The cutting plane method is polynomial for perfect matchings

The cutting plane approach to finding minimum-cost perfect matchings has been discussed by several authors over past decades. Its convergence has been an open question. We develop a cutting plane algorithm that converges in polynomial-time using only Edmonds’ blossom inequalities, and which maintains half-integral intermediate LP solutions supported by a disjoint union of odd cycles and edges. Our main insight is a method to retain only a subset of the previously added cutting planes based on their dual values. This allows us to quickly find violated blossom inequalities and argue convergence by tracking the number of odd cycles in the support of intermediate solution

The Cutting Plane Method is Polynomial for Perfect Matchings

The cutting plane approach to optimal matchings has been discussed by several authors over the past decades (e.g., Padberg and Rao '82, Grotschel and Holland '85, Lovasz and Plummer '86, Trick '87, Fischetti and Lodi '07) and its convergence has been an open question. We give a cutting plane algorithm that converges in polynomial-time using only Edmonds' blossom inequalities; it maintains half-integral intermediate LP solutions supported by a disjoint union of odd cycles and edges. Our main insight is a method to retain only a subset of the previously added cutting planes based on their dual values. This allows us to quickly find violated blossom inequalities and argue convergence by tracking the number of odd cycles in the support of intermediate solutions

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

On the Design, Analysis, and Implementation of Algorithms for Selected Problems in Graphs and Networks

This thesis studies three problems in network optimization, viz., the minimum spanning tree verification (MSTV) problem, the undirected negative cost cycle detection (UNCCD) problem, and the negative cost girth (NCG) problem. These problems find applications in several domains including program verification, proof theory, real-time scheduling, social networking, and operations research.;The MSTV problem is defined as follows: Given an undirected graph G = (V,E) and a spanning tree T, is T a minimum spanning tree of G? We focus on the case where the number of distinct edge weights is bounded. Using a bucketed data structure to organize the edge weights, we present an efficient algorithm for the MSTV problem, which runs in O (| E| + |V| · K) time, where K is the number of distinct edge weights. When K is a fixed constant, this algorithm runs in linear time. We also profile our MSTV algorithm with the current fastest known MSTV implementation. Our results demonstrate the superiority of our algorithm when K ≤ 24.;The UNCCD problem is defined as follows: Given an undirected graph G = (V,E) with arbitrarily weighted edges, does G contain a negative cost cycle? We discuss two polynomial time algorithms for solving the UNCCD problem: the b-matching approach and the T-join approach. We obtain new results for the case where the edge costs are integers in the range {lcub}--K ·· K{rcub}, where K is a positive constant. We also provide the first extensive empirical study that profiles the discussed UNCCD algorithms for various graph types, sizes, and experiments.;The NCG problem is defined as follows: Given a directed graph G = (V,E) with arbitrarily weighted edges, find the length, or number of edges, of the negative cost cycle having the least number of edges. We discuss three strongly polynomial NCG algorithms. The first NCG algorithm is known as the matrix multiplication approach in the literature. We present two new NCG algorithms that are asymptotically and empirically superior to the matrix multiplication approach for sparse graphs. We also provide a parallel implementation of the matrix multiplication approach that runs in polylogarithmic parallel time using a polynomial number of processors. We include an implementation profile to demonstrate the efficiency of the parallel implementation as we increase the graph size and number of processors. We also present an NCG algorithm for planar graphs that is asymptotically faster than the fastest topology-oblivious algorithm when restricted to planar graphs

System design for express airlines

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1988.Includes bibliographical references.by Michael R. Fisher, Jr.Ph.D

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Approximation contexts in addressing graph data structures

While the application of machine learning algorithms to practical problems has been expanded from fixed sized input data to sequences, trees or graphs input data, the composition of learning system has developed from a single model to integrated ones. Recent advances in graph based learning algorithms include: the SOMSD (Self Organizing Map for Structured Data), PMGraphSOM (Probability Measure Graph Self Organizing Map,GNN (Graph Neural Network) and GLSVM (Graph Laplacian Support Vector Machine). A main motivation of this thesis is to investigate if such algorithms, whether by themselves individually or modified, or in various combinations, would provide better performance over the more traditional artificial neural networks or kernel machine methods on some practical challenging problems. More succinctly, this thesis seeks to answer the main research question: when or under what conditions/contexts could graph based models be adjusted and tailored to be most efficacious in terms of predictive or classification performance on some challenging practical problems? There emerges a range of sub-questions including: how do we craft an effective neural learning system which can be an integration of several graph and non-graph based models? Integration of various graph based and non graph based kernel machine algorithms; enhancing the capability of the integrated model in working with challenging problems; tackling the problem of long term dependency issues which aggravate the performance of layer-wise graph based neural systems. This thesis will answer these questions. Recent research on multiple staged learning models has demonstrated the efficacy of multiple layers of alternating unsupervised and supervised learning approaches. This underlies the very successful front-end feature extraction techniques in deep neural networks. However much exploration is still possible with the investigation of the number of layers required, and the types of unsupervised or supervised learning models which should be used. Such issues have not been considered so far, when the underlying input data structure is in the form of a graph. We will explore empirically the capabilities of models of increasing complexities, the combination of the unsupervised learning algorithms, SOM, or PMGraphSOM, with or without a cascade connection with a multilayer perceptron, and with or without being followed by multiple layers of GNN. Such studies explore the effects of including or ignoring context. A parallel study involving kernel machines with or without graph inputs has also been conducted empirically