The Revolution in Graph Theoretic Optimization Problems

Spectral Graph Theory is the interplay between linear algebra and combinatorial graph theory. One application of this interplay is a nearly linear time solver for Symmetric Diagonally Dominate systems (SDD). This seemingly restrictive class of systems has received much interest in the last 15 years. Both algorithm design theory and practical implementations have made substantial progress. There is also a growing number of problems that can be efficiently solved using SDD solvers including: image segmentation, image denoising, finding solutions to elliptic equations, computing maximum flow in a graph, graph sparsification, and graphics. All these examples can be viewed as special case of convex optimization problems.

The Fiedler vector of a graph, namely the eigenvector corresponding to the second smallest eigenvalue of a graph Laplacian matrix, plays an important role in spectral graph theory with applications in problems such as graph bi-partitioning and envelope reduction. Algorithms designed to estimate this quantity usually rely on a priori knowledge of the entire graph, and employ techniques such as graph sparsification and power iterations, which have obvious shortcomings in cases where the graph is unknown, or changing dynamically. In this paper, we develop a framework in which we construct a stochastic process based on a set of interacting random walks on a graph and show that a suitably scaled version of our stochastic process converges to the Fiedler vector for a sufficiently large number of walks. Like other techniques based on exploratory random walks and on-the-fly computations, such as Markov Chain Monte Carlo (MCMC), our algorithm overcomes challenges typically faced by power iteration based approaches. But, unlike any existing random walk based method such as MCMCs where the focus is on the leading eigenvector, our framework with interacting random walks converges to the Fiedler vector (second eigenvector). We also provide numerical results to confirm our theoretical findings on different graphs, and show that our algorithm performs well over a wide range of parameters and the number of random walks. Simulations results over time varying dynamic graphs are also provided to show the efficacy of our random walk based technique in such settings. As an important contribution, we extend our results and show that our framework is applicable for approximating not just the Fiedler vector of graph Laplacians, but also the second eigenvector of any time reversible Markov Chain kernel via interacting random walks.

This survey accompanies the Josiah Williard Gibbs Lecture that I gave at the 2016 Joint Mathematics Meetings. It provides an introduction to three topics: algebraic and spectral graph theory, the sparsification of graphs, and the recent resolution of the Kadison–Singer Problem. The first and third are connected by the second.

The article deals with the problem of introducing cloud calculations into 10th-11th graders’ training to solve optimization problems in the context of the STEM-education concept. After analyzing existing programmes of optional courses on optimization problems, the programme of the optional course Optimization Problems has been developed and substantiated implying solution of problems by the cloud environment CoCalc. It is a routine calculating operation and not a mathematical model that is accentuated in the programme. It allows considering more problems which are close to reality without adapting the material while training 10th-11th graders. Besides, the mathematical apparatus of the course which is partially known to students as the knowledge acquired from such mathematics sections as the theory of probability, mathematical statistics, mathematical analysis and linear algebra is enough to master the suggested course. The developed course deals with a whole class of problems of conventional optimization which vary greatly. They can be associated with designing devices and technological processes, distributing limited resources and planning business functioning as well as with everyday problems of people. Devices, processes and situations to which a model of optimization problem is applied are called optimization problems. Optimization methods enable optimal solutions for mathematical models. The developed course is noted for building mathematical models and defining a method to be applied to finding an efficient solution.

Following this advice, we should look over the big questions, then doodle and sketch out some approaches. If you are an expert, then this is easy to do, but most people do not want to wait to become an expert before looking at interesting problems. Graph theory, our area of expertise, has many hard-to-solve questions. Some hark back to the recreational roots of the area yet still keep their mystery. These ‘‘acorns’’ can be planted on the backs of envelopes, on a blackboard, and over a coffee. Our goal is to collect some of these conjectures—arguably some of the most intriguing—in one place. We present ten conjectures in graph theory, and you can read about each one in at most ten minutes. As we live in the era of Twitter, all the conjectures we state are 140 characters or fewer (so ‘‘minute’’ here has a double meaning). We might even call these sketchy tweets, as we present examples for each conjecture that you can doodle on as you read. Hamming also references ambiguity: good researchers can work both on proving and disproving the same statement, so we approach the conjectures with an open mind. He also mentions that a good approach is to reframe the problem, and change the point of view. One example from Vizing’s Conjecture (which is discussed as our second-to-last conjecture in the following text), is the three-page paper [2] which, with a new way of thinking, reduced most of the published work of twenty years to a corollary of its main result! Given the size of modern graph theory, with its many smaller subfields (such as structural graph theory, random graphs, topological graph theory, graph algorithms, spectral graph theory, graph minors, and graph homomorphisms, to name a few), it would be impossible to list all, or even the bulk of the conjectures in the field. We are content instead to focus on a few family jewels, which have an intrinsic beauty and have provided some challenges for graph-theorists for at least two decades. There is something for everyone here, from undergraduate students taking their first course in graph theory, to seasoned researchers in the field. For additional reading on problems and conjectures in graph theory and other fields, see the Open Problem Garden maintained by IRMACS at Simon Fraser University [24]. We consider only finite and undirected graphs, with no multiple edges or loops (unless otherwise stated). We assume the reader has some basic familiarity with graphs and their terminology, including notions such as cycles, paths, complete graphs, complete bipartite graphs, vertex degrees, and connected graphs. We use the notation Cn for the cycle with n vertices, Pn for the path with n vertices, and Kn for the complete graph with n vertices. The complete bipartite graph with m and n vertices of the respective colors is denoted by Km,n. All the background we need can be found in any text in graph theory, such as those of Diestel [9] and West [41], or online (see for example [42]).

Variational quantum algorithms (VQAs) are promising hybrid quantum-classical methods designed to leverage the computational advantages of quantum computing while mitigating the limitations of current noisy intermediate-scale quantum (NISQ) hardware. Although VQAs have been demonstrated as proofs of concept, their practical utility in solving real-world problems—and whether quantum-inspired classical algorithms can match their performance—remains an open question. We present a novel application of the variational quantum linear solver (VQLS) and its classical neural quantum states-based counterpart, the variational neural linear solver (VNLS), as key components within a minimum map Newton solver for a complementarity-based rigid-body contact model. We demonstrate using the VNLS that our solver accurately simulates the dynamics of rigid spherical bodies during collision events. These results suggest that quantum and quantum-inspired linear algebra algorithms can serve as viable alternatives to standard linear algebra solvers for modelling certain physical systems.This article is part of the theme issue ‘Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms’.

Preface. 1 Introduction. 1.1 Two examples of combinatorial optimization. 1.2 Why study combinatorial optimization using statistical physics? 1.3 Textbooks. Bibliography. 2 Algorithms. 2.1 Pidgin Algol. 2.2 Iteration and recursion. 2.3 Divide-and-conquer. 2.4 Dynamic programming. 2.5 Backtracking. Bibliography. 3 Introduction to graphs. 3.1 Basic concepts and graph problems. 3.2 Basic graph algorithms. 3.3 Random graphs. Bibliography. 4 Introduction to complexity theory. 4.1 Turing machines. 4.2 Church's thesis. 4.3 Languages. 4.4 The halting problem. 4.5 Class P. 4.6 Class NP. 4.7 Definition of NP-completeness. 4.8 NP-complete problems. 4.9 Worst-case vs. typical-case complexity. Bibliography. 5 Statistical mechanics of the Ising model. 5.1 Phase transitions. 5.2 Some general notes on statistical mechanics. 5.3 The Curie-Weiss model of a ferromagnet. 5.4 The Ising model on a random graph. Bibliography. 6 Algorithms and numerical results for vertex covers. 6.1 Definitions. 6.2 Heuristic algorithms. 6.3 Branch-and-bound algorithm. 6.4 Results: Covering random graphs. 6.5 The leaf-removal algorithm. 6.6 Monte Carlo simulations. 6.7 Backbone. 6.8 Clustering of minimum vertex covers. Bibliography. 7 Statistical mechanics of vertex covers on a random graph. 7.1 Introduction. 7.2 The first-moment bound. 7.3 The hard-core lattice gas. 7.4 Replica approach. Bibliography. 8 The dynamics of vertex-cover algorithms. 8.1 The typical-case solution time of a complete algorithm. 8.2 The dynamics of generalized leaf-removal algorithms. 8.3 Random restart algorithms. Bibliography. 9 Towards new, statistical-mechanics motivated algorithms. 9.1 The cavity graph. 9.2 Warning propagation. 9.3 Belief propagation. 9.4 Survey propagation. 9.5 Numerical experiments on random graphs. Bibliography. 10 The satisfiability problem. 10.1 SAT algorithms. 10.2 Phase transitions in random K-SAT. 10.3 Typical-case dynamics of RandomWalkSAT. 10.4 Message-passing algorithms for SAT. Bibliography. 11 Optimization problems in physics. 11.1 Monte Carlo optimization. 11.2 Hysteric optimization. 11.3 Genetic algorithms. 11.4 Shortest paths and polymers in random media. 11.5 Maximum flows and random-field systems. 11.6 Submodular functions and free energy of Potts model. 11.7 Matchings and spin glasses. Bibliography. Index.

The maximum flow problem is a fundamental problem in graph theory and combinatorial optimization with a variety of important applications. Known distributed algorithms for this problem do not tolerate faults or adjust to dynamic changes in network topology. This paper presents the first distributed self-stabilizing algorithm for the maximum flow problem. Starting from an arbitrary state, the algorithm computes the maximum flow in a acyclic network in finitely many steps. Since the algorithm is self-stabilizing, it is inherently tolerant to transient faults and can automatically adjust to topology changes and to changes in other parameters of the problem. A slight modification of the original algorithm is also presented and it is conjectured that the new algorithm computes a maximum flow in arbitrary networks. >

The field of data mining increasingly adapts methods and algorithms from advanced matrix computations, graph theory and optimization. In these methods, the data is described using matrix representations (graphs are represented by their adjacency matrices) and the data mining problem is formulated as an optimization problem with matrix variables. With these, the data mining task becomes a process of minimizing or maximizing a desired objective function of matrix variables. Prominent examples include spectral clustering, non-negative matrix factorization, Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) related clustering and dimension reduction, tensor analysis and L-1 regularization. These matrix-formulated optimization-centric methodologies are rapidly becoming a significant part of data mining, and have evolved into a popular and rapidlyexpanding research area for solving challenging data mining problems. First, they are amenable to vigorous analysis and benefit from the well-established knowledge in linear algebra, graph theory and optimization accumulated through centuries. Second, they can be efficiently computed thanks to the mature software tools developed by scientific computing communities. Third, they are simple to implement and easy to

The maximum flow problem is a central problem in graph algorithms and optimization and OBDDs are one of the most common dynamic data structures for Boolean functions. Since in some applications graphs become larger and larger, a research branch has emerged which is concerned with the theoretical design and analysis of symbolic algorithms for classical graph problems on OBDD-represented graph instances. The algorithm for the maximum flow problem in 0-1 networks by Hachtel and Somenzi (1997) has been one of the first of these symbolic algorithms. Typically problems get harder when their input is represented symbolically, nevertheless not many concrete non-trivial lower bounds are known. Here, answering an open question posed by Sawitzki (2006) the first exponential lower bound on the space complexity of OBDD-based algorithms for the maximum flow problem in 0-1 networks is presented.

This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...

Steiner tree problem (STP) in graphs aims to find a tree of minimum weight in the graph that connects a given set of vertices. It is a classic NP-hard combinatorial optimization problem and has many applications in real world. Many approximate algorithms have been developed for STP, but they suffer from high computational complexity and weak worst-case solution guarantees, respectively. Heuristic algorithms are also developed. But each of them requires application domain knowledge to design and is only suitable for specific scenarios. In this paper, we design a novel framework Cherrypick based on novel graph embedding and deep reinforcement learning to tackle the STP. Given an STP instance, Cherrypick uses this embedding to encode its path-related information and sends the encoded graph to a deep reinforcement learning component based on deep Q network (DQN) to find solutions. We implement the Cherrypick and demonstrate its efficacy and efficiency with extensive experiments using real-world and synthetic datasets.

Finding depth-first-search (DFS) trees of graphs is one of the fundamental problems in graph theory which has many practical applications. However, there exists no NC parallel algorithm for this problem in general graphs. NC parallel algorithms for finding DFS trees for the interval graphs and circular-arcs graphs are presented. These algorithms take O(log n) time using kappa n processors on the EREW model, where kappa is the number of cliques in a graph of n nodes. >

In this dissertation I draw a connection between quantum adiabatic optimization, spectral graph theory, heat-diffusion, and sub-stochastic processes through the operators that govern these processes and their associated spectra. In particular, we study Hamiltonians which have recently become known as ``stoquastic'' or, equivalently, the generators of sub-stochastic processes. The operators corresponding to these Hamiltonians are of interest in all of the settings mentioned above. I predominantly explore the connection between the spectral gap of an operator, or the difference between the two lowest energies of that operator, and certain equilibrium behavior. In the context of adiabatic optimization, this corresponds to the likelihood of solving the optimization problem of interest. I will provide an instance of an optimization problem that is easy to solve classically, but leaves open the possibility to being difficult adiabatically. Aside from this concrete example, the work in this dissertation is predominantly mathematical and we focus on bounding the spectral gap. Our primary tool for doing this is spectral graph theory, which provides the most natural approach to this task by simply considering Dirichlet eigenvalues of subgraphs of host graphs. I will derive tight bounds for the gap of one-dimensional, hypercube, and general convex subgraphs. The techniques used will also adapt methods recently used by Andrews and Clutterbuck to prove the long-standing ``Fundamental Gap Conjecture''.

Extremal Problems in Discrete Geometry and Spectral Graph Theory