show that the algorithm of maintaining expander decompositions in graphs undergoing edge deletions directly by removing sparse cuts repeatedly can be made efficient. Formally, for an $m$-edge undirected graph $G$, we say a cut $(S, \overline{S})$ is $ϕ$-sparse if $|E_G(S, \overline{S})| < ϕ\cdot \min\{vol_G(S), vol_G(\overline{S})\}$. A $ϕ$-expander decomposition of $G$ is a partition of $V$ into sets $X_1, X_2, \ldots, X_k$ such that each cluster $G[X_i]$ contains no $ϕ$-sparse cut (meaning it is a $ϕ$-expander) with $\tilde{O}(ϕm)$ edges crossing between clusters. A natur…
We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G=(V,E) and a parameter k <= |E|, the small sparsest cut problem is to find a subset of vertices S with minimum conductance among all sets with volume at most k. Using ideas developed in local graph partitioning algorithms, we obtain the following bicriteria approximation algorithms for the small sparsest cut problem: - If there is a subset U with conductance ϕand vol(U) <= k, then there is a polynomial time algorithm to find a set S with conductance O(\sqrt{ϕ/ε}) and vol(S) <= k^{1+ε} for any …
We study deterministic algorithms for computing graph cuts, with focus on two fundamental problems: balanced sparse cut and $k$-vertex connectivity for small $k$ ($k=O(\polylog n)$). Both problems can be solved in near-linear time with randomized algorithms, but their previous deterministic counterparts take at least quadratic time. In this paper, we break this bound for both problems. Interestingly, achieving this for one problem crucially relies on doing so for the other. In particular, via a divide-and-conquer argument, a variant of the cut-matching game by [Khandekar et al.`07], and the lo…
A {\em local graph partitioning algorithm} finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph $G$, starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number $n$ of vertices in $G$. Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank. In this paper, we introduce a randomized local partitioning algorithm that finds a sparse cut by simulating the {\em volume…
We describe a new approximation algorithm for Max Cut. Our algorithm runs in $\tilde O(n^2)$ time, where $n$ is the number of vertices, and achieves an approximation ratio of $.531$. On instances in which an optimal solution cuts a $1-ε$ fraction of edges, our algorithm finds a solution that cuts a $1-4\sqrtε + 8ε-o(1)$ fraction of edges. Our main result is a variant of spectral partitioning, which can be implemented in nearly linear time. Given a graph in which the Max Cut optimum is a $1-ε$ fraction of edges, our spectral partitioning algorithm finds a set $S$ of vertices and a bipartition $…
Modeling information that resides on vertices of large graphs is a key problem in several real-life applications, ranging from social networks to the Internet-of-things. Signal Processing on Graphs and, in particular, graph wavelets can exploit the intrinsic smoothness of these datasets in order to represent them in a both compact and accurate manner. However, how to discover wavelet bases that capture the geometry of the data with respect to the signal as well as the graph structure remains an open question. In this paper, we study the problem of computing graph wavelet bases via sparse cuts …
Finding sparse cuts is an important tool in analyzing large-scale distributed networks such as the Internet and Peer-to-Peer networks, as well as large-scale graphs such as the web graph, online social communities, and VLSI circuits. In distributed communication networks, they are useful for topology maintenance and for designing better search and routing algorithms. In this paper, we focus on developing fast distributed algorithms for computing sparse cuts in networks. Given an undirected $n$-node network $G$ with conductance $ϕ$, the goal is to find a cut set whose conductance is close to $ϕ…
This paper proves strong lower bounds for distributed computing in the CONGEST model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can b…
We improve on random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a near-linear-time construction that transforms any graph on n vertices into an O(n\log n)-edge graph on the same vertices whose cuts have approximately the same value as the original graph's. In this new graph, for example, we can run the O(m^{3/2})-time maximum flow algorithm of Goldberg and Rao to find an s--t minimum cut in O(n^{3/2}) time. This corresponds to a (1+epsilon)-times minimum s--t cut in the original graph. In a similar way, we can approximate a sparsest cu…
Cheeger's inequality shows that any undirected graph $G$ with minimum nonzero normalized Laplacian eigenvalue $λ_G$ has a cut with conductance at most $O(\sqrt{λ_G})$. Qualitatively, Cheeger's inequality says that if the relaxation time of a graph is high, there is a cut that certifies this. However, there is a gap in this relationship, as cuts can have conductance as low as $Θ(λ_G)$. To better approximate the relaxation time of a graph, we consider a more general object. Instead of bounding the mixing time with cuts, we bound it with cuts in graphs obtained by Schur complementing out vertices…
We revisit classical connectivity problems in the CONGEST model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with $\widetilde{O}(1)$ rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of $Ω(Diam+\sqrt{n})$ rounds). Our main results are: (1) For $D$-diameter unweighted graphs with constant edge connectivity, we show an exact distributed deterministic computation of the minimum cut in $poly(D)$ rounds. This resolves one the open problems recently…
High-dimensional settings, where the data dimension ($d$) far exceeds the number of observations ($n$), are common in many statistical and machine learning applications. Methods based on $\ell_1$-relaxation, such as Lasso, are very popular for sparse recovery in these settings. Restricted Eigenvalue (RE) condition is among the weakest, and hence the most general, condition in literature imposed on the Gram matrix that guarantees nice statistical properties for the Lasso estimator. It is natural to ask: what families of matrices satisfy the RE condition? Following a line of work in this area, w…
We develop a framework for incorporating edge-dependent vertex weights (EDVWs) into the hypergraph minimum s-t cut problem. These weights are able to reflect different importance of vertices within a hyperedge, thus leading to better characterized cut properties. More precisely, we introduce a new class of hyperedge splitting functions that we call EDVWs-based, where the penalty of splitting a hyperedge depends only on the sum of EDVWs associated with the vertices on each side of the split. Moreover, we provide a way to construct submodular EDVWs-based splitting functions and prove that a hype…
We design algorithms for fitting a high-dimensional statistical model to a large, sparse network without revealing sensitive information of individual members. Given a sparse input graph $G$, our algorithms output a node-differentially-private nonparametric block model approximation. By node-differentially-private, we mean that our output hides the insertion or removal of a vertex and all its adjacent edges. If $G$ is an instance of the network obtained from a generative nonparametric model defined in terms of a graphon $W$, our model guarantees consistency, in the sense that as the number of …
We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, showing optimality of a simple spectral certificate. This evidence takes the form of a computationally-quiet planting: we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishabl…
