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1. Faster exact and approximation algorithms for packing and covering matroids via push-relabel

   https://doi.org/10.1137/1.9781611977912.82  

   Quanrud, Kent (January 2024, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024)

   Woodruff, David P. (Ed.)

   Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $$k$$ independent sets (that is, \emph{$$k$$-fold matroid union}), computing $$k$$ disjoint bases (a.k.a.\ \emph{matroid base packing}), and covering the elements by $$k$$ bases (a.k.a.\ \emph{matroid base covering}). These problems generalize naturally to integral and real-valued capacities on the elements. This work develops faster exact and/or approximation problems for these and some other closely related problems such as optimal reinforcement and matroid membership. We obtain improved running times both for general matroids in the independence oracle model and for the graphic matroid. The main thrust of our improvements comes from developing a faster and unifying \emph{push-relabel} algorithm for the integer-capacitated versions of these problems, building on previous work by [FM12]. We then build on this algorithm in two directions. First we develop a faster augmenting path subroutine for $$k$$-fold matroid union that, when appended to an approximation version of the push-relabel algorithm, gives a faster exact algorithm for some parameters of $$k$$. In particular we obtain a subquadratic-query running time in the uncapacitated setting for the three basic problems listed above. We also obtain faster approximation algorithms for these problems with real-valued capacities by reducing to small integral capacities via randomized rounding. To this end, we develop a new randomized rounding technique for base covering problems in matroids that may also be of independent interest. 

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2. Multi-Agent Maximization of a Monotone Submodular Function via Maximum Consensus

   https://doi.org/10.1109/CDC45484.2021.9682818  

   Rezazadeh, Navid; Kia, Solmaz S. (December 2021, IEEE Conference on Decision and Control)

   This paper studies distributed submodular optimization subject to partition matroid. We work in the value oracle model where the only access of the agents to the utility function is through a black box that returns the utility function value. The agents are communicating over a connected undirected graph and have access only to their own strategy set. As known in the literature, submodular maximization subject to matroid constraints is NP-hard. Hence, our objective is to propose a polynomial-time distributed algorithm to obtain a suboptimal solution with guarantees on the optimality bound. Our proposed algorithm is based on a distributed stochastic gradient ascent scheme built on the multilinear-extension of the submodular set function. We use a maximum consensus protocol to minimize the inconsistency of the shared information over the network caused by delay in the flow of information while solving for the fractional solution of the multilinear extension model. Furthermore, we propose a distributed framework of finding a set solution using the fractional solution. We show that our distributed algorithm results in a strategy set that when the team objective function is evaluated at worst case the objective function value is in 1−1/e−O(1/T) of the optimal solution in the value oracle model where T is the number of communication instances of the agents. An example demonstrates our results. 

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3. Minors of a random binary matroid

   Cooper, C.; Frieze, A.; Pegden, W. (January 2019, Random structures algorithms)

   Let $${\bf A}$$ be an $$n\times m$$ matrix over $$\mathbf{GF}_2$$ where each column consists of $$k$$ ones, and let $$M$$ be an arbitrary fixed binary matroid. The matroid growth rate theorem implies that there is a constant $$C_M$$ such that $$m\geq C_M n^2$$ implies that the binary matroid induced by {\bf A} contains $$M$$ as a minor. We prove that if the columns of $${\bf A}={\bf A}_{n,m,k}$$ are chosen \emph{randomly}, then there are constants $$k_M, L_M$$ such that $$k\geq k_M$$ and $$m\geq L_M n$$ implies that $${\bf A}$$ contains $$M$$ as a minor w.h.p. 

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4. Intersection Theory of Matroids: Variations on a Theme

   https://doi.org/10.1017/9781009490559.002  

   Ardila–Mantilla, Federico (June 2024, Cambridge University Press)

   Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh–Katz’s formula μ_k(M) = deg_M(α^{r−k}β^k) for the coefficients of the reduced characteristic polynomial of a matroid M as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes α and β in the Chow ring of M. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids. Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory. This survey was prepared for the Clay Lecture to be delivered at the 2024 British Combinatorics Conference. 

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5. A framework for quadratic form maximization over convex sets through nonconvex relaxations

   https://doi.org/10.1145/3406325.3451128  

   Bhattiprolu, Vijay; Lee, Euiwoong; Naor, Assaf (January 2021, STOC'21)

   null (Ed.)

   We investigate the approximability of the following optimization problem. The input is an n× n matrix A=(Aij) with real entries and an origin-symmetric convex body K⊂ ℝn that is given by a membership oracle. The task is to compute (or approximate) the maximum of the quadratic form ∑i=1n∑j=1n Aij xixj=⟨ x,Ax⟩ as x ranges over K. This is a rich and expressive family of optimization problems; for different choices of matrices A and convex bodies K it includes a diverse range of optimization problems like max-cut, Grothendieck/non-commutative Grothendieck inequalities, small set expansion and more. While the literature studied these special cases using case-specific reasoning, here we develop a general methodology for treatment of the approximability and inapproximability aspects of these questions. The underlying geometry of K plays a critical role; we show under commonly used complexity assumptions that polytime constant-approximability necessitates that K has type-2 constant that grows slowly with n. However, we show that even when the type-2 constant is bounded, this problem sometimes exhibits strong hardness of approximation. Thus, even within the realm of type-2 bodies, the approximability landscape is nuanced and subtle. However, the link that we establish between optimization and geometry of Banach spaces allows us to devise a generic algorithmic approach to the above problem. We associate to each convex body a new (higher dimensional) auxiliary set that is not convex, but is approximately convex when K has a bounded type-2 constant. If our auxiliary set has an approximate separation oracle, then we design an approximation algorithm for the original quadratic optimization problem, using an approximate version of the ellipsoid method. Even though our hardness result implies that such an oracle does not exist in general, this new question can be solved in specific cases of interest by implementing a range of classical tools from functional analysis, most notably the deep factorization theory of linear operators. Beyond encompassing the scenarios in the literature for which constant-factor approximation algorithms were found, our generic framework implies that that for convex sets with bounded type-2 constant, constant factor approximability is preserved under the following basic operations: (a) Subspaces, (b) Quotients, (c) Minkowski Sums, (d) Complex Interpolation. This yields a rich family of new examples where constant factor approximations are possible, which were beyond the reach of previous methods. We also show (under commonly used complexity assumptions) that for symmetric norms and unitarily invariant matrix norms the type-2 constant nearly characterizes the approximability of quadratic maximization. 

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