We study the problem of chasing positive bodies in $\ell_1$: given a sequence of bodies $K_{t}=\{x^{t}\in\mathbb{R}_{+}^{n}\mid C^{t}x^{t}\geq 1,P^{t}x^{t}\leq 1\}$ revealed online, where $C^{t}$ and $P^{t}$ are nonnegative matrices, the goal is to (approximately) maintain a point $x_t \in K_t$ such that $\sum_t \|x_t - x_{t-1}\|_1$ is minimized. This captures the fully-dynamic low-recourse variant of any problem that can be expressed as a mixed packing-covering linear program and thus also the fractional version of many central problems in dynamic algorithms such as set cover, load balancing, hyperedge orientation, minimum spanning tree, and matching. We give an $O(\log d)$-competitive algorithm for this problem, where $d$ is the maximum row sparsity of any matrix $C^t$. This bypasses and improves exponentially over the lower bound of $\sqrt{n}$ known for general convex bodies. Our algorithm is based on iterated information projections, and, in contrast to general convex body chasing algorithms, is entirely memoryless. We also show how to round our solution dynamically to obtain the first fully dynamic algorithms with competitive recourse for all the stated problems above; i.e. their recourse is less than the recourse of every other algorithm on every update sequence, up to polylogarithmic factors. This is a significantly stronger notion than the notion of absolute recourse in the dynamic algorithms literature.

翻译：我们研究$\ell_1$空间中追逐正凸体的问题：给定在线揭示的凸体序列$K_{t}=\{x^{t}\in\mathbb{R}_{+}^{n}\mid C^{t}x^{t}\geq 1,P^{t}x^{t}\leq 1\}$，其中$C^{t}$和$P^{t}$是非负矩阵，目标是（近似）维持一个点$x_t \in K_t$，使得$\sum_t \|x_t - x_{t-1}\|_1$最小化。这个问题刻画了任何可表示为混合打包-覆盖线性规划问题的完全动态低反悔变体，因此也涵盖了动态算法中许多核心问题的分数版本，例如集合覆盖、负载均衡、超边定向、最小生成树和匹配。我们为此问题给出了一个$O(\log d)$竞争比的算法，其中$d$是任意矩阵$C^t$的最大行稀疏度。这突破并指数级改进了针对一般凸体已知的$\sqrt{n}$下界。我们的算法基于迭代信息投影，与一般凸体追逐算法不同，它完全无记忆。我们还展示了如何动态地舍入我们的解，从而为上述所有问题首次获得具有竞争性反悔的完全动态算法；即，在每次更新序列上，它们的反悔都小于其他任何算法的反悔，且差距最多为多对数因子。这是比动态算法文献中绝对反悔概念显著更强的概念。