Vertex Separator Research Articles

HSMVS: heuristic search for minimum vertex separator on massive graphs

In graph theory, the problem of finding minimum vertex separator (MVS) is a classic NP-hard problem, and it plays a key role in a number of important applications in practice. The real-world massive graphs are of very large size, which calls for effective approximate methods, especially heuristic search algorithms. In this article, we present a simple yet effective heuristic search algorithm dubbed HSMVS for solving MVS on real-world massive graphs. Our HSMVS algorithm is developed on the basis of an efficient construction procedure and a simple yet effective vertex-selection heuristic. Experimental results on a large number of real-world massive graphs present that HSMVS is able to find much smaller vertex separators than three effective heuristic search algorithms, indicating the effectiveness of HSMVS. Further empirical analyses confirm the effectiveness of the underlying components in our proposed algorithm.

Space efficient algorithm for solving reachability using tree decomposition and separators

To solve reachability is to determine whether there is a path from one vertex to the other in a graph. Standard graph traversal algorithms such as DFS and BFS take linear time to solve reachability; however, their space complexity is also linear. On the other hand, Savitch's algorithm takes quasipolynomial time, although the space-bound is O(log2⁡n). In this paper, we study space-efficient algorithms for deciding reachability that runs in polynomial time.We show a polynomial-time algorithm that solves reachability in directed graphs using O(wlog⁡n) space. Our algorithm requires access to a tree decomposition of width w for the underlying undirected graph of the input. This requirement can be waived for graphs for which recursive balanced vertex separators can be computed space-efficiently.

Space-efficient algorithms for reachability in directed geometric graphs

The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem.For intersection graphs of Jordan regions, we show how to obtain a “good” vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and O(m1/2log⁡n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial time and O(m1/2log⁡n) space algorithm, where n and m are the number of vertices and edges, respectively. However, for unit contact disk graphs (penny graphs), we use a more involved technique and obtain a better algorithm. We show that for every ϵ>0, there exists a polynomial time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n1/4+ϵ) space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.

Graph-Based Deep Decomposition for Overlapping Large-Scale Optimization Problems

Decomposition methods play a critical role in cooperative co-evolutionary algorithms (CCEAs) for solving large-scale optimization problems. Although some well-performing decomposition methods have been designed based on the interactions among variables (IaV), their grouping accuracy is still limited due to the poor performance on the overlapping problems and the computational roundoff errors of IaV in the implementation. To deal with these limitations, a graph-based deep decomposition (GDD) method is proposed to obtain more accurate grouping results, especially for the overlapping problems. On the one hand, the GDD mines the IaV information and obtains the minimum vertex separator of the interaction graph of variables, so as to group variables deeply and recursively. On the other hand, the GDD has the ability of fault tolerance to deal with the computational roundoff errors of IaV and can improve the grouping accuracy. For better experimental studies of overlapping problems, a novel overlapping function generator is designed with the random and complicate overlap type, and two new metrics are proposed to evaluate the grouping accuracy. Comprehensive experiments show that GDD can greatly improve the grouping accuracy and help CCEAs perform better than other existing algorithms, especially on the overlapping problems. In addition, the GDD is highly fault tolerant and can divide problems accurately even on the inaccurate IaV.

Fast quantum circuit cutting with randomized measurements

We propose a new method to extend the size of a quantum computation beyond the number of physical qubits available on a single device. This is accomplished by randomly inserting measure-and-prepare channels to express the output state of a large circuit as a separable state across distinct devices. Our method employs randomized measurements, resulting in a sample overhead that is O~(4k/ε2), where ε is the accuracy of the computation and k the number of parallel wires that are "cut" to obtain smaller sub-circuits. We also show an information-theoretic lower bound of Ω(2k/ε2) for any comparable procedure. We use our techniques to show that circuits in the Quantum Approximate Optimization Algorithm (QAOA) with p entangling layers can be simulated by circuits on a fraction of the original number of qubits with an overhead that is roughly 2O(pκ), where κ is the size of a known balanced vertex separator of the graph which encodes the optimization problem. We obtain numerical evidence of practical speedups using our method applied to the QAOA, compared to prior work. Finally, we investigate the practical feasibility of applying the circuit cutting procedure to large-scale QAOA problems on clustered graphs by using a 30-qubit simulator to evaluate the variational energy of a 129-qubit problem as well as carry out a 62-qubit optimization.

Vertex separators, chordality and virtually free groups

In this paper, we consider some results obtained for graphs using minimal vertex separators and generalized chordality and translate them to the context of Geometric group theory. Using these new tools, we are able to give two new characterizations for a group to be virtually free. Furthermore, we prove that the Baumslag–Solitar group [Formula: see text] is [Formula: see text]-chordal for some [Formula: see text] if and only if [Formula: see text] and we give an application of generalized chordality to the study of the word problem.

Some results on connected vertex separators

Some results on connected vertex separators

ParGeMSLR: A parallel multilevel Schur complement low-rank preconditioning and solution package for general sparse matrices

This paper discusses parGeMSLR, a C++/MPI software library for the solution of sparse systems of linear algebraic equations via preconditioned Krylov subspace methods in distributed-memory computing environments. The preconditioner implemented in parGeMSLR is based on algebraic domain decomposition and partitions the symmetrized adjacency graph recursively into several non-overlapping partitions via a p-way vertex separator, where p is an integer multiple of the total number of MPI processes. From a numerical perspective, parGeMSLR builds a Schur complement approximate inverse preconditioner as the sum between the matrix inverse of the interface coupling matrix and a low-rank correction term. To reduce the cost associated with the computation of the approximate inverse matrices, parGeMSLR exploits a multilevel partitioning of the algebraic domain. The parGeMSLR library is implemented on top of the Message Passing Interface and can solve both real and complex linear systems. Furthermore, parGeMSLR can take advantage of hybrid computing environments with in-node access to one or more Graphics Processing Units. Finally, the parallel efficiency (weak and strong scaling) of parGeMSLR is demonstrated on a few model problems arising from discretizations of 3D Partial Differential Equations.

Space-Efficient Vertex Separators for Treewidth

For n-vertex graphs with treewidth k = O(n^{1/2-epsilon }) and an arbitrary epsilon >0, we present a word-RAM algorithm to compute vertex separators using only O(n) bits of working memory. As an application of our algorithm, we give an O(1)-approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in c^k n (log log n) log ^* n time using O(n) bits for some constant c > 0. Together with the result of Banerjee et al. (Proceedings of 21st international conference on computing and combinatorics (COCOON 2015). LNCS, vol 9198, Springer, pp 349–360, 2015. https://doi.org/10.1007/978-3-319-21398-9_28) we are able to compute a solution for all monadic-second-order problems (MSO) with O(n + tau (k) cdot p (log _{p} n) log n) bits in O(tau (k) cdot n^{2 + (2/log p)}) time where k is the treewidth of the given graph, p is some arbitrary parameter with 2 le p le n and tau is some function depending on the MSO formula. We finally use the tree decomposition obtained by our algorithm to solve Vertex Cover, Independent Set, Dominating Set, MaxCut and q-Coloring by using polynomial time and O(n) bits as long as the treewidth of the graph is smaller than c' log n for some problem dependent constant 0< c' < 1.

Hierarchical Orthogonal Factorization: Sparse Least Squares Problems

In this work, we develop a fast hierarchical solver for solving large, sparse least squares problems. We build upon the algorithm, spaQR (sparsified QR Gnanasekaran and Darve in SIAM J Matrix Anal Appl 43(1):94–123, 2022), that was developed by the authors to solve large sparse linear systems. Our algorithm is built on top of a Nested Dissection based multifrontal QR approach. We use low-rank approximations on the frontal matrices to sparsify the vertex separators at every level in the elimination tree. Using a two-step sparsification scheme, we reduce the number of columns and maintain the ratio of rows to columns in each front without introducing any additional fill-in. With this improvised scheme, we show that the runtime of the algorithm scales as \(\mathcal {O}(M \log N)\) and uses \(\mathcal {O}(M)\) memory to store the factorization. This is achieved at the expense of a small and controllable approximation error. The end result is an approximate factorization of the matrix stored as a sequence of sparse orthogonal and upper-triangular factors and hence easy to apply/solve with a vector. Numerical experiments compare the performance of the spaQR algorithm with direct multifrontal QR, an inner-outer iterative method, and CGLS iterative method with a diagonal preconditioner and Robust Incomplete Factorization (RIF) preconditioner.

A New Lower Bound on the Size of the Smallest Vertex Separator of a Graph

A New Lower Bound on the Size of the Smallest Vertex Separator of a Graph

On 3-degree 4-chordal graphs

A graph [Formula: see text] is a 3-degree 4-chordal graph if every cycle of length at least five has a chord and the degree of each vertex is at most three. In this paper, we investigate the structure of 3-degree 4-chordal graphs, which is a subclass of 3-degree graphs. We present a structural characterization based on minimal vertex separators and bi-connected components. Many classical problems are known to be NP-complete for 3-degree graphs, and 3-degree 4-chordal graphs are a nontrivial subclass of 3-degree graphs. Using our structural results, we present polynomial-time algorithms for many classical problems which are grouped into (i) subset problems (vertex cover and its variants, feedback vertex set, Steiner tree), (ii) Hamiltonicity and its variants, (iii) graph embedding problems (treewidth, minimum fill-in), and (iv) Coloring problems(rainbow coloring).

Analysis and Characterization of the Spread of COVID‐19 in Mexico through Complex Networks and Optimization Approaches

This work analyzes and characterizes the spread of the COVID‐19 disease in Mexico, using complex networks and optimization approaches. Specifically, we present two methodologies based on the principle of the rupture for the GC and Newton's law of motion to quantify the robustness and identify the Mexican municipalities whose population causes a fast spread of the SARS‐CoV‐2 virus. Specifically, the first methodology is based on several characteristics of the original version of the Vertex Separator Problem (VSP), and the second is based on a new mathematical model (NLM). By solving VSP, we can find nodes that cause the rupture of the giant component (GC). On the other hand, solving the NLM can find more influential nodes for the entire system’s development. Specifically, we present an analysis using a coupled social network model with information about the main characteristics of the contagion and deaths caused by COVID‐19 in Mexico for 19 months (January 2020–July 2021). This work aims to show through the approach of complex networks how the spread of the disease behaves, and, thus, researchers from other areas can delve into the characteristics that cause this behavior.

Casting Light on the Hidden Bilevel Combinatorial Structure of the Capacitated Vertex Separator Problem

Exploiting Bilevel Optimization Techniques to Disconnect Graphs into Small Components In order to limit the spread of possible viral attacks in a communication or social network, it is necessary to identify critical nodes, the protection of which disconnects the remaining unprotected graph into a bounded number of shores (subsets of vertices) of limited cardinality. In the article “'Casting Light on the Hidden Bilevel Combinatorial Structure of the Capacitated Vertex Separator Problem”, Furini, Ljubic, Malaguti, and Paronuzzi provide a new bilevel interpretation of the associated capacitated vertex separator problem and model it as a two-player Stackelberg game in which the leader interdicts (protects) the vertices, and the follower solves a combinatorial optimization problem on the resulting graph. Thanks to this bilevel interpretation, the authors derive different families of strengthening inequalities and show that they can be separated in polynomial time. The ideas exploited in their framework can also be extended to other vertex/edge deletion/insertion problems or graph partitioning problems by modeling them as two-player Stackelberg games to be solved through bilevel optimization.

An efficient algorithm for st-Connected Vertex Separator problem

In this paper, we give some basic notions concerning the st-connected vertex separator problem(st-CVS problem), then we give mathematical formula for st-CVS problem, then some special cases on some types of graphs, after then introduce an integer programming formulations for this problem. Also, we introduce a heuristic to solve this problem.

Deciding Hamiltonicity in Simple 2-Connected Graphs based on Minimal Vertex Separator

Deciding Hamiltonicity in Simple 2-Connected Graphs based on Minimal Vertex Separator

Methodology to quantify robustness in networks: case study—Higher Education System in Mexico

In this work, we propose a methodology to quantify robustness in networks. Specifically, the methodology is based on the resolution of the Vertex Separator Problem in order to find the set of nodes that the elimination of their links causes the rupture of the giant component. The methodology presented in this work was tested on a set of benchmark networks and on set of social networks modeled with the information about the main characteristics of the universities belonging to the Higher Education System in Mexico (HESM) for the period 2013–2017. The results show that the proposed methodology is able to quantify the robustness in several types of networks without using centrality measures or some other structural metric. On the other hand, regarding the HESM networks, we can assure that they are robust and that they are prone to generate communities or modules.

A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem

The minimum cut problem, MC, and the special case of the vertex separator problem, consists in partitioning the set of nodes of a graph G into k subsets of given sizes in order to minimize the number of edges cut after removing the k-th set. Previous work on approximate solutions uses, in increasing strength and expense: eigenvalue, semidefinite programming, SDP, and doubly nonnegative, DNN, bounding techniques. In this paper, we derive strengthened SDP and DNN relaxations, and we propose a scalable algorithmic approach for efficiently evaluating, theoretically verifiable, both upper and lower bounds. Our stronger relaxations are based on a new gangster set, and we demonstrate how facial reduction, FR, fits in well to allow for regularized relaxations. Moreover, the FR appears to be perfectly well suited for a natural splitting of variables, and thus for the application of splitting methods. Here, we adopt the strictly contractive Peaceman-Rachford splitting method, sPRSM. Further, we bring useful redundant constraints back into the subproblems, and show empirically that this accelerates sPRSM.In addition, we employ new strategies for obtaining lower bounds and upper bounds of the optimal value of MC from approximate iterates of the sPRSM thus aiding in early termination of the algorithm. We compare our approach with others in the literature on random datasets and vertex separator problems. This illustrates the efficiency and robustness of our proposed method.

The multi-terminal vertex separator problem: Branch-and-Cut-and-Price

We are given a graph G=(V∪T,E), with V∪T the set of vertices where T is a set of terminals and E the set of edges. The multi-terminal vertex separator problem consists in finding a subset of vertices S⊆V of minimum size intersecting all paths between every pair of terminals. In this paper we present three extended linear integer programming formulations for the multi-terminal vertex separator problem and we develop Branch-and-Price and Branch-and-Cut-and-Price algorithms. For each formulation we present the pricing problem, the branching scheme and the computation of the dual bound used during the column generation phase. Computational results are reported comparing the performance of the formulations on a set of instances.

Integer Laplacian eigenvalues of chordal graphs

In this paper, structural properties of chordal graphs are studied, establishing a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a sufficient condition for the cardinality of a maximal clique to appear as an integer Laplacian eigenvalue. Finally, we review two subclasses of chordal graphs, showing particular properties.