FROM INDEPENDENT SETS AND VERTEX COLORINGS TO ISOTROPIC SPACES AND ISOTROPIC DECOMPOSITIONS: ANOTHER BRIDGE BETWEEN GRAPHS AND ALTERNATING MATRIX SPACES

doi:10.1137/19M1299128

	FROM INDEPENDENT SETS AND VERTEX COLORINGS TO ISOTROPIC SPACES AND ISOTROPIC DECOMPOSITIONS: ANOTHER BRIDGE BETWEEN GRAPHS AND ALTERNATING MATRIX SPACES
	Bei, Xiaohui 1; Chen, Shiteng 2; Guan, Ji 2; Qiao, Youming 3; Sun, Xiaoming 4,5
刊名	SIAM JOURNAL ON COMPUTING
	2021
卷号	50 期号:3 页码:924-971
关键词	independent set vertex coloring alternating matrix spaces isotropic space isotropic decomposition exact exponential algorithms
ISSN号	0097-5397
DOI	10.1137/19M1299128
英文摘要	In the 1970s, Lovasz built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings [Proceedings of FCT, 1979, pp. 565-574]. A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the noncommutative rank problem [A. Garg et al., Proceedings of FOCS, 2016, pp. 109-117; G. Ivanyos, Y. Qiao, and K. V. Subrahmanyam, Comput. Complexity, 26 (2017), pp. 717-763]. In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex c-coloring problem reduce to the maximum isotropic space problem and the isotropic c-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.
资助项目	National Key R\&D Program of China[2018YFA0306701] ; National Natural Science Foundation of China[61832015] ; National Natural Science Foundation of China[61702489] ; National Natural Science Foundation of China[61832003] ; Key Research Program of Frontier Sciences, CAS[QYZDJ-SSW-SYS003] ; Australian Research Council[DE150100720] ; Australian Research Council[DP200100950] ; Strategic Priority Research Program of Chinese Academy of Sciences[XDB28000000] ; K. C. Wong Education Foundation
WOS研究方向	Computer Science ; Mathematics
语种	英语
出版者	SIAM PUBLICATIONS
WOS记录号	WOS:000674143400019
内容类型	期刊论文
源URL	[http://119.78.100.204/handle/2XEOYT63/17324]
专题	中国科学院计算技术研究所
通讯作者	Bei, Xiaohui
作者单位	1.Nanyang Technol Univ, Sch Phys & Math Sci, Singapore 637371, Singapore 2.Chinese Acad Sci, Inst Software, State Key Lab Comp Sci, Beijing, Peoples R China 3.Univ Technol Sydney, Ctr Quantum Software & Informat, Parramatta, NSW 2150, Australia 4.Chinese Acad Sci, Inst Comp Technol, Beijing, Peoples R China 5.Univ Chinese Acad Sci, Beijing, Peoples R China
推荐引用方式 GB/T 7714	Bei, Xiaohui,Chen, Shiteng,Guan, Ji,et al. FROM INDEPENDENT SETS AND VERTEX COLORINGS TO ISOTROPIC SPACES AND ISOTROPIC DECOMPOSITIONS: ANOTHER BRIDGE BETWEEN GRAPHS AND ALTERNATING MATRIX SPACES[J]. SIAM JOURNAL ON COMPUTING,2021,50(3):924-971.
APA	Bei, Xiaohui,Chen, Shiteng,Guan, Ji,Qiao, Youming,&Sun, Xiaoming.(2021).FROM INDEPENDENT SETS AND VERTEX COLORINGS TO ISOTROPIC SPACES AND ISOTROPIC DECOMPOSITIONS: ANOTHER BRIDGE BETWEEN GRAPHS AND ALTERNATING MATRIX SPACES.SIAM JOURNAL ON COMPUTING,50(3),924-971.
MLA	Bei, Xiaohui,et al."FROM INDEPENDENT SETS AND VERTEX COLORINGS TO ISOTROPIC SPACES AND ISOTROPIC DECOMPOSITIONS: ANOTHER BRIDGE BETWEEN GRAPHS AND ALTERNATING MATRIX SPACES".SIAM JOURNAL ON COMPUTING 50.3(2021):924-971.