Vertex Separator Research Articles

Exact and heuristic methods for the vertex separator problem

In this paper, we propose a practical and efficient methods to solve the vertex separator problem (VSP for short), based on branch-and-bound procedure, which uses linear programming, and a greedy algorithm. This problem arises in many areas of applications such as graph algorithms, communication networks, solving systems of equations, finite element and finite difference problems. We show, by computational experiments, that our approach is able to solve in short time large-scale instances of VSP from the literature to optimality or near optimality.

A branch-and-price algorithm for capacitated hypergraph vertex separation

We exactly solve the $${\mathcal {NP}}$$-hard combinatorial optimization problem of finding a minimum cardinality vertex separator with k (or arbitrarily many) capacitated shores in a hypergraph. We present an exponential size integer programming formulation which we solve by branch-and-price. The pricing problem, an interesting optimization problem on its own, has a decomposable structure that we exploit in preprocessing. We perform an extensive computational study, in particular on hypergraphs coming from the application of re-arranging a matrix into single-bordered block-diagonal form. Our experimental results show that our proposal complements the previous exact approaches in terms of applicability for larger k, and significantly outperforms them in the case $$k=\infty $$.

Computing the chromatic number using graph decompositions via matrix rank

Computing the smallest number q such that the vertices of a given graph can be properly q-colored, known as the chromatic number, is one of the oldest and most fundamental problems in combinatorial optimization. The q-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. For example, algorithms are known to solve the problem on graphs of treewidth tw in time O⁎(qtw), while a running time of O⁎((q−ε)tw) is impossible assuming the Strong Exponential Time Hypothesis (SETH). While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying q-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators.We present two algorithms for q-Coloring parameterized by cutwidth ctw: a deterministic one that runs in time O⁎(2ω⋅ctw), where ω is the square matrix multiplication exponent, and a randomized one with runtime O⁎(2ctw). In sharp contrast to earlier work, the running time is independent of q. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in O⁎((2−ε)ctw) time assuming SETH. Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an O⁎((⌊d/2⌋+1)pw) time randomized algorithm for q-Coloring on graphs of pathwidth pw and maximum degree d. Such a runtime was first obtained by Björklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no O⁎((⌊d/2⌋+1−ε)pw)-time algorithm exists assuming SETH.

Trade-offs between computation, communication, and synchronization in stencil-collective alternate update

In a computing platform composed of several homogeneous processors, any parallel schedule of an algorithm usually involves three basic costs: arithmetic throughput on each processor, data movement between processors, and synchronization latency for several processors. The trade-offs between these three costs could realistically reflect lower bounds on the execution time for an algorithm. Therefore, the trade-off analysis is important for evaluating the optimality of a proposed schedule, and often yields new insights in parallel optimization. In this paper, we focus on the trade-offs between computation, communication, and synchronization in the stencil-collective alternate update, which is often executed repeatedly by the complex workflow with multiple stages in most numerical methods, such as the conjugate gradient (CG) method, the nonlinear time integration method in the dynamical core of a global atmospheric general circulation model (AGCM), and so on. Firstly, in order to formalize a workflow with multiple different stages, a novel operator representation of parallel algorithms is proposed. Based on the operator representation, we find the minimum vertex separator of the dependency graph for a stencil-collective alternate update. This breakthrough brings us the opportunity to obtain the cost lower bounds. Next, the general trade-off theory of the stencil-collective alternate update is founded successfully, which extends the recent trade-off theory to a more general theoretical context. Finally, by applying the general theoretical result to several algorithms, namely CG method and the nonlinear time integration method in AGCM, we obtain the corresponding lower bounds of computational cost, communication throughput, and synchronization latency. It should be noted that the general theory can also be widely used to analyze other complex numerical methods in real-world applications.

Edge integrity of nearest neighbor graphs and separator theorems

We give upper bounds for the edge integrity and the vertex integrity of k-type nearest neighbor graphs in Rd. For n the order of a graph, examples are constructed to show that the bounds are best possible in the parameters n and k. The results follow from decompositions that are shown for certain graphs that possess small edge or vertex separators. A vertex separator theorem for nearest neighbor graphs is known, and an edge separator theorem is shown. We also apply the decompositions to obtain upper bounds for the edge integrity of planar graphs, graphs of fixed genus, and trees.

A weighted approach to the maximum cardinality bipartite matching problem with applications in geometric settings

We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph $G(A\cup B,E)$ with edge weights of $0$ or $1$. Let $w \leq n$ be an upper bound on the weight of any matching in $G$. Consider the subgraph induced by all the edges of $G$ with a weight $0$. Suppose every connected component in this subgraph has $O(r)$ vertices and $O(mr/n)$ edges. We present an algorithm to compute a maximum cardinality matching in $G$ in $\tilde{O}( m(\sqrt{w}+ \sqrt{r}+\frac{wr}{n}))$ time.When all the edge weights are $1$ (symmetrically when all weights are $0$), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in $O(m\sqrt{n})$ time. However, if we can carefully assign weights of $0$ and $1$ on its edges such that both $w$ and $r$ are sub-linear in $n$ and $wr=O(n^{\gamma})$ for $\gamma < 3/2$, then we can compute a maximum cardinality matching in $o(m\sqrt{n})$ time. Using our algorithm, we obtain a new $\tilde{O}(n^{4/3}/\varepsilon^3)$ time algorithm to compute an $\varepsilon$-approximate bottleneck matching of $A,B\subset \mathbb{R}^2$ and a $\frac{1}{\varepsilon^{O(d)}}n^{1+\frac{d-1}{2d-1}}\mathrm{poly}\log n$ time algorithm for computing an $\varepsilon$-approximate bottleneck matching in $d$-dimensions. All previous algorithms take $\Omega(n^{3/2})$ time.Our algorithm also applies to any graph $G(A \cup B,E)$ that has an easily computable balanced vertex separator of size $|V'|^{\delta}$, for every subgraph $G'(V',E')$ where $\delta\in [1/2,1)$. By applying our algorithm, we can compute a maximum matching in $\tilde{O}(mn^{\frac{\delta}{1+\delta}})$ time, improving upon the $O(m\sqrt{n})$ time taken by the HK-Algorithm.

On some combinatorial problems in cographs

The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper, we study some popular combinatorial problems restricted to cographs. We first present a structural characterization of minimal vertex separators in cographs. Further, we show that listing all minimal vertex separators and the complexity of some constrained vertex separators are polynomial-time solvable in cographs. We propose polynomial-time algorithms for connectivity augmentation problems and its variants in cographs, preserving the cograph property. Finally, using the dynamic programming paradigm, we present a generic framework to solve classical optimization problems such as the longest path, the Steiner path and the minimum leaf spanning tree problems restricted to cographs, our framework yields polynomial-time algorithms for all three problems.

The multi-terminal vertex separator problem: Polyhedral analysis and Branch-and-Cut

In this paper we consider a variant of the k-separator problem. Given a graph G=(V∪T,E) with V∪T the set of vertices, where T is a set of k terminals, the multi-terminal vertex separator problem consists in partitioning V∪T into k+1 subsets {S,V1,…,Vk} such that there is no edge between two different subsets Vi and Vj, each Vi contains exactly one terminal and the size of S is minimum. In this paper, we first show that the problem is NP-hard. Then we give two integer programming formulations for the problem. For one of these formulations, we investigate the related polyhedron and discuss its polyhedral structure. We describe some valid inequalities and characterize when these inequalities define facets. We also derive separation algorithms for these inequalities. Using these results, we develop a Branch-and-Cut algorithm for the problem, along with an extensive computational study.

Approximating Small Balanced Vertex Separators in Almost Linear Time

For a graph G with n vertices and m edges, we give a randomized Las Vegas algorithm that approximates a small balanced vertex separator of G in almost linear time. More precisely, we show the following, for any $$\frac{2}{3}\le \alpha <1$$ and any $$0<\varepsilon <1-\alpha $$ : If G contains an $$\alpha $$ -separator of size K, then our algorithm finds an $$(\alpha +\varepsilon )$$ -separator of size $${\mathcal {O}}(\varepsilon ^{-1}K^2\log ^{1+o(1)} n)$$ in time $${\mathcal {O}}(\varepsilon ^{-1}K^3m\log ^{2+o(1)} n)$$ w.h.p. In particular, if $$K\in {\mathcal {O}}(\hbox {polylog } n)$$ , then we obtain an $$(\alpha +\varepsilon )$$ -separator of size $${\mathcal {O}}(\varepsilon ^{-1}\hbox { polylog } n)$$ in time $${\mathcal {O}}(\varepsilon ^{-1}m\hbox { polylog } n)$$ w.h.p. The presented algorithm does not require knowledge of K.

Characterization by forbidden induced subgraphs of some subclasses of chordal graphs

Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set S is a vertex separator for vertices a and b if the removal of S of the graph separates a and b into distinct connected components. A graph G is chordal if and only if every minimal vertex separator is a clique. We study subclasses of chordal graphs defined by restrictions imposed on the intersections of its minimal separator cliques. Our goal is to characterize them by forbidden induced subgraphs. Some of these classes have already been studied such as chordal graphs in which two minimal separators have no empty intersection if and only if they are equal. Those graphs are known as strictly chordal graphs and they were first introduced as block duplicate graphs by Golumbic and Peled [Golumbic, M. C. and Peled, U. N., Block duplicate graphs and a hierarchy of chordal graphs, Discrete Applied Mathematics, 124 (2002) 67–71], they were also considered in [Kennedy, W., “Strictly chordal graphs and phylogenetic roots”, Master Thesis, University of Alberta, 2005] and [De Caria, P. and Gutiérrez, M., On basic chordal graphs and some of its subclasses, Discrete Applied Mathematics, 210 (2016) 261–276], showing that strictly chordal graphs are exactly the (gem, dart)-free graphs.

Non-inclusion and other subclasses of chordal graphs

In this work we assess subclasses of chordal graphs focusing on special structures such as maximal cliques and minimal vertex separators. A subclass is presented and characterized, the non-inclusion chordal graphs, graphs in which there is no proper containment between any two minimal vertex separators. A hierarchical structure of classes is outlined, showing that some well-known subclasses of chordal graphs are subclasses of the newly defined class. In order to characterize non-inclusion chordal graphs, a supporting structure, the clique-bipartite graph of a chordal graph is defined.

Partitioning a graph into small pieces with applications to path transversal

Given a graph $$G = (V, E)$$ and an integer $$k \in \mathbb {N}$$, we study $$k$$-Vertex Separator (resp. $$k$$-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has less than k vertices. We also study $$k$$-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. Our main results are the following improved approximation algorithms.

Vulnerability of Subclasses of Chordal Graphs

In this paper, we introduce a new invariant that supports an accurate evaluation of the connectivity of graphs belonging to some subclasses of chordal graphs, allowing the establishment of a total ordering of the elements of the class. It is based on the minimal vertex separators of the graph, and, as so, its computation is performed in linear time. Results about the behavior of the new invariant and a comparison with the toughness are presented for block graphs.

An I/O-Efficient Algorithm for Computing Vertex Separators on Multi-Dimensional Grid Graphs and Its Applications

A vertex separator, in general, refers to a set of vertices whose removal disconnects the original graph into subgraphs possessing certain nice properties. Such separators have proved useful for solving a variety of graph problems. The core contribution of the paper is an I/O-efficient algorithm that finds, on any $d$-dimensional grid graph, a small vertex separator which mimics the well-known separator of [Maheshwari and Zeh, SICOMP'08] for planar graphs. We accompany our algorithm with two applications. First, by integrating our separators with existing algorithms, we strengthen the current state-of-the-art results of three fundamental problems on 2D grid graphs: finding connected components, finding single source shortest paths, and breadth-first search. Second, we show how our separator-algorithm can be deployed to perform density-based clustering on $d$-dimensional points I/O-efficiently.

A multilevel bilinear programming algorithm for the vertex separator problem

The Vertex Separator Problem for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets that satisfy specified size constraints. The Vertex Separator Problem was formulated in the paper 10.1016/j.ejor.2014.05.042 as a continuous (non-concave/non-convex) bilinear quadratic program. In this paper, we develop a more general continuous bilinear program which incorporates vertex weights, and which applies to the coarse graphs that are generated in a multilevel compression of the original Vertex Separator Problem. We develop a method for improving upon a given vertex separator by applying a Mountain Climbing Algorithm to the bilinear program using an incidence vector for the separator as a starting guess. Sufficient conditions are developed under which the algorithm can improve upon the starting guess after at most two iterations. The refinement algorithm is augmented with a perturbation technique to enable escapes from local optima and is embedded in a multilevel framework for solving large scale instances of the problem. The multilevel algorithm is shown through computational experiments to perform particularly well on communication and collaboration networks.

Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is δ -hyperbolic if every geodesic triangle is δ -thin. In this paper, we study the relation between vertex separator sets, certain chordality properties that generalize being chordal and the hyperbolicity of the graph. We also give a characterization of being quasi-isometric to a tree in terms of chordality and prove that this condition also characterizes being hyperbolic, when restricted to triangles, and having stable geodesics, when restricted to bigons.

The min-cut and vertex separator problem

We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order 2n+1 which provide a good compromise between quality of the bound and computational effort to actually compute it. Here, n is the order of the graph. Our numerical results indicate that the new bounds are quite strong and can be computed for graphs of medium size (n approx 300) with reasonable effort of a few minutes of computation time. Further, we exploit those bounds to obtain bounds on the size of the vertex separators. A vertex separator is a subset of the vertex set of a graph whose removal splits the graph into two disconnected subsets. We also present an elegant way of convexifying non-convex quadratic problems by using semidefinite programming. This approach results with bounds that can be computed with any standard convex quadratic programming solver.

Event‐triggered non‐fragile filtering of linear systems with a structure separated approach

Considering filter gains with additive interval variations, this study is devoted to the non-fragile H ∞ filtering of linear systems with an event-triggered transmission. To take into account the influence of these variations thoroughly, a structured vertex separator is adopted to alleviate the computation burden. Combining the Finsler lemma, conditions of designing the non-fragile filter to ensure the required H ∞ performance are formed by linear matrix inequalities. Compared to the norm-bounded method to handle the variations, the proposed method could provide less conservative results. Simulations are executed to show the validity of the proposed method.

New characterizations of Gallai’s i-triangulated graphs

The i-triangulated graphs, introduced by Tibor Gallai in the early 1960s, have a number of characterizations—one of the simplest is that every odd cycle of length 5 or more has noncrossing chords. A variety of new characterizations are proved, starting with a simple forbidden induced subgraph characterization and including the following two:(1) Every 2-connected induced subgraph H is bipartite or chordal or, for every induced even cycle C of H, the intersection of the neighborhoods in H of all the vertices of C induces a complete multipartite subgraph.(2) For every 2-connected induced subgraph H that is not bipartite, every cardinality-2 minimal vertex separator of H induces an edge.

Decomposing a graph into expanding subgraphs

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Three examples of our results are the following: A classical result of Lipton, Rose and Tarjan from 1979 states that if is a hereditary family of graphs and every graph in has a vertex separator of size , then every graph in has O(n) edges. We construct a hereditary family of graphs with vertex separators of size such that not all graphs in the family have O(n) edges. Trevisan and Arora‐Barak‐Steurer have recently shown that given a graph G, one can remove only 1% of its edges to obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are essentially best possible, even when one is allowed to remove 99% of G's edges. Sudakov and the second author have recently shown that every graph with average degree d contains an n‐vertex subgraph with average degree at least and vertex expansion . We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1). The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super‐linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.