Correlation bounds for fields and matroids

Let $G$ be a finite connected graph, and let $T$ be a spanning tree of $G$ chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events $e_1 \in T$ and $e_2 \in T$ are negatively correlated for any distinct edges $e_1$ and $e_2$. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events $e \in B$, where $B$ is a randomly chosen basis of a matroid. As an application, we prove Mason's conjecture that the number of $k$-element independent sets of a matroid forms an ultra-log-concave sequence in $k$.Comment: 16 pages. Supersedes arXiv:1804.0307

Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

Algorithms as Mechanisms: The Price of Anarchy of Relax-and-Round

Many algorithms that are originally designed without explicitly considering incentive properties are later combined with simple pricing rules and used as mechanisms. The resulting mechanisms are often natural and simple to understand. But how good are these algorithms as mechanisms? Truthful reporting of valuations is typically not a dominant strategy (certainly not with a pay-your-bid, first-price rule, but it is likely not a good strategy even with a critical value, or second-price style rule either). Our goal is to show that a wide class of approximation algorithms yields this way mechanisms with low Price of Anarchy. The seminal result of Lucier and Borodin [SODA 2010] shows that combining a greedy algorithm that is an $\alpha$-approximation algorithm with a pay-your-bid payment rule yields a mechanism whose Price of Anarchy is $O(\alpha)$. In this paper we significantly extend the class of algorithms for which such a result is available by showing that this close connection between approximation ratio on the one hand and Price of Anarchy on the other also holds for the design principle of relaxation and rounding provided that the relaxation is smooth and the rounding is oblivious. We demonstrate the far-reaching consequences of our result by showing its implications for sparse packing integer programs, such as multi-unit auctions and generalized matching, for the maximum traveling salesman problem, for combinatorial auctions, and for single source unsplittable flow problems. In all these problems our approach leads to novel simple, near-optimal mechanisms whose Price of Anarchy either matches or beats the performance guarantees of known mechanisms.Comment: Extended abstract appeared in Proc. of 16th ACM Conference on Economics and Computation (EC'15

Positive configuration space

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang-Mills scattering amplitudes, which will be discussed in a sequel.Comment: 46 pages; citations adde

Notes on Feynman Integrals and Renormalization

I review various aspects of Feynman integrals, regularization and renormalization. Following Bloch, I focus on a linear algebraic approach to the Feynman rules, and I try to bring together several renormalization methods found in the literature from a unifying point of view, using resolutions of singularities. In the second part of the paper, I briefly sketch the work of Belkale, Brosnan resp. Bloch, Esnault and Kreimer on the motivic nature of Feynman integrals.Comment: 39

Hardness and Approximation of Submodular Minimum Linear Ordering Problems

The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $f(\cdot)$ due to an ordering $\sigma$ of the items (say $[n]$), i.e., $\min_{\sigma} \sum_{i\in [n]} f(E_{i,\sigma})$, where $E_{i,\sigma}$ is the set of items mapped by $\sigma$ to indices $[i]$. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata, Tetali, and Tripathi [ITT2012], using Lov\'asz extension of submodular functions. We show a $(2-\frac{1+\ell_{f}}{1+|E|})$-approximation for monotone submodular MLOP where $\ell_{f}=\frac{f(E)}{\max_{x\in E}f(\{x\})}$ satisfies $1 \leq \ell_f \leq |E|$. Our theory provides new approximation bounds for special cases of the problem, in particular a $(2-\frac{1+r(E)}{1+|E|})$-approximation for the matroid MLOP, where $f = r$ is the rank function of a matroid. We further show that minimum latency vertex cover (MLVC) is $\frac{4}{3}$-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest