1,076 research outputs found

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Clique‐width: Harnessing the power of atoms

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    Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class if they are so on the atoms (graphs with no clique cut-set) of . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph is -free if is not an induced subgraph of , and it is -free if it is both -free and -free. A class of -free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for -free graphs, as evidenced by one known example. We prove the existence of another such pair and classify the boundedness of clique-width on -free atoms for all but 18 cases

    Finding a Maximum Restricted tt-Matching via Boolean Edge-CSP

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    The problem of finding a maximum 22-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum tt-matching which excludes specified complete tt-partite subgraphs, where tt is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete tt-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most 2t12t-1. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum 22-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most t+1t+1. Our algorithms are obtained from exploiting the discrete structure of restricted tt-matchings and employing an algorithm for the Boolean edge-CSP.Comment: 20 pages, 2 figure

    Speed limits and locality in many-body quantum dynamics

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    We review the mathematical speed limits on quantum information processing in many-body systems. After the proof of the Lieb-Robinson Theorem in 1972, the past two decades have seen substantial developments in its application to other questions, such as the simulatability of quantum systems on classical or quantum computers, the generation of entanglement, and even the properties of ground states of gapped systems. Moreover, Lieb-Robinson bounds have been extended in non-trivial ways, to demonstrate speed limits in systems with power-law interactions or interacting bosons, and even to prove notions of locality that arise in cartoon models for quantum gravity with all-to-all interactions. We overview the progress which has occurred, highlight the most promising results and techniques, and discuss some central outstanding questions which remain open. To help bring newcomers to the field up to speed, we provide self-contained proofs of the field's most essential results.Comment: review article. 93 pages, 10 figures, 1 table. v2: minor change

    The Potts model and the independence polynomial:Uniqueness of the Gibbs measure and distributions of complex zeros

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    Part 1 of this dissertation studies the antiferromagnetic Potts model, which originates in statistical physics. In particular the transition from multiple Gibbs measures to a unique Gibbs measure for the antiferromagnetic Potts model on the infinite regular tree is studied. This is called a uniqueness phase transition. A folklore conjecture about the parameter at which the uniqueness phase transition occurs is partly confirmed. The proof uses a geometric condition, which comes from analysing an associated dynamical system.Part 2 of this dissertation concerns zeros of the independence polynomial. The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the independence polynomial is related to phase transitions in terms of the analycity of the free energy and plays an important role in the design of efficient algorithms to approximately compute evaluations of the independence polynomial. Chapter 5 directly relates the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. This is done by moreover relating the set of zeros of the independence polynomial to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios. Chapter 6 studies boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. It is shown that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori

    Multi-agent Learning For Game-theoretical Problems

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    Multi-agent systems are prevalent in the real world in various domains. In many multi-agent systems, interaction among agents is inevitable, and cooperation in some form is needed among agents to deal with the task at hand. We model the type of multi-agent systems where autonomous agents inhabit an environment with no global control or global knowledge, decentralized in the true sense. In particular, we consider game-theoretical problems such as the hedonic coalition formation games, matching problems, and Cournot games. We propose novel decentralized learning and multi-agent reinforcement learning approaches to train agents in learning behaviors and adapting to the environments. We use game-theoretic evaluation criteria such as optimality, stability, and resulting equilibria

    Statistical methods for gene selection and genetic association studies

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    This dissertation includes five Chapters. A brief description of each chapter is organized as follows. In Chapter One, we propose a signed bipartite genotype and phenotype network (GPN) by linking phenotypes and genotypes based on the statistical associations. It provides a new insight to investigate the genetic architecture among multiple correlated phenotypes and explore where phenotypes might be related at a higher level of cellular and organismal organization. We show that multiple phenotypes association studies by considering the proposed network are improved by incorporating the genetic information into the phenotype clustering. In Chapter Two, we first illustrate the proposed GPN to GWAS summary statistics. Then, we assess contributions to constructing a well-defined GPN with a clear representation of genetic associations by comparing the network properties with a random network, including connectivity, centrality, and community structure. The network topology annotations based on the sparse representations of GPN can be used to understand the disease heritability for the highly correlated phenotypes. In applications of phenome-wide association studies, the proposed GPN can identify more significant pairs of genetic variant and phenotype categories. In Chapter Three, a powerful and computationally efficient gene-based association test is proposed, aggregating information from different gene-based association tests and also incorporating expression quantitative trait locus information. We show that the proposed method controls the type I error rates very well and has higher power in the simulation studies and can identify more significant genes in the real data analyses. In Chapter Four, we develop six statistical selection methods based on the penalized regression for inferring target genes of a transcription factor (TF). In this study, the proposed selection methods combine statistics, machine learning , and convex optimization approach, which have great efficacy in identifying the true target genes. The methods will fill the gap of lacking the appropriate methods for predicting target genes of a TF, and are instrumental for validating experimental results yielding from ChIP-seq and DAP-seq, and conversely, selection and annotation of TFs based on their target genes. In Chapter Five, we propose a gene selection approach by capturing gene-level signals in network-based regression into case-control association studies with DNA sequence data or DNA methylation data, inspired by the popular gene-based association tests using a weighted combination of genetic variants to capture the combined effect of individual genetic variants within a gene. We show that the proposed gene selection approach have higher true positive rates than using traditional dimension reduction techniques in the simulation studies and select potentially rheumatoid arthritis related genes that are missed by existing methods

    International environmental cooperation and climate change laws: A quantitative analysis

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    The increasing number of IEAs has induced a complex web of interdependent relationships among countries. This thesis mainly studies the international environmental cooperation network created by IEAs and countries’ adoption of national climate change laws by combing theories and methods from network science, economic and political economics and international relations. Specifically, I will outline four projects concerned with IEAs and climate change laws. In the first project, I construct a statistically significant international environmental cooperation network among countries and study its emergency and evolution by investigating its structural properties. The results reveal that the popularity of environmental agreements led to the emergence of an environmental cooperation network and document how collaboration is accelerating. The second and third projects concern the meso-organisation of international environmental cooperation. Specifically, the second project studies the community structure of the environmental cooperation network. Community detection is conducted, and results show that environmental cooperation presents regionalisation. In the third project, I study the core-periphery structure of international environmental cooperation by investigating the nestedness and rich clubs arising from country-treaty relationships. Furthermore, the cooperation complexity is analysed based on methods from economic complexity to further assess country-treaty relationships. I develop a new measure to quantify the diversification of countries’ commitment to environmental treaties. Results show that European countries lie at the core of international environmental cooperation with the highest diversification of commitment. In addition, countries’ diversification of commitment is significantly iv correlated with environmental performance within countries. In the fourth project, I turn to national climate change laws to explore factors influencing the burst of countries’ adoption behaviours. I show that scientific consensus, COPs, and natural disasters are significantly and positively associated with the burst of countries’ adoption behaviours
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