Chasing Positive Bodies

TL;DR

We address the problem of chasing positive bodies in the $oldsymbol{ll}_1$ norm, capturing the fully dynamic recourse of a broad class of packing–covering LPs. The main contribution is an $Oigl(rac{1}{
} rac{d}{
}igr)$-competitive algorithm that maintains a point in $K_t^{1+
}$ using memoryless KL-projection updates to violated halfspaces, bypassing general convex-body lower bounds. A refined dual analysis removes the aspect-ratio dependence, yielding an $Oigl(rac{1}{
} rac{d}{
}igr)$ bound without $oldsymbol{elta}$-growth. The approach extends to rounding strategies that convert the fractional online solution into fully dynamic, competitive-recourse algorithms for Set Cover, Bipartite Matching, Load Balancing, and MST, via stabilizers and subsequent absolute-recourse subroutines. These results provide the first fully dynamic, competitive-recourse algorithms for these central problems and demonstrate the practical impact of competitive recourse in dynamic combinatorial optimization.

Abstract

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.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.1: Warmup Bound
• Lemma 3.2
• proof
• Lemma 3.3: Bounding movement in terms of $y$
• proof
• Lemma 3.4: Bounding $z$ in terms of $y$
• proof
• Lemma 3.5