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CLOSEDNESS OF INTEGER HULLS OF SIMPLE CONIC SETS

SIAM JOURNAL ON DISCRETE MATHEMATICS 2016年第1期30卷 70-99页

作者： Moran, Diego A. Dey, Santanu S. Virginia Polytech Inst & State Univ Dept Ind & Syst Engn Blacksburg VA 24061 USA Univ Adolfo Ibanez Sch Business Diagonal Las Torres 2640 Santiago Chile Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn Atlanta GA 30332 USA

Let C be a full-dimensional pointed closed convex cone in R-m obtained by taking the conic hull of a strictly convex set. Given A is an element of Q(mxn1), B is an element of Q(mxn2), and b is an element of Q(m), a simple conic mixed-integer set (SCMIS) is a set of the form {(x, y) is an element of Z(n1) x R-n2 vertical bar Ax + By - b is an element of C}. In this paper, we give a complete characterization of the closedness of convex hulls of SCMISs. Under certain technical conditions on the cone C, we show that the closedness characterization can be used to construct a polynomial-time algorithm to check the closedness of convex hulls of SCMISs. Moreover, we also show that the Lorentz cone satisfies these technical conditions. In the special case of pure integer problems, we present sufficient conditions, which can be checked in polynomial time, to verify the closedness of intersection of SCMISs.

关键词： closedness polynomial-time algorithm mixed-integer convex programming

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Identification of the anchor points in FDH models

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2015年第3期246卷 936-943页

作者： Soleimani-damaneh, Majid Mostafaee, Amin Univ Tehran Coll Sci Sch Math Stat & Comp Sci Tehran Iran Islamic Azad Univ Tehran North Branch Dept Math Coll Sci Tehran Iran

This paper investigates the anchor points in nonconvex Data Envelopment Analysis (DEA), called Free Disposal Hull (FDH), technologies. We develop the concept of anchor points under various returns to scale as-sumptions in FDH models. A necessary and sufficient condition for characterizing the anchor points is provided. Since the set of anchor points is a subset of the set of extreme units, a definition of extreme units in non-convex technologies as well as a new method for obtaining these units are given. Finally, a polynomial-time algorithm for identification of the anchor points in FDH models is provided. Obtaining both extreme units and anchor points is done via calculating only some ratios, without solving any mathematical programming problem. (C) 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.

关键词： Data Envelopment Analysis (DEA) FDH models Anchor point Returns to scale polynomial-time algorithm

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Some algorithmic results for finding compatible spanning circuits in edge-colored graphs

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2020年第4期40卷 1008-1019页

作者： Guo, Zhiwei Broersma, Hajo Li, Ruonan Zhang, Shenggui Northwestern Polytech Univ Sch Math & Stat Xian 710129 Shaanxi Peoples R China Univ Twente Fac EEMCS POB 217 NL-7500 AE Enschede Netherlands Northwestern Polytech Univ Xian Budapest Joint Res Ctr Combinator Xian 710129 Shaanxi Peoples R China

A compatible spanning circuit in a (not necessarily properly) edge-colored graphGis a closed trail containing all vertices ofGin which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobas and Erdos from the 1970s.

关键词： Edge-colored graph Compatible spanning circuit NP-complete problem polynomial-time algorithm

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Subset feedback vertex set on graphs of bounded independent set size

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THEORETICAL COMPUTER SCIENCE 2020年 814卷 177-188页

作者： Papadopoulos, Charis Tzimas, Spyridon Univ Ioannina Dept Math Ioannina Greece

The (WEIGHTED) SUBSET FEEDBACK VERTEX SET problem is a generalization of the classical FEEDBACK VERTEX SET problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although SUBSET FEEDBACK VERTEX SET and FEEDBACK VERTEX SET exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that WEIGHTED FEEDBACK VERTEX SET can be solved in polynomial time. We provide a dichotomy result with respect to the size alpha OF a maximum independent set. In particular we show that WEIGHTED SUBSET FEEDBACK VERTEX SET can be solved in polynomial time for graphs with alpha <= 3, whereas we prove that the problem remains NP-hard for graphs with alpha >= 4. Moreover, we show that the (unweighted) SUBSET FEEDBACK VERTEX SET problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time n(O)(alpha). To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size. NODE MULTIWAY CUT is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for SUBSET FEEDBACK VERTEX SET, we settle the complexity of NODE MULTIWAY CUT as well as its variants where nodes are weighted and/or the terminals are deletable, for every value of the given independent set number. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Subset feedback vertex set Node multiway cut polynomial-time algorithm NPcompleteness W[1]-hardness Graphs of bounded independent set size

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Convex optimisation-based joint channel and power allocation scheme for orthogonal frequency division multiple access networks

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IET COMMUNICATIONS 2015年第1期9卷 28-32页

作者： Liu, Peng Li, Jiandong Li, Hongyan Meng, Yun Xidian Univ State Key Lab ISN Xian 710071 Shaanxi Peoples R China Xidian Univ Inst Informat Sci Xian 710071 Shaanxi Peoples R China

This study concerns joint channel and power allocation scheme for multi-user orthogonal frequency division multiple access system. The author's highlight is margin adaptive (MA) resource allocation problem namely minimising the total transmit power of users with rate requirement constraints. MA is generally provable NP-hard;the typical methods are either to relax and round, or to fix the transmission mode of users (e.g. modulation and coding). Differently, they reorganise MA problem with only power variables left and design a novel relaxation scheme to enable the convexity. The polynomial-time algorithm-interior-point method-is employed to solve the relaxation problem and the theoretical complexity is further presented. Simulation results demonstrate that the author's scheme can provide high energy efficiency compared with the existing methods, 100% relative error bounds with respect to the optimum in most cases, and low computational complexity.

关键词： OFDM modulation frequency division multiple access channel allocation convex programming constraint theory radio access networks communication complexity joint channel and power allocation scheme multiuser orthogonal frequency division multiple access system margin adaptive resource allocation problem rate requirement constraints NP-hard problem MA problem power variables relaxation scheme polynomial-time algorithm relative error bounds computational complexity convex optimisation multiple access networks interior point method

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Two disjoint shortest paths problem with non-negative edge length

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OPERATIONS RESEARCH LETTERS 2019年第1期47卷 66-69页

作者： Kobayashi, Yusuke Sako, Ryo Kyoto Univ Kyoto Japan Univ Tsukuba Tsukuba Ibaraki Japan

In the two disjoint shortest paths problem (2-DSPP), the input is a graph (or a digraph) and its vertex pairs (s(1), t(1)) and (S-2, t(2)), and the objective is to find two vertex-disjoint paths P-1 and P-2 such that P-i is a shortest path from s(i) to t(i) for i = 1, 2, if they exist. In this paper, we give a first polynomial-time algorithm for the undirected version of the 2-DSPP with an arbitrary non-negative edge length function. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Disjoint paths Shortest path polynomial-time algorithm

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COUNTING THE NUMBER OF MINIMUM CUTS IN UNDIRECTED MULTIGRAPHS

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IEEE TRANSACTIONS ON RELIABILITY 1991年第5期40卷 610-614页

作者： NAGAMOCHI, H SUN, Z IBARAKI, T TOYOHASHI UNIV TECHNOL DEPT INFORMAT & COMP SCITOYOHASHI 441JAPAN

The problem of counting the number of cuts with the minimum cardinality in an undirected multigraph arises in various applications such as testing the super-lambda-ness of a graph and calculating upper and lower bounds on the probabilistic connectedness of a stochastic graph G in which edges are subject to failure. This paper shows that the number \C(G)\ of cuts with the minimum cardinality lambda(G) in a multiple graph G = (V, E) can be computed in O(\E\ + lambda(G)\V\2 + lambda(G)\C(G) parallel-to V\) time.

关键词： UNDIRECTED GRAPH MULTIGRAPH EDGE-CONNECTIVITY MINIMUM CUT polynomial-time algorithm SPANNING FOREST SPANNING SUBGRAPH EDGE CONTRACTION SUPER-LAMBDA

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Graph Bipartization Problem with Applications to Via Minimization in VLSI Design

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INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE 2023年第4期34卷 347-361页

作者： Lin, Lan Lin, Yixun Tongji Univ Sch Elect & Informat Engn Shanghai 200092 Peoples R China Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China

The bipartization problem for a graph G asks for finding a subset S of V(G) such that the induced subgraph G[S] is bipartite and vertical bar S vertical bar is maximized. This problem has significant applications in the via minimization of VLSI design. The problem has been proved NP-complete and the fixed parameter solvability has been known in the literature. This paper presents several polynomial-time algorithms for special graph families, such as split graphs, co-bipartite graphs, chordal graphs, and permutation graphs.

关键词： Bipartite induced subgraph odd-cycle transversal via minimization VLSI design polynomial-time algorithm

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Vertex and edge covers with clustering properties: Complexity and algorithms

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JOURNAL OF DISCRETE algorithmS 2009年第2期7卷 149-167页

作者： Fernau, Henning Manlove, David F. Univ Trier FB Abt Informat 4 D-54286 Trier Germany Univ Glasgow Dept Comp Sci Glasgow G12 8QQ Lanark Scotland

We consider the concepts of a t-total vertex cover and a t-total edge cover (t >= 1), which generalise the notions of a vertex cover and an edge cover, respectively. A t-total vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has at least t vertices (edges). These definitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NP-completeness and approximability results (both upper and lower bounds) and FPT algorithms for problems concerned with finding the minimum size of a t-total vertex cover, t-total edge cover and connected vertex cover, in particular improving on a previous FPT algorithm for the latter problem. (C) 2008 Elsevier B. V. All rights reserved.

关键词： t-total vertex cover Connected vertex cover t-total edge cover polynomial-time algorithm NP-completeness Approximation algorithm FPT algorithm

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HEURISTIC algorithm FOR FINDING THE MAXIMUM INDEPENDENT SET

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CYBERNETICS AND SYSTEMS ANALYSIS 2012年第5期48卷 673-680页

作者： Plotnikov, A. D. Volodymyr Dahl East Ukraine Natl Univ Minist Educ & Sci Youth & Sport Ukraine Lugansk Ukraine

A heuristic algorithm is developed for finding the maximum independent set of vertices in an undirected graph. To this end, the technique of finite partially ordered sets is used, in particular, the technique of partitioning such a set into a minimum number of chains. A special digraph is constructed and a solution algorithm is proposed on the basis of a hypothesis about its properties. Some experimental data are presented for well-known examples.

关键词： heuristic algorithm maximum independent set clique NP-hard NP-complete class NP polynomial-time algorithm finite partially ordered set

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