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A polynomial arc-search interior-point algorithm for convex quadratic programming

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2011年第1期215卷 25-38页

作者： Yang, Yaguang NRC Res Off Rockville MD 20850 USA

Arc-search is developed for linear programming in [24,25]. The algorithms search for optimizers along an ellipse that is an approximation of the central path. In this paper, the arc-search method is applied to primal-dual path-following interior-point method for convex quadratic programming. A simple algorithm with iteration complexity O(root nlog(1/epsilon)) is devised. Several improvements on the simple algorithm, which improve computational efficiency, increase step length, and further reduce duality gap in every iteration, are then proposed and implemented. It is intuitively clear that the iteration with these improvements will reduce the duality gap more than the iteration of the simple algorithm without the improvements, though it is hard to show how much these improvements reduce the complexity bound. The proposed algorithm is implemented in MATLAB and tested on quadratic programming problems originating from [13]. The result is compared to the one obtained by LOQO in [22] The proposed algorithm uses fewer iterations in all these problems and the number of total iterations is 27% fewer than the one obtained by LOQO. This preliminary result shows that the proposed algorithm is promising. Published by Elsevier B.V.

关键词： Arc-search Convex quadratic programming Interior-point method polynomial algorithm

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Better polynomial algorithms for scheduling unit-length jobs with bipartite incompatibility graphs on uniform machines

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BULLETIN OF THE POLISH ACADEMY OF SCIENCES-TECHNICAL SCIENCES 2019年第1期67卷 31-36页

作者： Pikies, T. Kubale, M. Gdansk Univ Technol ETI Fac Dept Algorithms & Syst Modelling Gabriela Narutowicza 11-12 PL-80233 Gdansk Poland

The goal of this paper is to explore and to provide tools for the investigation of the problems of unit-length scheduling of incompatible jobs on uniform machines. We present two new algorithms that are a significant improvement over the known algorithms. The first one is algorithm 2 which is 2-approximate for the problem Qm vertical bar p(j) =1,G = bisubquartic vertical bar C-max. The second one is algorithm 3 which is 4-approximate for the problem Qm vertical bar p(j) =1,G = bisubquarticl vertical bar Sigma C-j, where m is an element of {2, 3, 4). The theory behind the proposed algorithms is based on the properties of 2-coloring with maximal coloring width, and on the properties of ideal machine, an abstract machine that we introduce in this paper.

关键词： approximation algorithm graph coloring incompatible job polynomial algorithm scheduling uniform machine unit-time job

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A polynomial Arc-Search Interior-Point algorithm for Linear Programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2013年第3期158卷 859-873页

作者： Yang, Yaguang NRC Res Off Rockville MD 20850 USA

In this paper, ellipsoidal estimations are used to track the central path of linear programming. A higher-order interior-point algorithm is devised to search the optimizers along the ellipse. The algorithm is proved to be polynomial with the best complexity bound for all polynomial algorithms and better than the best known bound for higher-order algorithms.

关键词： Arc-search Interior-point method polynomial algorithm Linear programming

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A polynomial recognition of unit forms using graph-based strategies

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DISCRETE APPLIED MATHEMATICS 2019年 253卷 61-72页

作者： Alves, Jesmmer Castongay, Diane Brustle, Thomas Inst Fed Goiano Campus Morrinhos Morrinhos Morrinhos Go Brazil Univ Fed Goias inst Informat Goiania Go Brazil Univ Sherbrooke Fac Sci Sherbrooke PQ Canada

The units forms are algebraic expressions that have important role in representation theory of algebras. We identified that existing algorithms have exponential time complexity for weakly nonnegative and weakly positive types. In this paper we introduce a polynomial algorithm for the recognition of weakly nonnegative unit forms. The related algorithm identifies hypercritical restrictions in a given unit form, testing every subgraph of 9 vertices of the unit form associated graph. By adding Depth First Search approach, a similar strategy could be used in the recognition of weakly positive unit forms. We also present the most popular methods to decide whether or not a unit form is weakly nonnegative or weakly positive, we analyze their time complexity and we compare the results with our algorithms. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Integral quadratic form Unit form Critical unit form Hypercritical unit form polynomial algorithm Graph

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polynomial-time Classification of Skew-symmetrizable Matrices with a Positive Definite Quasi-Cartan Companion

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FUNDAMENTA INFORMATICAE 2021年第4期181卷 313-337页

作者： Perez, Claudia Rivera, Daniel Univ Autonoma Estado Morelos Ctr Invest Ciencias Av Univ 1001 Cuernavaca Morelos Mexico

Skew-symmetrizable matrices play an essential role in the classification of cluster algebras. We prove that the problem of assigning a positive definite quasi-Cartan companion to a skew-symmetrizable matrix is in polynomial class P. We also present an algorithm to determine the finite type Delta is an element of {A(n), D-n, B-n, C-n, E-6, E-7, E-8, F-4, G(2)} of a cluster algebra associated to the mutation-equivalence class of a connected skew-symmetrizable matrix B, if it has one.

关键词： Cluster algebra positive quasi-Cartan companion polynomial algorithm biconnected component triconnected component

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Connected max cut is polynomial for graphs without the excluded minor K5\e

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2020年第4期40卷 869-875页

作者： Chaourar, Brahim Imam Mohammad Ibn Saud Islamic Univ IMSIU POB 90950 Riyadh 11623 Saudi Arabia

Given a graph G = (V, E), a connected cut delta(U) is the set of edges of E linking all vertices of U to all vertices of V\U such that the induced subgraphs G[U] and G[V\U] are connected. Given a positive weight function w defined on E, the connected maximum cut problem (CMAX CUT) is to find a connected cut Omega such that w(Omega) is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs, and thus for graph without the excluded minor K-5. In this paper, we prove that CMAX CUT is polynomial for the class of graphs without the excluded minor K-5\e, denoted by G(K-5\e). We deduce two quadratic time algorithms: one for the minimum cut problem in G( K-5\e) without computing the maximum flow, and another one for Hamilton cycle problem in the class of graphs without the two excluded minors the prism P-6 and K-3,K-3. This latter problem is NP-complete for maximal planar graphs.

关键词： Max cut Connected max cut polynomial algorithm Min cut Graphs without the excluded minor K-5\e Hamilton cycle problem

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THE RECOGNITION OF SIMPLE-TRIANGLE GRAPHS AND OF LINEAR-INTERVAL ORDERS IS polynomial

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2015年第3期29卷 1150-1185页

作者： Mertzios, George B. Univ Durham Sch Engn & Comp Sci Durham DH1 3LH England

Intersection graphs of geometric objects have been extensively studied, due to both their interesting structure and their numerous applications;prominent examples include interval graphs and permutation graphs. In this paper we study a natural graph class that generalizes both interval and permutation graphs, namely simple-triangle graphs. Simple-triangle graphs-also known as PI (point-interval) graphs-are the intersection graphs of triangles that are defined by a point on a line L-1 and an interval on a parallel line L-2. They lie naturally between permutation and trapezoid graphs, which are the intersection graphs of line segments between L-1 and L-2 and of trapezoids between L-1 and L-2, respectively. Although various efficient recognition algorithms for permutation and trapezoid graphs are well known to exist, the recognition of simple-triangle graphs has remained an open problem since their introduction by Corneil and Kamula three decades ago. In this paper we resolve this problem by proving that simple-triangle graphs can be recognized in polynomial time. Given a graph G with n vertices, such that its complement (G) over bar has m edges, our algorithm runs in O(n(2)m) time. As a consequence, our algorithm also solves a longstanding open problem in the area of partial orders, namely, the recognition of linear-interval orders, i.e., of partial orders P = P-1 boolean AND P-2, where P-1 is a linear order and P-2 is an interval order. This is one of the first results on recognizing partial orders P that are the intersection of orders from two different classes P-1 and P-2. In complete contrast to this, partial orders P which are the intersection of orders from the same class P have been extensively investigated, and in most cases the complexity status of these recognition problems has been already established.

关键词： intersection graph PI graph recognition problem partial order polynomial algorithm

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The Longest Path Problem Is polynomial on Cocomparability Graphs

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algorithmICA 2013年第1期65卷 177-205页

作者： Ioannidou, Kyriaki Nikolopoulos, Stavros D. Univ Ioannina Dept Comp Sci GR-45110 Ioannina Greece

The longest path problem is the problem of finding a path of maximum length in a graph. As a generalization of the Hamiltonian path problem, it is NP-complete on general graphs and, in fact, on every class of graphs that the Hamiltonian path problem is NP-complete. polynomial solutions for the longest path problem have recently been proposed for weighted trees, Ptolemaic graphs, bipartite permutation graphs, interval graphs, and some small classes of graphs. Although the Hamiltonian path problem on cocomparability graphs was proved to be polynomial almost two decades ago, the complexity status of the longest path problem on cocomparability graphs has remained open;actually, the complexity status of the problem has remained open even on the smaller class of permutation graphs. In this paper, we present a polynomial-time algorithm for solving the longest path problem on the class of cocomparability graphs. Our result resolves the open question for the complexity of the problem on such graphs, and since cocomparability graphs form a superclass of both interval and permutation graphs, extends the polynomial solution of the longest path problem on interval graphs and provides polynomial solution to the class of permutation graphs.

关键词： Longest path problem Cocomparability graphs Permutation graphs polynomial algorithm Complexity

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2-Stack Sorting is polynomial

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THEORY OF COMPUTING SYSTEMS 2017年第3期60卷 552-579页

作者： Pierrot, Adeline Rossin, Dominique Univ Paris Saclay CNRS Univ Paris Sud LRI F-91405 Orsay France Univ Paris Saclay CNRS Ecole Polytech LIX F-92128 Palaiseau France

In this article, we give a polynomial algorithm to decide whether a given permutation sigma is sortable with two stacks in series. This is indeed a longstanding open problem which was first introduced by Knuth (1973). He introduced the stack sorting problem as well as permutation patterns which arises naturally when characterizing permutations that can be sorted with one stack. When several stacks in series are considered, few results are known. There are two main different problems. The first one is the complexity of deciding if a permutation is sortable or not, the second one being the characterization and the enumeration of those sortable permutations. We hereby prove that the first problem lies in P by giving a polynomial algorithm to solve it. This article relies on Pierrot and Rossin (2013) in which 2-stack pushall sorting is defined and studied.

关键词： Stack Sort Permutation Pattern polynomial algorithm Combinatorics algorithms Permutation patterns Stack-sorting

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polynomial-TIME algorithmS FOR REGULAR SET-COVERING AND THRESHOLD SYNTHESIS

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DISCRETE APPLIED MATHEMATICS 1985年第1期12卷 57-69页

作者： PELED, UN SIMEONE, B UNIV ROME LA SAPIENZA DEPT STATROMEITALY

A set-covering problem is called regular if a cover always remains a cover when any column in it is replaced by an earlier column. From the input of the problem - the coefficient matrix of the set-covering inequalities - it is possible to check in polynomial time whether the problem is regular or can be made regular by permuting the columns. If it is, then all the minimal covers are generated in polynomial time, and one of them is an optimal solution. The algorithm also yields an explicit bound for the number of minimal covers. These results can be used to check in polynomial time whether a given set-covering problem is equivalent to some knapsack problem without additional variables, or equivalently to recognize positive threshold functions in polynomial time. However, the problem of recognizing when an arbitrary Boolean function is threshold is NP-complete. It is also shown that the list of maximal non-covers is essentially the most compact input possible, even if it is known in advance that the problem is regular.

关键词： Set-covering regular knapsack polynomial algorithm NP-completeness

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