Abstract
In the 1970s, Lovász 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.
Highlights
1.1 Between graphs and matrix spaces: some known bridgesThe bridge between perfect matchings and full-rank matricesIt is well-known that some graph-theoretic problems reduce to certain problems about linear spaces of matrices
We focus on undirected simple graphs, it is natural, as Tutte and Lovász did with perfect matchings, to work with alternating matrix spaces
Emboldened by Theorem 3, we propose to view isotropic spaces and decompositions as linear algebraic analogues of independent sets and vertex colorings, and study these two structures from the perspectives of graph theory and algorithms
Summary
It is well-known that some graph-theoretic problems reduce to certain problems about linear spaces of matrices. Derandomizing the corresponding algorithm for general linear spaces of matrices – not necessarily those of the form BG or AG – is known as the symbolic determinant identity testing problem, and turns out to be of fundamental significance in complexity theory, as that would imply strong circuit lower bounds which are considered to be beyond current techniques [28, 65]. The problem of testing whether a matrix space has a shrunk subspace arises naturally from several mathematical and computational displines, including algebraic complexity, non-commutative algebra, invariant theory, and analysis [48, 49, 61]. Not surprisingly this problem has had several names. For [61], Ivanyos et al need the polynomial upper bound [39], which in turn relies crucially on the regularity lemma developed in [60]
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