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Log-Concave Polynomials I: Entropy and a Deterministic Approximation Algorithm for Counting Bases of Matroids [article]

Nima Anari and Shayan Oveis Gharan and Cynthia Vinzant
2018 arXiv   pre-print
We give a deterministic polynomial time 2^O(r)-approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r.  ...  For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the corresponding polynomials.  ...  We introduce a general optimization-based algorithm for approximate counting involving discrete objects, and show that our algorithm performs well for matroids and their intersections.  ... 
arXiv:1807.00929v2 fatcat:wxr2ywkgobhipdj4dhotur5a7q

Maximizing Determinants under Matroid Constraints [article]

Vivek Madan, Aleksandar Nikolov, Mohit Singh, Uthaipon Tantipongpipat
2020 arXiv   pre-print
To show the approximation guarantee, we utilize recent works on strongly log-concave polynomials and show new relationships between different convex programs studied for the problem.  ...  The current best results include an e^2k-estimation for any matroid of rank k and a (1+ϵ)^d-approximation for a uniform matroid of rank k> d+d/ϵ, where the rank k> d denotes the desired size of the optimal  ...  Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids.  ... 
arXiv:2004.07886v1 fatcat:2z4i3t4wujggvpwd4lst7m6uf4

A Tight Analysis of Bethe Approximation for Permanent [article]

Nima Anari, Alireza Rezaei
2019 arXiv   pre-print
We prove that the permanent of nonnegative matrices can be deterministically approximated within a factor of √(2)^n in polynomial time, improving upon the previous deterministic approximations.  ...  Our bound is tight, and when combined with previously known inequalities lower bounding the permanent, fully resolves the quality of Bethe approximation for permanent.  ...  We remark that entropy-based bounds have recently found applications in counting bases of matroids and matroid intersections [AOV18] ; perfect matchings are a special case of matroid intersections.  ... 
arXiv:1811.02933v2 fatcat:7hxms2beffdynjxhgl5pygrthi

Entropy, Optimization and Counting [article]

Mohit Singh, Nisheeth K. Vishnoi
2013 arXiv   pre-print
Subsequently, we show how algorithms for (approximately) counting the underlying discrete set can be translated into efficient algorithms to (approximately) compute max-entropy distributions.  ...  In this paper we study the problem of computing max-entropy distributions over a discrete set of objects subject to observed marginals.  ...  Thus, the number of calls to the counting oracle by the ellipsoid algorithm of Theorem 2.13 is bounded by a polynomial in m, log R and log 1 /ε.  ... 
arXiv:1304.8108v1 fatcat:mnwysuaehvfwfkv34jat7ne3ce

Fully-Dynamic Submodular Cover with Bounded Recourse [article]

Anupam Gupta, Roie Levin
2020 arXiv   pre-print
The algorithm guarantees a total recourse of O(log(c_max/ c_min)·∑_t≤ Tg_t(N)), where c_max,c_min are the largest/smallest costs of elements in N.  ...  This structured class includes set-coverage functions, so our algorithm matches the known O(log n)-competitiveness and O(1) recourse guarantees for fully-dynamic SetCover.  ...  There is a deterministic algorithm for Fully-Dynamic Metric MinimumStein-erTree that achieves a competitive ratio of O(1) and an amortized recourse bound of O(log D), where D is the ratio of the maximum  ... 
arXiv:2009.00800v1 fatcat:4vznx6vtynbdbmjgwuejwbcfmu

The Distance Approach to Approximate Combinatorial Counting [article]

Alexander Barvinok, Alex Samorodnitsky
2000 arXiv   pre-print
As an application, we present a new randomized polynomial time algorithm which approximates the permanent of a 0-1 matrix by solving a small number of Assignment problems.  ...  We develop general methods to obtain fast (polynomial time) estimates of the cardinality of a combinatorially defined set via solving some randomly generated optimization problems on the set.  ...  Gromov and B. Sudakov for many helpful discussions.  ... 
arXiv:math/0005263v1 fatcat:2z4jliry3jcedhrcjwb2dmpwgq

Deep Submodular Functions [article]

Jeffrey Bilmes, Wenruo Bai
2017 arXiv   pre-print
We start with an overview of a class of submodular functions called SCMMs (sums of concave composed with non-negative modular functions plus a final arbitrary modular).  ...  We show that, for any integer k>0, there are k-layer DSFs that cannot be represented by a k'-layer DSF for any k'<k.  ...  Given a submodular function for the loss, as was done in [92] , then the greedy algorithm offers the standard 1 − 1/e approximation guarantee for LAI.  ... 
arXiv:1701.08939v1 fatcat:zvbpgq7d2feqrantvl7hi34jum

Graph Cuts with Interacting Edge Costs - Examples, Approximations, and Algorithms [article]

Stefanie Jegelka
2016 arXiv   pre-print
In this paper, we connect these applications via the generic formulation of "cooperative graph cuts", for which we study complexity, algorithms, and connections to polymatroidal network flows.  ...  We study an extension of the classical graph cut problem, wherein we replace the modular (sum of edge weights) cost function by a submodular set function defined over graph edges.  ...  Hence, for the deterministic variant of Algorithm 1, we can only show the following approximation bound: Lemma 8.  ... 
arXiv:1402.0240v4 fatcat:wznkbacpcnashntirqnv5z24um

Tight Bounds for Approximate Carathéodory and Beyond [article]

Vahab Mirrokni, Renato Paes Leme, Adrian Vladu, Sam Chiu-wai Wong
2015 arXiv   pre-print
We give a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope's vertices.  ...  As simple extensions of our method, we then provide new algorithms for submodular function minimization and SVM training.  ...  Given an n-element matroid by M of rank r and a fractional point x * inside its base polytope, our algorithm produces a sparse distribution D over matroid bases such that marginals are approximately preserve  ... 
arXiv:1512.08602v1 fatcat:mmivrsy3qrf7tmn3y4f2bp3asm

Sensor Planning for Large Numbers of Robots [article]

Micah Corah
2021 arXiv   pre-print
Specifically, consider the problem of how to select motions for a team of robots to maximize a notion of sensing quality (the sensing objective) over the near future, say by maximizing the amount of unknown  ...  Please refer to the thesis for the full abstract.* After a disaster, locating and extracting victims quickly is critical because mortality rises rapidly after the first two days.  ...  Looking toward applications in multi-robot sensing, the best known algorithm for submodular maximization with a matroid constraint has adaptivity: O(log |U | log(rankI )) [12, 42] .  ... 
arXiv:2102.04054v1 fatcat:zgg75lo6wfdwvekn5twqpmi7oe

Proportional Volume Sampling and Approximation Algorithms for A-Optimal Design [article]

Aleksandar Nikolov and Mohit Singh and Uthaipon Tao Tantipongpipat
2018 arXiv   pre-print
In this paper, we introduce proportional volume sampling to obtain improved approximation algorithms for A-optimal design.  ...  Our main result is to obtain improved approximation algorithms for the A-optimal design problem by introducing the proportional volume sampling algorithm.  ...  Damian Straszak and Nisheeth K. Vishnoi. Real stable polynomials and matroids: optimization and counting.  ... 
arXiv:1802.08318v5 fatcat:dirb5tbukzavho356x7s23nftm

Parameter estimation for Gibbs distributions [article]

David G. Harris, Vladimir Kolmogorov
2021 arXiv   pre-print
We illustrate with improved algorithms for counting connected subgraphs and perfect matchings in a graph. We develop a key subroutine to estimate the partition function Z.  ...  This improves over a prior algorithm of Huber (2015) which computes a single point estimate Z(β_max) using O( q log n( log q + loglog n + ε^-2)) samples.  ...  Acknowledgments We thank Heng Guo for helpful explanations of algorithms for sampling connected subgraphs and matchings, Maksym Serbyn for bringing to our attention the Wang-Landau algorithm and its use  ... 
arXiv:2007.10824v5 fatcat:qwpxu7k53nac3blbprmabtgo7i

Near-optimal Approximate Discrete and Continuous Submodular Function Minimization [article]

Brian Axelrod, Yang P. Liu, Aaron Sidford
2019 arXiv   pre-print
Further, we leverage a generalization of this result to obtain efficient algorithms for minimizing a broad class of nonconvex functions.  ...  In this paper we provide improved running times and oracle complexities for approximately minimizing a submodular function.  ...  Acknowledgements We thank Deeparnab Chakrabarty, Yin Tat Lee, Sahil Singla, Kevin Tian, and Sam Chiu-wai Wong for helpful conversations.  ... 
arXiv:1909.00171v1 fatcat:npzhhspxsvg6jkye67zt5yiytm

Defending Elections Against Malicious Spread of Misinformation [article]

Bryan Wilder, Yevgeniy Vorobeychik
2018 arXiv   pre-print
Nevertheless, we give provable polynomial-time approximation algorithms for computing the defender's minimax optimal strategy across a range of settings, encompassing different population structures as  ...  We study a zero-sum game between an attacker, who attempts to subvert an election by propagating a fake new story or other misinformation over a set of advertising channels, and a defender who attempts  ...  Acknowledgments: This work was partially supported by the National Science Foundation (CNS-1640624, IIS-1649972, and IIS-1526860), Office of Naval Research (N00014-15-1-2621), and Army Research Office  ... 
arXiv:1809.05521v2 fatcat:2cf4a26p6bdn3hbnxmkdmhnahe

Graphical Models, Exponential Families, and Variational Inference

Martin J. Wainwright, Michael I. Jordan
2007 Foundations and Trends® in Machine Learning  
The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.  ...  In various applied fields including bioinformatics, speech processing, image processing and control theory, statistical models have long been formulated in terms of graphs, and algorithms for computing  ...  Acknowledgments A large number of people contributed to the gestation of this survey, and it is a pleasure to acknowledge them here.  ... 
doi:10.1561/2200000001 fatcat:3f33bwasgvg5ndjfqezocaxxfa
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