The notion of duality is a key element in understanding the interplay between the stability and chromatic numbers of a graph. This notion is a central aspect in the celebrated theory of perfect graphs and is further and deeply developed in the context of the Lovász theta function and its equivalent characterizations and variants. The main achievement of this paper is the introduction of a new family of norms, providing upper bounds for the stability number, that are obtained through duality from the norms motivated by Hoffman's lower bound for the chromatic number, and which achieve the (complementary) Lovász theta function at their optimum. As a consequence, our norms make it formal that Hoffman's bound for the chromatic number and the Delsarte--Hoffman ratio bound for the stability number are indeed dual. Further, we show that our new bounds strengthen the convex quadratic bounds for the stability number studied by Luz and Schrijver, and which achieve the Lovász theta function at their optimum. One of the key observations regarding weighted versions of these bounds is that, for any upper bound for the stability number of a graph which is a positive definite monotone gauge function, its gauge dual is a lower bound on the fractional chromatic number, and conversely. Our presentation is elementary and accessible to a wide audience.

Total dual integrality is a powerful and unifying concept in polyhedral combinatorics and integer programming that enables the refinement of geometric min-max relations given by linear programming strong duality into combinatorial min-max theorems. The definition of a linear inequality system being totally dual integral (TDI) revolves around the existence of optimal dual solutions that are integral and thus naturally applies to a host of combinatorial optimization problems that are cast as integer programs whose linear program (LP) relaxations have the TDIness property. However, when combinatorial problems are formulated using more general convex relaxations, such as semidefinite programs (SDPs), it is not at all clear what an appropriate notion of integrality in the dual program is, thus inhibiting the generalization of the theory to more general forms of structured convex optimization. (In fact, we argue that the rank-one constraint usually added to SDP relaxations is not adequate in the dual SDP.) In this paper, we propose a notion of total dual integrality for SDPs that generalizes the notion for LPs, by relying on an “integrality constraint" for SDPs that is primal-dual symmetric. A key ingredient for the theory is a generalization to compact convex sets of a result of Hoffman for polytopes, fundamental for generalizing the polyhedral notion of total dual integrality introduced by Edmonds and Giles. We study the corresponding theory applied to SDP formulations for stable sets in graphs using the Lovász theta function and show that total dual integrality in this case corresponds to the underlying graph being perfect. We also relate dual integrality of an SDP formulation for the maximum cut problem to bipartite graphs. Total dual integrality for extended formulations naturally comes into play in this context.

Many fast graph algorithms begin by preprocessing the graph to improve its sparsity. A common form of this is spectral sparsification, which involves removing and reweighting the edges of the graph while approximately preserving its spectral properties. This task has a more general linear algebraic formulation in terms of approximating sums of rank-one matrices. This article considers a more general task of approximating sums of symmetric, positive semidefinite matrices of arbitrary rank. We present two deterministic, polynomial time algorithms for solving this problem. The first algorithm applies the pessimistic estimators of Wigderson and Xiao, and the second involves an extension of the method of Batson, Spielman, and Srivastava. These algorithms have several applications, including sparsifiers of hypergraphs, sparse solutions to semidefinite programs, sparsifiers of unique games, and graph sparsifiers with various auxiliary constraints.