Extremal sequences of polynomial complexity

The joint spectral radius of a bounded set of $d \times d$ real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called \emph{extremal} if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer $p \geq 1$, there exists a pair of square matrices of dimension $2^p(2^{p+1}-1)$ for which every extremal sequence has subword complexity at least $2^{-p^2}n^p$.Comment: 15 page

Polynomial complexity despite the fermionic sign

It is commonly believed that in quantum Monte Carlo approaches to fermionic many- body problems, the infamous sign problem generically implies prohibitively large computational times for obtaining thermodynamic-limit quantities. We point out that for convergent Feynman diagrammatic series evaluated with the Monte Carlo algorithm of [Rossi, arXiv:1612.05184], the computational time increases only polynomially with the inverse error on thermodynamic-limit quantities

New Shortest Lattice Vector Problems of Polynomial Complexity

The Shortest Lattice Vector (SLV) problem is in general hard to solve, except for special cases (such as root lattices and lattices for which an obtuse superbase is known). In this paper, we present a new class of SLV problems that can be solved efficiently. Specifically, if for an $n$-dimensional lattice, a Gram matrix is known that can be written as the difference of a diagonal matrix and a positive semidefinite matrix of rank $k$ (for some constant $k$), we show that the SLV problem can be reduced to a $k$-dimensional optimization problem with countably many candidate points. Moreover, we show that the number of candidate points is bounded by a polynomial function of the ratio of the smallest diagonal element and the smallest eigenvalue of the Gram matrix. Hence, as long as this ratio is upper bounded by a polynomial function of $n$, the corresponding SLV problem can be solved in polynomial complexity. Our investigations are motivated by the emergence of such lattices in the field of Network Information Theory. Further applications may exist in other areas.Comment: 13 page

Byzantine Agreement with Optimal Early Stopping, Optimal Resilience and Polynomial Complexity

We provide the first protocol that solves Byzantine agreement with optimal early stopping ($\min\{f+2,t+1\}$ rounds) and optimal resilience ($n>3t$) using polynomial message size and computation. All previous approaches obtained sub-optimal results and used resolve rules that looked only at the immediate children in the EIG (\emph{Exponential Information Gathering}) tree. At the heart of our solution are new resolve rules that look at multiple layers of the EIG tree.Comment: full version of STOC 2015 abstrac

Optimal CSMA-based Wireless Communication with Worst-case Delay and Non-uniform Sizes

Carrier Sense Multiple Access (CSMA) protocols have been shown to reach the full capacity region for data communication in wireless networks, with polynomial complexity. However, current literature achieves the throughput optimality with an exponential delay scaling with the network size, even in a simplified scenario for transmission jobs with uniform sizes. Although CSMA protocols with order-optimal average delay have been proposed for specific topologies, no existing work can provide worst-case delay guarantee for each job in general network settings, not to mention the case when the jobs have non-uniform lengths while the throughput optimality is still targeted. In this paper, we tackle on this issue by proposing a two-timescale CSMA-based data communication protocol with dynamic decisions on rate control, link scheduling, job transmission and dropping in polynomial complexity. Through rigorous analysis, we demonstrate that the proposed protocol can achieve a throughput utility arbitrarily close to its offline optima for jobs with non-uniform sizes and worst-case delay guarantees, with a tradeoff of longer maximum allowable delay

Full one-loop amplitudes from tree amplitudes

We establish an efficient polynomial-complexity algorithm for one-loop calculations, based on generalized $D$-dimensional unitarity. It allows automated computations of both cut-constructible {\it and} rational parts of one-loop scattering amplitudes from on-shell tree amplitudes. We illustrate the method by (re)-computing all four-, five- and six-gluon scattering amplitudes in QCD at one-loop.Comment: 27 pages, revte