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We display an application of the notions of kernelization and data reduction from parameterized complexity to proof complexity: Specifically, we show that the existence of data reduction rules for a parameterized problem having (a). a…

Computational Complexity · Computer Science 2021-04-29 Gabriel Istrate , Cosmin Bonchis , Adrian Craciun

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive…

Computational Complexity · Computer Science 2019-05-10 Eva-Maria C. Hols , Stefan Kratsch , Astrid Pieterse

We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to…

Optimization and Control · Mathematics 2016-05-24 Steffen Borgwardt , Shmuel Onn

Nowhere dense classes of graphs are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness,…

Discrete Mathematics · Computer Science 2018-09-06 Stephan Kreutzer , Roman Rabinovich , Sebastian Siebertz

We introduce a new technique for proving kernelization lower bounds, called cross-composition. A classical problem L cross-composes into a parameterized problem Q if an instance of Q with polynomially bounded parameter value can express the…

Computational Complexity · Computer Science 2015-03-17 Hans L. Bodlaender , Bart M. P. Jansen , Stefan Kratsch

We study the algorithmic properties of the graph class Chordal-ke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. We discover that a number of…

Data Structures and Algorithms · Computer Science 2020-06-23 Fedor V. Fomin , Petr A. Golovach

As a general type of machine learning approach, artificial neural networks have established state-of-art benchmarks in many pattern recognition and data analysis tasks. Among various kinds of neural networks architectures, polynomial neural…

Machine Learning · Computer Science 2022-09-07 Chao Pan , Chuanyi Zhang

In binary polynomial optimization, the goal is to find a binary point maximizing a given polynomial function. In this paper, we propose a novel way of formulating this general optimization problem, which we call factorized binary polynomial…

Optimization and Control · Mathematics 2024-07-08 Alberto Del Pia

Kernel-based learning algorithms are widely used in machine learning for problems that make use of the similarity between object pairs. Such algorithms first embed all data points into an alternative space, where the inner product between…

Machine Learning · Statistics 2017-09-21 Amir-Hossein Karimi

We present a geometric formulation of the Multiple Kernel Learning (MKL) problem. To do so, we reinterpret the problem of learning kernel weights as searching for a kernel that maximizes the minimum (kernel) distance between two convex…

Machine Learning · Computer Science 2014-03-18 John Moeller , Parasaran Raman , Avishek Saha , Suresh Venkatasubramanian

Kernel methods are considered an effective technique for on-line learning. Many approaches have been developed for compactly representing the dual solution of a kernel method when the problem imposes memory constraints. However, in…

Machine Learning · Computer Science 2016-07-21 Giovanni Da San Martino , Nicolò Navarin , Alessandro Sperduti

The study of networks has witnessed an explosive growth over the past decades with several ground-breaking methods introduced. A particularly interesting -- and prevalent in several fields of study -- problem is that of inferring a function…

Kernel-based methods enjoy powerful generalization capabilities in handling a variety of learning tasks. When such methods are provided with sufficient training data, broadly-applicable classes of nonlinear functions can be approximated…

Machine Learning · Statistics 2017-12-29 Fatemeh Sheikholeslami , Dimitris Berberidis , Georgios B. Giannakis

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial…

Data Structures and Algorithms · Computer Science 2015-03-20 Hans L. Bodlaender , Bart M. P. Jansen , Stefan Kratsch

This paper discusses an approach for incorporating prior physical knowledge into the neural network to improve data efficiency and the generalization of predictive models. If the dynamics of a system approximately follows a given…

Neural and Evolutionary Computing · Computer Science 2020-05-29 Andrei Ivanov , Uwe Iben , Anna Golovkina

Kernel methods are among the most popular techniques in machine learning. From a frequentist/discriminative perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the…

Machine Learning · Statistics 2012-04-17 Mauricio A. Alvarez , Lorenzo Rosasco , Neil D. Lawrence

Krentel [J. Comput. System. Sci., 36, pp.490--509] presented a framework for an NP optimization problem that searches an optimal value among exponentially-many outcomes of polynomial-time computations. This paper expands his framework to a…

Quantum Physics · Physics 2007-05-23 Tomoyuki Yamakami

In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G-S is a tree. The problem is NP-complete and even NP-hard to…

Data Structures and Algorithms · Computer Science 2013-10-01 Archontia C. Giannopoulou , Daniel Lokshtanov , Saket Saurabh , Ondrej Suchy

Kernel density estimation is a well known method involving a smoothing parameter (the bandwidth) that needs to be tuned by the user. Although this method has been widely used the bandwidth selection remains a challenging issue in terms of…

Statistics Theory · Mathematics 2019-02-05 Suzanne Varet , Claire Lacour , Pascal Massart , Vincent Rivoirard

General factors are a generalization of matchings. Given a graph $G$ with a set $\pi(v)$ of feasible degrees, called a degree constraint, for each vertex $v$ of $G$, the general factor problem is to find a (spanning) subgraph $F$ of $G$…

Discrete Mathematics · Computer Science 2024-05-24 Shuai Shao , Stanislav Živný