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Hypergraph Cuts with General Splitting Functions [article]

Nate Veldt and Austin R. Benson and Jon Kleinberg
2020 arXiv   pre-print
to the node-weighted multiway cut problem in graphs, again subject to a submodularity property.  ...  In this case, we show that the general hypergraph s-t cut problem can be reduced to a tractable graph s-t cut problem if and only if the splitting functions are submodular.  ...  (b) For cardinality-based splitting functions, separating one node by itself costs w 1 = 1 and splits with two nodes together have penalty w 2 .  ... 
arXiv:2001.02817v1 fatcat:tfjnelvk5ngrxalvxdbenhw2by

Experimental Design for Cost-Aware Learning of Causal Graphs [article]

Erik M. Lindgren, Murat Kocaoglu, Alexandros G. Dimakis, Sriram Vishwanath
2018 arXiv   pre-print
We consider the minimum cost intervention design problem: Given the essential graph of a causal graph and a cost to intervene on a variable, identify the set of interventions with minimum total cost that  ...  can learn any causal graph with the given essential graph.  ...  We see that the ε-class intervals can be partitioned into t + 1 contiguous regions, and the δ-class can be partitioned into ( 2t 2 ) − t 2 + t contiguous regions.  ... 
arXiv:1810.11867v1 fatcat:rur6epvmxfccxk6gmpmakydcsm

An Integer Programming Formulation of the Parsimonious Loss of Heterozygosity Problem

Daniele Catanzaro, Martine Labbe, Bjarni V. Halldorsson
2013 IEEE/ACM Transactions on Computational Biology & Bioinformatics  
Index Terms-clique partitioning, submodular functions, polymatroid rank functions, undirected catch-point interval graph, combinatorial optimization, mixed integer programming, computational biology, loss  ...  Moreover, we show that the PLOHP can be formulated as a specific version of the clique partition problem in a particular class of graphs called undirected catch-point interval graphs and we prove its general  ...  Then, due to the nature of f α , it is possible to obtain a lower cost partition into cliques of G by just breaking C into |C| cliques of cardinality 1.  ... 
doi:10.1109/tcbb.2012.138 pmid:24407298 fatcat:cd2eirjc3vcahop4pendvyxgci

On MAP Inference by MWSS on Perfect Graphs [article]

Adrian Weller, Tony S. Jebara
2013 arXiv   pre-print
We derive new results for this approach, including a general decomposition theorem for MRFs of any order and number of labels, extensions of results for binary pairwise models with submodular cost functions  ...  A promising, recent technique is to reduce the problem to finding a maximum weight stable set (MWSS) on a derived weighted graph, which if perfect, allows inference in polynomial time.  ...  Also cost functions are the negative of our ψs, thus submodular cost functions are equivalent to supermodular ψs.  ... 
arXiv:1309.6872v1 fatcat:v4xp57f3mfgrfn55zootz6nlqi

SAS-Assisted Coexistence-Aware Dynamic Channel Assignment in CBRS Band [article]

Xuhang Ying, Milind Buddhikot, Sumit Roy
2018 arXiv   pre-print
The proposed conflict graph allows us to formulate PA CA and GAA CA with binary conflicts as max-cardinality and max-reward CA, respectively.  ...  We show that the utility function is submodular, and the problem is an instance of matroid-constrained submodular maximization.  ...  Then M = (V, I) is a matroid, called a partition matroid. 2) Matroid-Constrained Submodular Maximization: As we can see, the constraint in the max-utility CA formulation in Eq. (3) defines a partition  ... 
arXiv:1805.06053v2 fatcat:dgs3ptz3anfatcf7oxqxirzb44

An Improved Greedy Algorithm for Subset Selection in Linear Estimation [article]

Shamak Dutta, Nils Wilde, Stephen L. Smith
2022 arXiv   pre-print
The solution quality improves with a finer grid resolution but at the cost of increased computation.  ...  method to reduce the computational complexity, or conversely to increase solution quality, of the greedy algorithm by considering a search space consisting only of prediction locations and centroids of cliques  ...  First, we find the centroids of maximal cliques in a graph with nodes as prediction locations.  ... 
arXiv:2203.16070v1 fatcat:f42l3ugt6ndirct7ovqnk4jyd4

Minimizing Sparse High-Order Energies by Submodular Vertex-Cover

Andrew Delong, Olga Veksler, Anton Osokin, Yuri Boykov
2012 Neural Information Processing Systems  
Instead of adding variables, we transform the original problem into a comparatively small instance of submodular vertex-cover.  ...  Several approaches are based, for example, on generalized message-passing, or on transformation to a pairwise model with extra 'auxiliary' variables.  ...  In the bipartite submodular vertex-cover (SVC-B) problem, the graph nodes V can be partitioned into sets J , K so the binary variables are u ∈ {0, 1} J , v ∈ {0, 1} K and we solve (SVC-B) minimize f (u  ... 
dblp:conf/nips/DelongVOB12 fatcat:e3vdy2bzxrgs7fh7l7hkuozppi

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

Stefanie Jegelka
2016 arXiv   pre-print
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.  ...  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.  ...  Our instance is a graph with n = 10 modes, shown in Figure 6 . The graph edges are partitioned into n/2 sets, indicated by colors.  ... 
arXiv:1402.0240v4 fatcat:wznkbacpcnashntirqnv5z24um

Submodular Order Functions and Assortment Optimization [article]

Rajan Udwani
2021 arXiv   pre-print
While the objectives in assortment optimization are known to be non-submodular (and non-monotone) even for simple choice models, we show that they are compatible with the notion of submodular order.  ...  Consequently, we obtain new and in some cases the first constant factor guarantee for constrained assortment optimization in fundamental choice models.  ...  Consider an instance of the cardinality constrained problem for a π submodular function.  ... 
arXiv:2107.02743v3 fatcat:aku6tt3khrg37jyvbalpewul5u

Submodularity In Machine Learning and Artificial Intelligence [article]

Jeff Bilmes
2022 arXiv   pre-print
We discuss submodular combinatorial information functions, and how submodularity is useful for clustering, data partitioning, parallel machine learning, active and semi-supervised learning, probabilistic  ...  We offer a plethora of submodular definitions; a full description of a number of example submodular functions and their generalizations; example discrete constraints; a discussion of basic algorithms for  ...  The general approach is to partition the set into blocks.  ... 
arXiv:2202.00132v1 fatcat:sp4b3ww3ajdxvfgigp7xw4f4yq

Inhomogeneous Hypergraph Clustering with Applications [article]

Pan Li, Olgica Milenkovic
2017 arXiv   pre-print
We prove that inhomogeneous partitioning produces a quadratic approximation to the optimal solution if the inhomogeneous costs satisfy submodularity constraints.  ...  A widely used method for hypergraph partitioning relies on minimizing a normalized sum of the costs of partitioning hyperedges across clusters.  ...  The weight w e (S) indicates the cost of cutting/partitioning the hyperedge e into two subsets, S and e/S.  ... 
arXiv:1709.01249v4 fatcat:z2fprzxx7ved3ppy3fowgmxh3e

Max-Sum Diversification, Monotone Submodular Functions and Dynamic Updates [article]

Allan Borodin, Aadhar Jain, Hyun Chul Lee, Yuli Ye
2016 arXiv   pre-print
It is NP-hard even with the triangle inequality.  ...  We study a broad class of problems where the distances are a metric, where the constraint is given by independence in a matroid, where quality is determined by a monotone submodular function, and diversity  ...  In a partition matroid, the universe U is partitioned into sets S 1 , . . . , S m and the independent sets S satisfy S = ∪ 1≤i≤m S i with |S i | ≤ k i for some given bounds k i on each part of the partition  ... 
arXiv:1203.6397v3 fatcat:3phfoyvnazeqnbdyaprqbvhjui

Markov Random Field modeling, inference & learning in computer vision & image understanding: A survey

Chaohui Wang, Nikos Komodakis, Nikos Paragios
2013 Computer Vision and Image Understanding  
In this paper, we present a comprehensive survey of Markov Random Fields (MRFs) in computer vision and image understanding, with respect to the modeling, the inference and the learning.  ...  While MRFs were introduced into the computer vision eld about two decades ago, they started to become a ubiquitous tool for solving visual perception problems around the turn of the millennium following  ...  Wang was with the Vision Lab at  ... 
doi:10.1016/j.cviu.2013.07.004 fatcat:d4ruu3u4gvg3dmud7xhni5gpbq

Graphical Models: Queries, Complexity, Algorithms

Martin C. Cooper, Simon de Givry, Thomas Schiex, Markus Bläser, Christophe Paul
2020 Symposium on Theoretical Aspects of Computer Science  
We restrict ourselves to functions of discrete variables but try to cover a variety of models that are not always considered as "Graphical Models", ranging from functions with Boolean variables and Boolean  ...  We use a simple algebraic semi-ring based framework for generality, define associated queries, relationships between graphical models, complexity results, and families of algorithms, with their associated  ...  Restrictions which are not exclusively concerned with the graph, nor exclusively concerned with the language of cost functions define what are known as hybrid classes.  ... 
doi:10.4230/lipics.stacs.2020.4 dblp:conf/stacs/CooperGS20 fatcat:rqyjypnqozd47frmaobbcpo5ei

Sensor Planning for Large Numbers of Robots [article]

Micah Corah
2021 arXiv   pre-print
However, greedy algorithms typically force robots to make decisions sequentially so that planning time grows with the number of robots.  ...  Fortunately, common sensing objectives benefit from well-known monotonicity properties (e.g. submodularity), and greedy algorithms can exploit these monotonicity properties to solve the receding-horizon  ...  Problem 2 (Cardinality-constrained submodular maximization). Any instance of Prob. 1 where (U , I ) is a uniform matroid is an instance of cardinality-constrained submodular maximization.  ... 
arXiv:2102.04054v1 fatcat:zgg75lo6wfdwvekn5twqpmi7oe
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