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submodular functions and Perfect Graphs

MATHEMATICS OF OPERATIONS RESEARCH 2025年第1期50卷 1-781 C2页

作者： Abrishami, Tara Chudnovsky, Maria Dibek, Cemil Vuskovic, Kristina Univ Hamburg Dept Math D-20148 Hamburg Germany Princeton Univ Dept Math Princeton NJ 08544 USA Princeton Univ Dept Operat Res & Financial Engn Princeton NJ 08544 USA Univ Leeds Sch Comp Leeds LS2 9JT England

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm.

关键词： perfect graphs submodular functions maximum weight independent set even sets

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submodular functions: LEARNABILITY, STRUCTURE, AND OPTIMIZATION

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SIAM JOURNAL ON COMPUTING 2018年第3期47卷 703-754页

作者： Balcan, Maria-Florina Harvey, Nicholas J. A. Carnegie Mellon Univ Sch Comp Sci Pittsburgh PA 15213 USA Univ British Columbia Dept Comp Sci Vancouver BC V6T 1Z1 Canada

submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we study submodular functions from a learning theoretic angle. We provide algorithms for learning submodular functions, as well as lower bounds on their learnability. In doing so, we uncover several novel structural results revealing ways in which submodular functions can be both surprisingly structured and surprisingly unstructured. We provide several concrete implications of our work in other domains including algorithmic game theory and combinatorial optimization. At a technical level, this research combines ideas from many areas, including learning theory (distributional learning and PAC-style analyses), combinatorics and optimization (matroids and submodular functions), and pseudorandomness (lossless expander graphs).

关键词： submodular functions distributional learning matroids algorithmic game theory optimization PAC model gross substitutes

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submodular functions: Optimization and Approximation

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International Congress of Mathematicians (ICM 2010)

作者： Iwata, Satoru Kyoto Univ Math Sci Res Inst Kyoto 6068502 Japan

ISBN: (纸本)9789814324342

submodular functions are discrete analogue of convex functions, arising in various fields of applied mathematics including game theory, information theory, and queueing theory. This survey aims at providing an overview on fundamental properties of submodular functions and recent algorithmic developments of their optimization and approximation. For submodular function minimization, the ellipsoid method had long been the only polynomial algorithm until combinatorial strongly polynomial algorithms appeared a decade ago. On the other hand, for submodular function maximization, which is NP-hard and known to refuse any polynomial algorithms, constant factor approximation algorithms have been developed with applications to combinatorial auction, machine learning, and social networks. In addition, an efficient method has been developed for approximating submoduar functions everywhere, which leads to a generic framework of designing approximation algorithms for combinatorial optimization problems with submodular costs. In some specific cases, however, one can devise better approximation algorithms.

关键词： submodular functions discrete optimization approximation algorithms

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Metric learning with submodular functions

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NEUROCOMPUTING 2020年 416卷 328-339页

作者： Pan, Jiajun Le Capitaine, Hoel Univ Nantes CNRS UMR LS2N F-6004 Nantes France

Most of the metric learning mainly focuses on using single feature weights with L-p norms, or the pair of features with Mahalanobis distances to learn the similarities between the samples, while ignoring the potential value of higher-order interactions in the feature space. In this paper, we investigate the possibility of learning weights to coalitions of features whose cardinality can be greater than two, with the help of set-functions. With the more particular property of submodular set-function, we propose to define a metric for continuous features based on Lovasz extension of submodular functions, and then present a dedicated metric learning approach. According to the submodular constraints, it naturally leads to a higher complexity price so that we use the xi-additive fuzzy measure to decrease this complexity, by reducing the order of interactions that are taken into account. This approach finally gives a computationally, feasible problem. Experiments on various datasets show the effectiveness of the approach. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Metric learning Nonlinear metric submodular functions

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MAXIMIZING NON-MONOTONE submodular functions

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SIAM JOURNAL ON COMPUTING 2011年第4期40卷 1133-1153页

作者： Feige, Uriel Mirrokni, Vahab S. Vondrak, Jan Weizmann Inst Sci Dept Comp Sci & Appl Math IL-76100 Rehovot Israel Google Res New York NY 10011 USA IBM Almaden Res Ctr San Jose CA 95120 USA

submodular maximization generalizes many important problems including Max Cut in directed and undirected graphs and hypergraphs, certain constraint satisfaction problems, and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard. In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative (non-monotone) submodular functions. In particular, we give a deterministic local-search 1/3-approximation and a randomized 2/5-approximation algorithm for maximizing nonnegative submodular functions. We also show that a uniformly random set gives a 1/4-approximation. For symmetric submodular functions, we show that a random set gives a 1/2-approximation, which can also be achieved by deterministic local search. These algorithms work in the value oracle model, where the submodular function is accessible through a black box returning f(S) for a given set S. We show that in this model, a (1/2 + epsilon)-approximation for symmetric submodular functions would require an exponential number of queries for any fixed epsilon > 0. In the model where f is given explicitly (as a sum of nonnegative submodular functions, each depending only on a constant number of elements), we prove NP-hardness of (5/6 + epsilon)-approximation in the symmetric case and NP-hardness of (3/4 + epsilon)-approximation in the general case.

关键词： approximation algorithms combinatorial optimization submodular functions submodular function maximization local search algorithms information-theoretic lower bounds

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Inverse problems of submodular functions on digraphs

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2000年第3期104卷 559-575页

作者： Cai, M Yang, X Li, Y Chinese Acad Sci Inst Syst Sci Beijing Peoples R China Chinese Acad Sci Lab Management Decis & Informat Syst Beijing Peoples R China

In this paper, we study the inverse problem of submodular functions on digraphs. Given a feasible solution x* for a linear program generated by a submodular function defined on digraphs, we try to modify the coefficient vector c of the objective function, optimally and within bounds, such that x* becomes an optimal solution of the linear program. It is shown that the problem can be formulated as a combinatorial linear program and can be transformed further into a minimum cost circulation problem. Hence, it can be solved in strongly polynomial time. We also give a necessary and sufficient condition for the feasibility of the problem. Finally, we extend the discussion to the version of the inverse problem with multiple feasible solutions.

关键词： inverse problems submodular functions digraphs minimum cost circulation strongly polynomial algorithms

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Efficient minimization of higher order using monotonic Boolean functions submodular functions

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DISCRETE APPLIED MATHEMATICS 2017年 220卷 1-19页

作者： Ramalingam, Srikumar Russell, Chris Ladicky, L'ubor Torr, Philip H. S. Univ Utah Sch Comp Salt Lake City UT 84112 USA Alan Turing Inst Warwick England Univ Edinburgh Edinburgh EH8 9YL Midlothian Scotland ETH Zurich Switzerland Univ Oxford Oxford England

submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision, and many others. The general solver has a complexity of O(n(3) log(2) n.E + n(4) log (0(1)) n) where E is the time required to evaluate the function and n is the number of variables (Lee et al., 2015). On the other hand, many computer vision and machine learning problems are defined over special subclasses of submodular functions that can be written as the sum of many submodular cost functions defined over cliques containing only a few variables. In such functions, the pseudo-Boolean (or polynomial) representation (Boros and Hammer, 2002) of these subclasses are of degree (or order, or clique size) k where k << n. In this work, we develop efficient algorithms for the minimization of this useful subclass of submodular functions. To do this, we define novel mapping that transform submodular functions of order k into quadratic ones. The underlying idea is to use auxiliary variables to model the higher order terms and the transformation is found using a carefully constructed linear program. In particular, we model the auxiliary variables as monotonic Boolean functions, allowing us to obtain a compact transformation using as few auxiliary variables as possible. The transformed quadratic function can be efficiently minimized using the standard max-flow algorithm with a time complexity of O((n+m)(3)) where m is the total number of auxiliary variables involved in transforming all the higher order terms to quadratic ones. Specifically, we show that our approach for fourth order function requires only 2 auxiliary variables in contrast to 30 or more variables used in existing approaches. In the general case, we give art upper bound for the number or auxiliary variables required to transform a function of order k using Dedekind number, which is substantially lower than the existing bound of 2(2k). (C) 2016 Elsevier B.V. All rights res

关键词： submodular functions Quadratic pseudo-Boolean functions Monotonic Boolean functions Dedekind number Max-flow/mincut algorithm

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Minimizing symmetric submodular functions

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MATHEMATICAL PROGRAMMING 1998年第1-2期82卷 3-12页

作者： Queyranne, M Univ British Columbia Fac Commerce & Business Adm Vancouver BC V6T 1Z2 Canada

We describe a purely combinatorial algorithm which, given a submodular set function f on a finite set V, finds a nontrivial subset A of V minimizing f[A] + f[V \ A].:This algorithm, an extension of the Nagamochi-Ibaraki minimum cut algorithm as simplified by Steer and Wagner [M. Steer, F. Wagner, A simple min cut algorithm, Proceedings of the European Symposium on Algorithms ESA '94, LNCS 855, Springer, Berlin, 1994, pp. 141-147] and by Frank [A. Frank, On the edge-connectivity algorithm of Nagamochi and Ibaraki, Laboratoire Artemis, IMAG, Universite J. Fourier, Grenbole, 1994], minimizes any symmetric submodular function using O(/V/(3)) calls to a function value oracle. (C) 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

关键词： symmetric submodular function minimization submodular function minimization symmetric submodular functions submodular functions submodular systems

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Improved Streaming Algorithms for Maximizing Monotone submodular functions under a Knapsack Constraint

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ALGORITHMICA 2021年第3期83卷 879-902页

作者： Huang, Chien-Chung Kakimura, Naonori CNRS Ecole Normale Super Paris France Keio Univ Yokohama Kanagawa Japan

In this paper, we consider the problem of maximizing amonotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a (0.5 - epsilon)-approximate solution in O( K epsilon(-1)) space, where K is the knapsack capacity (Badanidiyuru et al. KDD 2014). The approximation ratio is recently shown to be optimal (Feldman et al. STOC 2020). In thiswork, we propose a (0.4 - epsilon)-approximation algorithm for the knapsackconstrained problem, using space that is a polynomial of K and epsilon. This improves on the previous best ratio of 0.363 - epsilon with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one.

关键词： submodular functions Streaming algorithm Approximation algorithm

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Streaming Algorithms for Maximizing Monotone submodular functions Under a Knapsack Constraint

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ALGORITHMICA 2020年第4期82卷 1006-1032页

作者： Huang, Chien-Chung Kakimura, Naonori Yoshida, Yuichi Univ PSL DIENS Ecole Normale Super Paris France Keio Univ Dept Math Yokohama Japan Natl Inst Informat Tokyo Japan

In this paper, we consider the problem of maximizing amonotone submodular function subject to a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access only to a small fraction of the data stored in primary memory. For this problem, we propose a (0.363 epsilon)-approximation algorithm, requiring only a single pass through the data;moreover, we propose a (0.4- epsilon)-approximation algorithm requiring a constant number of passes through the data. The required memory space of both algorithms depends only on the size of the knapsack capacity and epsilon.

关键词： submodular functions Single-pass streaming Multiple-pass streaming Constant approximation

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