195 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game

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    In the past 20 years, increasing complexity in real world data has lead to the study of higher-order data models based on partitioning hypergraphs. However, hypergraph partitioning admits multiple formulations as hyperedges can be cut in multiple ways. Building upon a class of hypergraph partitioning problems introduced by Li & Milenkovic, we study the problem of minimizing ratio-cut objectives over hypergraphs given by a new class of cut functions, monotone submodular cut functions (mscf's), which captures hypergraph expansion and conductance as special cases. We first define the ratio-cut improvement problem, a family of local relaxations of the minimum ratio-cut problem. This problem is a natural extension of the Andersen & Lang cut improvement problem to the hypergraph setting. We demonstrate the existence of efficient algorithms for approximately solving this problem. These algorithms run in almost-linear time for the case of hypergraph expansion, and when the hypergraph rank is at most O(1)O(1). Next, we provide an efficient O(logn)O(\log n)-approximation algorithm for finding the minimum ratio-cut of GG. We generalize the cut-matching game framework of Khandekar et. al. to allow for the cut player to play unbalanced cuts, and matching player to route approximate single-commodity flows. Using this framework, we bootstrap our algorithms for the ratio-cut improvement problem to obtain approximation algorithms for minimum ratio-cut problem for all mscf's. This also yields the first almost-linear time O(logn)O(\log n)-approximation algorithms for hypergraph expansion, and constant hypergraph rank. Finally, we extend a result of Louis & Makarychev to a broader set of objective functions by giving a polynomial time O(logn)O\big(\sqrt{\log n}\big)-approximation algorithm for the minimum ratio-cut problem based on rounding 22\ell_2^2-metric embeddings.Comment: Comments and feedback welcom

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    A survey of parameterized algorithms and the complexity of edge modification

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    The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

    バンディット問題における環境適応的リグレット解析

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    京都大学新制・課程博士博士(情報学)甲第24939号情博第850号京都大学大学院情報学研究科システム科学専攻(主査)准教授 本多 淳也, 教授 田中 利幸, 教授 鹿島 久嗣学位規則第4条第1項該当Doctor of InformaticsKyoto UniversityDFA

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Partitioning Hypergraphs is Hard: Models, Inapproximability, and Applications

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    We study the balanced kk-way hypergraph partitioning problem, with a special focus on its practical applications to manycore scheduling. Given a hypergraph on nn nodes, our goal is to partition the node set into kk parts of size at most (1+ϵ)nk(1+\epsilon)\cdot \frac{n}{k} each, while minimizing the cost of the partitioning, defined as the number of cut hyperedges, possibly also weighted by the number of partitions they intersect. We show that this problem cannot be approximated to within a n1/polyloglognn^{1/\text{poly} \log\log n} factor of the optimal solution in polynomial time if the Exponential Time Hypothesis holds, even for hypergraphs of maximal degree 2. We also study the hardness of the partitioning problem from a parameterized complexity perspective, and in the more general case when we have multiple balance constraints. Furthermore, we consider two extensions of the partitioning problem that are motivated from practical considerations. Firstly, we introduce the concept of hyperDAGs to model precedence-constrained computations as hypergraphs, and we analyze the adaptation of the balanced partitioning problem to this case. Secondly, we study the hierarchical partitioning problem to model hierarchical NUMA (non-uniform memory access) effects in modern computer architectures, and we show that ignoring this hierarchical aspect of the communication cost can yield significantly weaker solutions.Comment: Published in the 35th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2023

    Quantum Thermal State Preparation

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    Preparing ground states and thermal states is of key importance to simulating quantum systems on a quantum computer. Despite the hope for practical quantum advantage in quantum simulation, popular approaches like variational circuits or adiabatic algorithms appear to face serious difficulties. Monte-Carlo style quantum Gibbs samplers have emerged as an alternative, but prior proposals have been unsatisfactory due to technical obstacles related to energy-time uncertainty. We introduce simple continuous-time quantum Gibbs samplers that overcome these obstacles by efficiently simulating Nature-inspired quantum Master Equations (Liouvillians) utilizing the operator Fourier transform. In addition, we construct the first provably accurate and efficient algorithm for preparing certain purified Gibbs states (called thermal field double states in high-energy physics) of rapidly thermalizing systems; this algorithm also benefits from a Szegedy-type quadratic improvement with respect to the mixing time. Our algorithms' cost has a favorable dependence on temperature, accuracy, and the mixing time (or spectral gap) of the relevant Liouvillians. We contribute to the theory of thermalization by developing a general analytic framework that handles energy uncertainty through non-asymptotic secular approximation and approximate detailed balance, establishing our approximation guarantees and, as a byproduct yielding the first rigorous proof of finite-time thermalization for physically derived Liouvillians. Given the success of the classical Metropolis algorithm and the ubiquity of thermodynamics, we anticipate that quantum Gibbs sampling will become an indispensable tool in quantum computing.Comment: 68 pages, 11 figure

    A Subquadratic Bound for Online Bisection

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    In the online bisection problem one has to maintain a partition of nn elements into two clusters of cardinality n/2n/2. During runtime, an online algorithm is given a sequence of requests, each being a pair of elements: an inter-cluster request costs one unit while an intra-cluster one is free. The algorithm may change the partition, paying a unit cost for each element that changes its cluster. This natural problem admits a simple deterministic O(n2)O(n^2)-competitive algorithm [Avin et al., DISC 2016]. While several significant improvements over this result have been obtained since the original work, all of them either limit the generality of the input or assume some form of resource augmentation (e.g., larger clusters). Moreover, the algorithm of Avin et al. achieves the best known competitive ratio even if randomization is allowed. In this paper, we present a first randomized online algorithm that breaks this natural barrier and achieves a competitive ratio of O~(n27/14)\tilde{O}(n^{27/14}) without resource augmentation and for an arbitrary sequence of requests

    Tight Approximations for Graphical House Allocation

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    The Graphical House Allocation (GHA) problem asks: how can nn houses (each with a fixed non-negative value) be assigned to the vertices of an undirected graph GG, so as to minimize the sum of absolute differences along the edges of GG? This problem generalizes the classical Minimum Linear Arrangement problem, as well as the well-known House Allocation Problem from Economics. Recent work has studied the computational aspects of GHA and observed that the problem is NP-hard and inapproximable even on particularly simple classes of graphs, such as vertex disjoint unions of paths. However, the dependence of any approximations on the structural properties of the underlying graph had not been studied. In this work, we give a nearly complete characterization of the approximability of GHA. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, and bounded-degree planar graphs. For each of these graph classes, we then prove matching lower bounds, showing that in each case, no significant improvement can be attained unless P = NP. We also present general approximation ratios as a function of structural parameters of the underlying graph, such as treewidth; these match the tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we investigate the special case of bounded-degree trees in some detail. We first refute a conjecture by Hosseini et al. [2023] about the structural properties of exact optimal allocations on binary trees by means of a counterexample on a depth-33 complete binary tree. This refutation, together with our hardness results on trees, might suggest that approximating the optimal envy even on complete binary trees is infeasible. Nevertheless, we present a linear-time algorithm that attains a 33-approximation on complete binary trees
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