Adaptive Monte Carlo Search for Conjecture Refutation in Graph Theory

Graph theory is an interdisciplinary field of study that has various applications in mathematical modeling and computer science. Research in graph theory depends on the creation of not only theorems but also conjectures. Conjecture-refuting algorithms attempt to refute conjectures by searching for counterexamples to those conjectures, often by maximizing certain score functions on graphs. This study proposes a novel conjecture-refuting algorithm, referred to as the adaptive Monte Carlo search (AMCS) algorithm, obtained by modifying the Monte Carlo tree search algorithm. Evaluated based on its success in finding counterexamples to several graph theory conjectures, AMCS outperforms existing conjecture-refuting algorithms. The algorithm is further utilized to refute six open conjectures, two of which were chemical graph theory conjectures formulated by Liu et al. in 2021 and four of which were formulated by the AutoGraphiX computer system in 2006. Finally, four of the open conjectures are strongly refuted by generalizing the counterexamples obtained by AMCS to produce a family of counterexamples. It is expected that the algorithm can help researchers test graph-theoretic conjectures more effectively.Comment: 27 pages, 11 figures, 3 tables; Milo Roucairol pointed out that both of our papers used an incorrect formula for the harmonic of a graph. The revised Conjecture 4 was able to be refuted. This paper and the GitHub repository have been updated accordingl

Spectral Reconstruction and Isomorphism of graphs using variable neighbourhood search

The Euclidean distance between the eigenvalue sequences of graphs G and H, on the same number of vertices, is called the spectral distance between G and H. This notion is the basis of a heuristic algorithm for reconstructing a graph with prescribed spectrum. By using a graph Γ constructed from cospectral graphs G and H, we can ensure that G and H are isomorphic if and only if the spectral distance between Γ  and G+K2 is zero. This construction is exploited to design a heuristic algorithm for testing graph isomorphism. We present preliminary experimental results obtained by implementing these algorithms in conjunction with a meta-heuristic known as a variable neighbourhood search

Addressing Computational Bottlenecks in Higher-Order Graph Matching with Tensor Kronecker Product Structure

Graph matching, also known as network alignment, is the problem of finding a correspondence between the vertices of two separate graphs with strong applications in image correspondence and functional inference in protein networks. One class of successful techniques is based on tensor Kronecker products and tensor eigenvectors. A challenge with these techniques are memory and computational demands that are quadratic (or worse) in terms of problem size. In this manuscript we present and apply a theory of tensor Kronecker products to tensor based graph alignment algorithms to reduce their runtime complexity from quadratic to linear with no appreciable loss of quality. In terms of theory, we show that many matrix Kronecker product identities generalize to straightforward tensor counterparts, which is rare in tensor literature. Improved computation codes for two existing algorithms that utilize this new theory achieve a minimum 10 fold runtime improvement.Comment: 14 pages, 2 pages Supplemental, 5 figure

Modulus of edge covers and stars

This paper explores the modulus (discrete $p$-modulus) of the family of edge covers on a discrete graph. This modulus is closely related to that of the larger family of fractional edge covers; the modulus of the latter family is guaranteed to approximate the modulus of the former within a multiplicative factor based on the length of the shortest odd cycle in the graph. The bounds on edge cover modulus can be computed efficiently using a duality result that relates the fractional edge covers to the family of stars

Positive Definite Kernels in Machine Learning

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as reproducing kernel Hibert spaces, the natural extension of the set of functions $\{k(x,\cdot),x\in\mathcal{X}\}$ associated with a kernel $k$ defined on a space $\mathcal{X}$. We discuss at length the construction of kernel functions that take advantage of well-known statistical models. We provide an overview of numerous data-analysis methods which take advantage of reproducing kernel Hilbert spaces and discuss the idea of combining several kernels to improve the performance on certain tasks. We also provide a short cookbook of different kernels which are particularly useful for certain data-types such as images, graphs or speech segments.Comment: draft. corrected a typo in figure

Mixed-Integer Programming Approaches to Generalized Submodular Optimization and its Applications

Submodularity is an important concept in integer and combinatorial optimization. A classical submodular set function models the utility of selecting homogenous items from a single ground set, and such selections can be represented by binary variables. In practice, many problem contexts involve choosing heterogenous items from more than one ground set or selecting multiple copies of homogenous items, which call for extensions of submodularity. We refer to the optimization problems associated with such generalized notions of submodularity as Generalized Submodular Optimization (GSO). GSO is found in wide-ranging applications, including infrastructure design, healthcare, online marketing, and machine learning. Due to the often highly nonlinear (even non-convex and non-concave) objective function and the mixed-integer decision space, GSO is a broad subclass of challenging mixed-integer nonlinear programming problems. In this tutorial, we first provide an overview of classical submodularity. Then we introduce two subclasses of GSO, for which we present polyhedral theory for the mixed-integer set structures that arise from these problem classes. Our theoretical results lead to efficient and versatile exact solution methods that demonstrate their effectiveness in practical problems using real-world datasets

Free-Shape Polygonal Object Localization

Polygonal objects are prevalent in man-made scenes. Early approaches to detecting them relied mainly on geometry while subsequent ones also incorporated appearance-based cues. It has recently been shown that this could be done fast by searching for cycles in graphs of line-fragments, provided that the cycle scoring function can be expressed as additive terms attached to individual fragments. In this paper, we propose an approach that eliminates this restriction. Given a weighted line-fragment graph, we use its cyclomatic number to partition the graph into managebly-sized sub-graphs that preserve nodes and edges with a high weight and are most likely to contain object contours. Object contours are then detected as maximally scoring elementary circuits enumerated in each sub-graph. Our approach can be used with any cycle scoring function and multiple candidates that share line fragments can be found. This is unlike in other approaches that rely on a greedy approach to finding candidates. We demonstrate that our approach significantly outperforms the state-of-the-art for the detection of building rooftops in aerial images and polygonal object categories from ImageNet