Sparse Cuts Research Articles

Additive Sparsification of CSPs

Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC’96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA’17] and non-Boolean domains by Butti and Živný [SIDMA’20]. Bansal, Svensson and Trevisan [FOCS’19] introduced a weaker notion of sparsification termed “additive sparsification”, which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P :{ 0,1} k → { 0,1} of a fixed arity k , we show that CSP( P ) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP( P ) admits an additive sparsifier for any predicate P : D k → { 0,1} of a fixed arity k on an arbitrary finite domain D .

A Stochastic Optimization Approach to Energy-Efficient Underground Timetabling Under Uncertain Dwell and Running Times

We consider a problem from the context of energy-efficient underground railway timetabling, in which an existing timetable draft is improved by slightly changing departure and running times. In practice, synchronization between accelerating and braking trains to utilize regenerative braking plays a major role for the energy efficiency of a timetable. Because deviations from a planned timetable may lead to unnecessarily high energy consumption during actual operation, we include operational uncertainties in our model to create a timetable that remains energy efficient, even if deviations from the nominal timetable occur. To solve the problem, we use a scenario expansion model in conjunction with a Benders decomposition approach. As an alternative to solving the Benders subproblems, we present a heuristic sparse cut that can be computed efficiently. The resulting sparse-cut heuristic produces high-quality solutions on a set of real-world instances stemming from the Nürnberg underground system, outperforming the integrated mixed-integer programming approach as well as the basic Benders approach. Additionally, we evaluate two static recovery strategies—shortening dwell times as well as shortening dwell and running times—to determine the cost and benefit of handling delays using a simple static rule. In our experiments, we are able to reduce the energy consumption by up to 9.4% and confirm that delay recovery via shortening dwell times is an energy-efficient and effective way to increase punctuality at low cost in terms of energy. Funding: This research was supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy through the Center for Analytics – Data – Applications (ADA-Center) within the framework of “BAYERN DIGITAL II” (20-3410-2-9-8).

Graph Algorithm Based Submodular Function for Sparsest Cut Problem

Sparsest cut problems are very important graph partitions, which have been widely applied in expander graphs, Markov chains, and image segmentation. In this paper, we study the edge-weighted version of the Sparse Cut Problem, which minimizes the ratio of the total weight of edges between blocks and the total weight of edges incident to vertices in one block. We first prove that the problem is even NP-hard for an edge-weighted graph with bridges. Then, we combine and generalize submodular functions and principal partition to design a graph algorithm to improve the initial bipartition, which runs in polynomial time by using network flow as its subroutines.

Faster Cut Sparsification of Weighted Graphs

A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of (1pm epsilon ). This paper considers computing cut sparsifiers of weighted graphs of size O(nlog (n)/epsilon ^2). Our algorithm computes such a sparsifier in time O(mcdot min (alpha (n)log (m/n),log (n))), both for graphs with polynomially bounded and unbounded integer weights, where alpha (cdot ) is the functional inverse of Ackermann’s function. This improves upon the state of the art by Benczúr and Karger (SICOMP, 2015), which takes O(mlog ^2 (n)) time. For unbounded weights, this directly gives the best known result for cut sparsification. Together with preprocessing by an algorithm of Fung et al. (SICOMP, 2019), this also gives the best known result for polynomially-weighted graphs. Consequently, this implies the fastest approximate min-cut algorithm, both for graphs with polynomial and unbounded weights. In particular, we show that it is possible to adapt the state of the art algorithm of Fung et al. for unweighted graphs to weighted graphs, by letting the partial maximum spanning forest (MSF) packing take the place of the Nagamochi–Ibaraki forest packing. MSF packings have previously been used by Abraham et al. (FOCS, 2016) in the dynamic setting, and are defined as follows: an M-partial MSF packing of G is a set mathcal {F}={F_1, ldots , F_M}, where F_i is a maximum spanning forest in G{setminus } bigcup _{j=1}^{i-1}F_j. Our method for computing (a sufficient estimation of) the MSF packing is the bottleneck in the running time of our sparsification algorithm.

Quasipolynomial Multicut-mimicking Networks and Kernels for Multiway Cut Problems

We show the existence of an exact mimicking network of k O ( log k ) edges for minimum multicuts over a set of terminals in an undirected graph, where k is the total capacity of the terminals, i.e., the sum of the degrees of the terminal vertices. Furthermore, using the best available approximation algorithm for Small Set Expansion , we show that a mimicking network of k O ( log3 k ) edges can be computed in randomized polynomial time. As a consequence, we show quasipolynomial kernels for several problems, including Edge Multiway Cut , Group Feedback Edge Set for an arbitrary group, and Edge Multicut parameterized by the solution size and the number of cut requests. The result combines the matroid-based irrelevant edge approach used in the kernel for s -Multiway Cut with a recursive decomposition and sparsification of the graph along sparse cuts. This is the first progress on the kernelization of Multiway Cut problems since the kernel for s -Multiway Cut for constant value of s (Kratsch and Wahlström, FOCS 2012).

Bounds on maximum concurrent flow in random bipartite graphs

Algorithms that exploit the duality between maximum concurrent flow and sparse cuts in a graph can effectively discover hierarchical community structures in social networks. We analyze the maximum concurrent flow on random bipartite graphs and compare the results to two bounds based on graph structure.. The minimum degree bound is based on a sparse cut that removes a node of minimum degree. The other is the “gridlock” bound, where all edges are saturated with flow. We show that the gridlock bound is more constraining when the minimum degree of the graph is sufficiently large, and provide a way to calculate this bound on graphs of diameter three, which occurs with high probability on larger random bipartite graphs.

Sparsification of Binary CSPs

A cut $\varepsilon$-sparsifier of a weighted graph $G$ is a re-weighted subgraph of $G$ of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of $\varepsilon$. Since their introduction by Bencz\'ur and Karger [STOC'96], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIDMA'17] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains.

Improved Guarantees for Vertex Sparsification in Planar Graphs

Graph sparsification aims at compressing large graphs into smaller ones while preserving important characteristics of the input graph. In this work we study vertex sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. We focus on the following notions: (1) Given a digraph $G=(V,E)$ and terminal vertices $K \subset V$ with $|K| = k$, a (vertex) reachability sparsifier of $G$ is a digraph $H=(V_H,E_H)$, $K \subset V_H$ that preserves all reachability information among terminal pairs. Let $|V_H|$ denote the size of $H$. In this work we introduce the notion of reachability-preserving minors (RPMs), i.e., we require $H$ to be a minor of $G$. We show any directed graph $G$ admits an RPM $H$ of size $O(k^3)$, and if $G$ is planar, then the size of $H$ improves to $O(k^{2} \log k)$. We complement our upper bound by showing that there exists an infinite family of grids such that any RPM must have $\Omega(k^{2})$ vertices. (2) Given a weighted undirected graph $G=(V,E)$ and terminal vertices $K$ with $|K|=k$, an exact (vertex) cut sparsifier of $G$ is a graph $H$ with $K \subset V_H$ that preserves the value of minimum cuts separating any bipartition of $K$. We show that planar graphs with all the $k$ terminals lying on the same face admit exact cut sparsifiers of size $O(k^{2})$ that are also planar. Our result extends to flow and distance sparsifiers. It improves the previous best-known bound of $O(k^22^{2k})$ for cut and flow sparsifiers by an exponential factor and matches an $\Omega(k^2)$ lower-bound for this class of graphs.

Cheeger constants, structural balance, and spectral clustering analysis for signed graphs

We introduce a family of multi-way Cheeger-type constants {hkσ,k=1,2,…,n} on a signed graph Γ=(G,σ) such that hkσ=0 if and only if Γ has k balanced connected components. These constants are switching invariant and bring together in a unified viewpoint a number of important graph-theoretical concepts, including the classical Cheeger constant, those measures of bipartiteness introduced by Desai-Rao, Trevisan, Bauer–Jost, respectively, on unsigned graphs, and the frustration index (originally called the line index of balance by Harary) on signed graphs. We further unify the (higher-order or improved) Cheeger and dual Cheeger inequalities for unsigned graphs as well as the underlying algorithmic proof techniques by establishing their corresponding versions on signed graphs. In particular, we develop a spectral clustering method for finding k almost-balanced subgraphs, each defining a sparse cut. The proper metric for such a clustering is the metric on a real projective space. We also prove estimates of the extremal eigenvalues of signed Laplace matrix in terms of number of signed triangles (3-cycles).

Complete Minors in Graphs Without Sparse Cuts

AbstractWe show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d = d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order $$\begin{equation*} \Omega\left( \sqrt{\frac{n d}{\log d}} \right). \end{equation*}$$As a corollary we determine the order of a largest complete minor one can guarantee in $d$-regular graphs for which the 2nd largest eigenvalue is bounded away from $d/2$, in $(d/n, o(d))$-jumbled graphs, and in random $d$-regular graphs, for almost all $d = d(n)$.

A General Framework for Graph Sparsification

We present a general framework for constructing cut sparsifiers in undirected graphs---weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative facto...

Analysis of Sparse Cutting Planes for Sparse MILPs with Applications to Stochastic MILPs

In this paper, we present an analysis of the strength of sparse cutting planes for mixed integer linear programs (MILP) with sparse formulations. We examine three kinds of problems: packing problems, covering problems, and more general MILPs with the only assumption that the objective function is nonnegative. Given an MILP instance of one of these three types, assume that we decide on the support of cutting planes to be used and the strongest inequalities on these supports are added to the linear programming relaxation. We present bounds on the ratio of optimal value of the LP relaxation after adding cuts and the optimal value of the MILP that depends only on the sparsity structure of the constraint matrix and the support of sparse cuts selected, that is, these bounds are completely data independent. These results also shed light on the strength of single-scenario cuts for two-stage stochastic MILPs.

Single Pass Spectral Sparsification in Dynamic Streams

We present the first single pass algorithm for computing spectral sparsifiers for graphs in the dynamic semi-streaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph $G$, our algorithm maintains a randomized linear sketch of the incidence matrix of $G$ into dimension $O (\frac{1}{\epsilon^2}n{polylog} (n))$. Using this sketch, at any point, the algorithm can output a $(1 \pm \epsilon)$ spectral sparsifier for $G$ with high probability. While $O (\frac{1}{\epsilon^2} n{polylog}(n))$ space algorithms are known for computing cut sparsifiers in dynamic streams [K. J. Ahn, S. Guha, and A. McGregor, in Proceedings of the 31st ACM Symposium on Principles of Database Systems, 2012, pp. 5--14; A. Goel, M. Kapralov, and I. Post, \hrefhttp://arXiv.org/abs/1203.4900 arXiv:1203.4900, 2002] and spectral sparsifiers in insertion-only streams [J. A. Kelner and A. Levin, Theory Comput. Syst., 53 (2013), pp. 243--262], prior to our work, the best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension $\Omega (\frac{1}{\epsilon^2}n^{5/3})$ [K. J. Ahn, S. Guha, and A. McGregor, in Proceedings of the 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2013, pp. 1--10]. To achieve our result, we show that using a coarse sparsifier for $G$ and a linear sketch of $G$'s incidence matrix, it is possible to sample edges by effective resistance, obtaining a spectral sparsifier of arbitrary precision. Sampling from the sketch requires a novel application of $\ell_2/\ell_2$ sparse recovery, a natural extension of the $\ell_0$ methods used for cut sparsifiers in [K. J. Ahn, S. Guha, and A. McGregor, in Proceedings of the 31st ACM Symposium on Principles of Database Systems, 2012, pp. 5--14]. Recent work on row sampling for matrix approximation gives a recursive approach for obtaining the required coarse sparsifiers [G. L. Miller and R. Peng, ŭlhttp://arXiv.org/abs/1211.2713v1, 2012]. Under certain restrictions, our approach also extends to the problem of maintaining a spectral approximation for a general matrix $A^\top A$ given a stream of updates to rows in $A$.

Almost Optimal Local Graph Clustering Using Evolving Sets

Spectral partitioning is a simple, nearly linear time algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. A 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 [Spielman and Teng 2013; Andersen et al. 2006]. In this article, we introduce a simple randomized local partitioning algorithm that finds a sparse cut by simulating the volume-biased evolving set process , which is a Markov chain on sets of vertices. We prove that for any ϵ > 0, and any set of vertices A that has conductance at most φ, for at least half of the starting vertices in A our algorithm will output (with constant probability) a set of conductance O (√φ /ϵ). We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is vol( A ) ϵ φ -1/2 polylog( n ), where vol( A ) = Σ v ∈ A d ( v ) is the volume of the set A . This gives an algorithm with the same guarantee (up to a constant factor) as the Cheeger's inequality that runs in time slightly superlinear in the size of the output. This is the first sublinear (in the size of the input) time algorithm with almost the same guarantee as the Cheeger's inequality. In comparison, the best previous local partitioning algorithm, by Andersen et al. [2006], has a worse approximation guarantee of O (√φ log n ) and a larger ratio of φ -1 polylog( n ) between the complexity and output volume. As a by-product of our results, we prove a bicriteria approximation algorithm for the expansion profile of any graph. For 0 < k ≤ vol( V )/2, let φ( k ) : min S : vol( S ) ≤ k φ( S ). There is a polynomial time algorithm that, for any k , ϵ > 0, finds a set S of volume vol( S ) ≤ O ( k 1 + ϵ ) and expansion φ( S )≤ O (√φ ( k )/ϵ). As a new technical tool, we show that for any set S of vertices of a graph, a lazy t -step random walk started from a randomly chosen vertex of S will remain entirely inside S with probability at least (1 - φ( S )/2) t . This itself provides a new lower bound to the uniform mixing time of any finite state reversible Markov chain.

Metric extension operators, vertex sparsifiers and Lipschitz extendability

We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/ log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log2 k/ log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we obtain a lower bound of $$\Omega \left( {\sqrt {\log k/\log \log k} } \right)$$ for flow sparsifiers and a lower bound of $$\Omega \left( {\sqrt {\log k} /\log \log k} \right)$$ for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist $$\tilde O\left( {\sqrt {\log k} } \right)$$ cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than $$\tilde \Omega \left( {\sqrt {\log k} } \right)$$ would imply a negative answer to this question.

Multiway Spectral Partitioning and Higher-Order Cheeger Inequalities

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero. It has been conjectured that an analogous characterization holds for higher multiplicities: There are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut. We resolve this conjecture positively. Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into R k , and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size ≈ n / k and λ k , the k th smallest eigenvalue of the normalized Laplacian, where n is the number of vertices. In particular, we show that in every graph there are at least k /2 disjoint sets (one of which will have size at most 2 n / k ), each having expansion at most O (√λ k log k ). Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result. The √log k bound is tight, up to constant factors, for the “noisy hypercube” graphs.

A local spectral method for graphs: with applications to improving graph partitions and exploring data graphs locally

The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many applications, however, graphs that arise have several local regions of interest, and the second eigenvector will typically fail to provide information fine-tuned to each local region. In this paper, we introduce a locally-biased analogue of the second eigenvector, and we demonstrate its usefulness at highlighting local properties of data graphs in a semi-supervised manner. To do so, we first view the second eigenvector as the solution to a constrained optimization problem, and we incorporate the local information as an additional constraint; we then characterize the optimal solution to this new problem and show that it can be interpreted as a generalization of a Personalized PageRank vector; and finally, as a consequence, we show that the solution can be computed in nearly-linear time. In addition, we show that this locally-biased vector can be used to compute an approximation to the best partition near an input seed set in a manner analogous to the way in which the second eigenvector of the Laplacian can be used to obtain an approximation to the best partition in the entire input graph. Such a primitive is useful for identifying and refining clusters locally, as it allows us to focus on a local region of interest in a semi-supervised manner. Finally, we provide a detailed empirical evaluation of our method by showing how it can applied to finding locally-biased sparse cuts around an input vertex seed set in social and information networks.

Graph Sparsification by Effective Resistances

We present a nearly linear time algorithm that produces high-quality spectral sparsifiers of weighted graphs. Given as input a weighted graph $G=(V,E,w)$ and a parameter $\epsilon>0$, we produce a weighted subgraph $H=(V,\tilde{E},\tilde{w})$ of $G$ such that $|\tilde{E}|=O(n\log n/\epsilon^2)$ and all $x\in\mathbb{R}^V$ satisfy $(1-\epsilon)\sum_{uv\in E}\,(x(u)-x(v))^2w_{uv}\leq\sum_{uv\in\tilde{E}}\,(x(u)-x(v))^2\tilde{w}_{uv}\leq(1+\epsilon)\sum_{uv\in E}\,(x(u)-x(v))^2w_{uv}$. This improves upon the spectral sparsifiers constructed by Spielman and Teng, which had $O(n\log^{c}n)$ edges for some large constant $c$, and upon the cut sparsifiers of Benczur and Karger, which only satisfied these inequalities for $x\in\{0,1\}^V$. A key ingredient in our algorithm is a subroutine of independent interest: a nearly linear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in $O(\log n)$ time.

Linear time algorithms for finding sparsest cuts in various graph classes

Abstract [ S , S ¯ ] denotes the set of edges with exactly one end vertex in S. The density of an edge cut [ S , S ¯ ] is | [ S , S ¯ ] | / ( | S | | S ¯ | ) . A sparsest cut is an edge cut with minimum density. We characterize the sparsest cuts for unit interval graphs, complete bipartite graphs and cactus graphs. For all of these classes, the characterizations lead to linear time algorithms to find a sparsest cut.

Inoculation strategies for victims of viruses and the sum-of-squares partition problem

We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install anti-virus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being stopped by some intermediate node. We prove many game theoretic properties of the model, including an easily applied characterization of Nash equilibria, culminating in our showing that a centralized solution can give a much better total cost than an equilibrium solution. Though it is NP-hard to compute such a social optimum, we show that the problem can be reduced to a previously unconsidered combinatorial problem that we call the sum-of-squares partition problem. Using a greedy algorithm based on sparse cuts, we show that this problem can be approximated to within a factor of O ( log 1.5 n ) .