Given a matrix $A$ and vector $b$ with polynomial entries in $d$ real variables $\delta=(\delta_1,\ldots,\delta_d)$ we consider the following notion of feasibility: the pair $(A,b)$ is locally feasible if there exists an open neighborhood $U$ of $0$ such that for every $\delta\in U$ there exists $x$ satisfying $A(\delta)x\ge b(\delta)$ entry-wise. For $d=1$ we construct a polynomial time algorithm for deciding local feasibility. For $d \ge 2$ we show local feasibility is NP-hard. This also gives the first polynomial-time algorithm for the asymptotic linear program problem introduced by Jeroslow in 1973. As an application (which was the primary motivation for this work) we give a computer-assisted proof of ergodicity of the following elementary 1D cellular automaton: given the current state $\eta_t \in \{0,1\}^{\mathbb{Z}}$ the next state $\eta_{t+1}(n)$ at each vertex $n\in \mathbb{Z}$ is obtained by $\eta_{t+1}(n)= \text{NAND}\big(\text{BSC}_\delta(\eta_t(n-1)), \text{BSC}_\delta(\eta_t(n))\big)$. Here the binary symmetric channel $\text{BSC}_\delta$ takes a bit as input and flips it with probability $\delta$ (and leaves it unchanged with probability $1-\delta$). It is shown that there exists $\delta_0>0$ such that for all $0<\delta<\delta_0$ the distribution of $\eta_t$ converges to a unique stationary measure irrespective of the initial condition $\eta_0$. We also consider the problem of broadcasting information on the 2D-grid of noisy binary-symmetric channels $\text{BSC}_\delta$, where each node may apply an arbitrary processing function to its input bits. We prove that there exists $\delta_0'>0$ such that for all noise levels $0<\delta<\delta_0'$ it is impossible to broadcast information for any processing function, as conjectured by Makur, Mossel and Polyanskiy.

翻译：给定具有$d$个实变量$\delta=(\delta_1,\ldots,\delta_d)$的多元多项式系数的矩阵$A$和向量$b$，我们考虑如下可行性概念：若存在$0$的开邻域$U$使得对每个$\delta\in U$都存在$x$满足逐项不等式$A(\delta)x\ge b(\delta)$，则称$(A,b)$是局部可行的。对$d=1$情形，我们构造了判定局部可行性的多项式时间算法；对$d\ge 2$情形，我们证明局部可行性是NP难的。这也给出了Jeroslow于1973年提出的渐近线性规划问题的首个多项式时间算法。作为应用（此项研究的主要动机），我们通过计算机辅助证明给出了以下基本一维元胞自动机的遍历性：给定当前状态$\eta_t \in \{0,1\}^{\mathbb{Z}}$，每个顶点$n\in \mathbb{Z}$的下一个状态$\eta_{t+1}(n)$由规则$\eta_{t+1}(n)= \text{NAND}\big(\text{BSC}_\delta(\eta_t(n-1)), \text{BSC}_\delta(\eta_t(n))\big)$获得。这里二元对称信道$\text{BSC}_\delta$以概率$\delta$翻转输入比特（以概率$1-\delta$保持不变）。证明存在$\delta_0>0$使得对所有$0<\delta<\delta_0$，无论初始条件$\eta_0$如何，$\eta_t$的分布收敛到唯一平稳测度。我们还考虑了在带有噪声的二维二元对称信道网格$\text{BSC}_\delta$上广播信息的问题，其中每个节点可对其输入比特应用任意处理函数。我们证明存在$\delta_0'>0$使得对所有噪声水平$0<\delta<\delta_0'$，任何处理函数都无法实现信息广播，这一结论验证了Makur、Mossel和Polyanskiy的猜想。