Pub Date : 1900-01-01DOI: 10.1137/1.9780898718805.ch16
M. Laurent
We compare several semideenite relaxations for the cut polytope obtained by applying the lift and project methods of Lovv asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem. This relaxation Q t (G) can be deened as the projection on the edge subspace of the set F t (n), which consists of the matrices indexed by all subsets of 1; n] of cardinality t + 1 with the same parity as t + 1 and having the property that their (I; J)-th entry depends only on the symmetric diierence of the sets I and J. The set F 0 (n) is the basic semideenite relaxation of max-cut consisting of the semideenite matrices of order n with an all ones diagonal, while F n?2 (n) is the 2 n?1-dimensional simplex with the cut matrices as vertices. We show the following geometric properties: If Y 2 F t (n) has rank t + 1, then Y can be written as a convex combination of at most 2 t cut matrices, extending a result of Anjos and Wolkowicz for the case t = 1; any 2 t+1 cut matrices form a face of F t (n) for t = 0; 1; n ? 2. The class L t of the graphs G for which Q t (G) is the cut polytope of G is shown to be closed under taking minors. The graph K 7 is a forbidden minor for membership in L 2 , while K 3 and K 5 are the only minimal forbidden minors for the classes L 0 and L 1 , respectively.
我们比较了用Lovv asz和Schrijver以及Lasserre的提升和投影方法得到的切割多晶体的几种半长晶石弛豫。我们证明了将Lasserre构造应用于最大切割问题的节点公式时获得了最紧的松弛。这个松弛Q t (G)可以理解为集合F t (n)在边缘子空间上的投影，该集合由所有1的子集所索引的矩阵组成;n]与t + 1的奇偶性相同，并且具有它们的(I;集合f0 (n)是最大切的基本半恒量松弛，由n阶半恒量矩阵组成，对角线为全一，而F n?2 (n)等于2n ?以切割矩阵为顶点的一维单纯形。我们证明了以下几何性质:如果Y 2 F t (n)的秩为t + 1，则Y可以写成最多2个t切矩阵的凸组合，推广了Anjos和Wolkowicz在t = 1情况下的结果;当t = 0时，任意2个t+1切割矩阵形成F t (n)的面;1;n ?2. 图G的一类L t，其中Q t (G)是G的切多面体，证明了该类L t在取余子下是闭的。图k7是l2中成员的禁止子图，而k3和k5分别是l0和l1类的唯一最小禁止子图。
{"title":"Semidefinite Relaxations for Max-Cut","authors":"M. Laurent","doi":"10.1137/1.9780898718805.ch16","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch16","url":null,"abstract":"We compare several semideenite relaxations for the cut polytope obtained by applying the lift and project methods of Lovv asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem. This relaxation Q t (G) can be deened as the projection on the edge subspace of the set F t (n), which consists of the matrices indexed by all subsets of 1; n] of cardinality t + 1 with the same parity as t + 1 and having the property that their (I; J)-th entry depends only on the symmetric diierence of the sets I and J. The set F 0 (n) is the basic semideenite relaxation of max-cut consisting of the semideenite matrices of order n with an all ones diagonal, while F n?2 (n) is the 2 n?1-dimensional simplex with the cut matrices as vertices. We show the following geometric properties: If Y 2 F t (n) has rank t + 1, then Y can be written as a convex combination of at most 2 t cut matrices, extending a result of Anjos and Wolkowicz for the case t = 1; any 2 t+1 cut matrices form a face of F t (n) for t = 0; 1; n ? 2. The class L t of the graphs G for which Q t (G) is the cut polytope of G is shown to be closed under taking minors. The graph K 7 is a forbidden minor for membership in L 2 , while K 3 and K 5 are the only minimal forbidden minors for the classes L 0 and L 1 , respectively.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124064375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1137/1.9780898718805.ch10
D. Naddef
An optical viewing device having a stabilized mirror in the objective lens system, the arrangement being such that the conditions for stabilization of the image against vibration about an axis perpendicular to the line of sight of the device is 2 2s 1 1 - + = r fr m WHERE R IS A COUPLING RATIO; S IS THE DISTANCE ALONG THE OPTICAL AXIS BETWEEN THE MIRROR AND A PART OF THE OBJECTIVE LENS SYSTEM THAT LIES IN FRONT OF THE REFLECTING SURFACE; F IS THE FOCAL LENGTH OF THAT PART OF THE OBJECTIVE LENS SYSTEM THAT LIES IN FRONT OF THE REPLECTING SURFACE; AND M IS THE OVERALL MAGNIFICATION OF THE COMPLETE OPTICAL SYSTEM OF THE DEVICE.
一种在物镜系统中具有稳定反射镜的光学观察装置，其排列使得图像在垂直于该装置视线的轴上不受振动的稳定条件为2 2s 1 1 - + = r fr m，其中r为耦合比;S为反射镜与位于反射面前的部分物镜系统沿光轴的距离;F为物镜系统中位于反射面前面的部分的焦距;m是设备完整光学系统的总放大倍率。
{"title":"The Domino Inequalities for the Symmetric Traveling Salesman Problem","authors":"D. Naddef","doi":"10.1137/1.9780898718805.ch10","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch10","url":null,"abstract":"An optical viewing device having a stabilized mirror in the objective lens system, the arrangement being such that the conditions for stabilization of the image against vibration about an axis perpendicular to the line of sight of the device is 2 2s 1 1 - + = r fr m WHERE R IS A COUPLING RATIO; S IS THE DISTANCE ALONG THE OPTICAL AXIS BETWEEN THE MIRROR AND A PART OF THE OBJECTIVE LENS SYSTEM THAT LIES IN FRONT OF THE REFLECTING SURFACE; F IS THE FOCAL LENGTH OF THAT PART OF THE OBJECTIVE LENS SYSTEM THAT LIES IN FRONT OF THE REPLECTING SURFACE; AND M IS THE OVERALL MAGNIFICATION OF THE COMPLETE OPTICAL SYSTEM OF THE DEVICE.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122073395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1137/1.9780898718805.ch2
L. Wolsey
{"title":"Time for Old and New Faces","authors":"L. Wolsey","doi":"10.1137/1.9780898718805.ch2","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch2","url":null,"abstract":"","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131171925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1137/1.9780898718805.ch18
R. Bixby, M. Fenelon, Zonghao Gu, E. Rothberg, Roland Wunderling
{"title":"Mixed-Integer Programming: A Progress Report","authors":"R. Bixby, M. Fenelon, Zonghao Gu, E. Rothberg, Roland Wunderling","doi":"10.1137/1.9780898718805.ch18","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch18","url":null,"abstract":"","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"250 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131947999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1137/1.9780898718805.ch6
C. Gentile, U. Haus, M. Köppe, G. Rinaldi, R. Weismantel
In this paper some operations are described that transform every graph into a perfect graph by replacing nodes with sets of new nodes. The transformation is done in such a way that every stable set in the perfect graph corresponds to a stable set in the original graph. These operations yield a purely combinatorial augmentation procedure for finding a maximum weighted stable set in a graph. Starting with a stable set in a given graph one defines a simplex type tableau whose associated basic feasible solution is the incidence vector of the stable set. In an iterative fashion, non-basic columns that would lead to pivoting into non-integral basic feasible solutions, are replaced by new columns that one can read off from special graph structures such as odd holes, odd antiholes, and various generalizations. Eventually, either a pivot leading to an integral basic feasible solution is performed, or the optimality of the current solution is proved.
{"title":"On the Way to Perfection: Primal Operations for Stable Sets in Graphs","authors":"C. Gentile, U. Haus, M. Köppe, G. Rinaldi, R. Weismantel","doi":"10.1137/1.9780898718805.ch6","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch6","url":null,"abstract":"In this paper some operations are described that transform every graph into a perfect graph by replacing nodes with sets of new nodes. The transformation is done in such a way that every stable set in the perfect graph corresponds to a stable set in the original graph. These operations yield a purely combinatorial augmentation procedure for finding a maximum weighted stable set in a graph. Starting with a stable set in a given graph one defines a simplex type tableau whose associated basic feasible solution is the incidence vector of the stable set. In an iterative fashion, non-basic columns that would lead to pivoting into non-integral basic feasible solutions, are replaced by new columns that one can read off from special graph structures such as odd holes, odd antiholes, and various generalizations. Eventually, either a pivot leading to an integral basic feasible solution is performed, or the optimality of the current solution is proved.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132027741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1137/1.9780898718805.ch5
J. Fonlupt
The clique-rank of a perfect graph G introduced by Fonlupt and Sebö is the linear rank of the incidence matrix of the maximum cliques of G. We study this rank for 3-chromatic perfect graphs. We prove that if, in addition, G is diamond-free, G has two distinct colorations. An immediate consequence is that the Strong Perfect Graph Conjecture holds for diamond-free graphs and for graphs with clique number equal to three. The proofs use both linear algebra and combinatorial arguments.
{"title":"The Clique-Rank of 3-Chromatic Perfect Graphs","authors":"J. Fonlupt","doi":"10.1137/1.9780898718805.ch5","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch5","url":null,"abstract":"The clique-rank of a perfect graph G introduced by Fonlupt and Sebö is the linear rank of the incidence matrix of the maximum cliques of G. We study this rank for 3-chromatic perfect graphs. We prove that if, in addition, G is diamond-free, G has two distinct colorations. An immediate consequence is that the Strong Perfect Graph Conjecture holds for diamond-free graphs and for graphs with clique number equal to three. The proofs use both linear algebra and combinatorial arguments.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116872107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：