Online Maximum Independent Set of Hyperrectangles

TL;DR

This work analyzes the online maximum independent set problem for intersection graphs of axis-aligned hyperrectangles in $\mathbb{R}^d$ across multiple input orders and adversary models. It develops optimal online algorithms (notably DetGreedy) for unit hypercubes, $\sigma$-bounded hypercubes, unit-volume hyperrectangles, and arbitrary hypercubes under adaptive adversaries, and provides tight or near-tight bounds for oblivious adversaries, including a novel SelectiveGreedy$_k$ approach for $\sigma$-bounded cases. The dominating-order setting yields an optimal online strategy (DetGreedy) against adaptive offline adversaries, while the oblivious-adversary analysis reveals valuable trade-offs via Greedy$_p$ and SelectiveGreedy$_k$ with explicit competitive ratios. Collectively, the results map the competitive landscape for online MIS of hyperrectangles, offer algorithms with provable guarantees, and suggest practical directions for extending online geometric packing problems to higher dimensions and varied input models.

Abstract

The maximum independent set problem is a classical NP-hard problem in theoretical computer science. In this work, we study a special case where the family of graphs considered is restricted to intersection graphs of sets of axis-aligned hyperrectangles and the input is provided in an online fashion. We prove results for several adversary models, classes of hyperrectangles, and restrictions on the order of the input. Under the adaptive offline and adaptive online adversary models, we find the optimal online algorithm for unit hypercubes, $σ$-bounded hypercubes, unit-volume hyperrectangles, and arbitrary hypercubes, in both non-dominated and arbitrary order. Under the oblivious adversary model, we prove bounds on the competitive ratio of an optimal online algorithm for the same classes of hyperrectangles and input orders, and we find algorithms that are optimal up to constant factors. For input in dominating order, we find the optimal online algorithm for arbitrary hyperrectangles under all adversary models. We conclude by discussing several promising directions for future work.

Figures (17)

• Figure 1: An example of an oblivious adversary strategy for unit hypercubes in non-dominated order.
• Figure 2: An example of an oblivious adversary strategy for unit hypercubes in arbitrary order.
• Figure 3: An example of an adaptive online adversary strategy for unit-volume hyperrectangles in non-dominated order.
• Figure 4: An optimal arrangement of $n=6$ unit-area rectangles for $p=0.50$.
• Figure 5: An example of an oblivious adversary strategy for unit-volume hyperrectangles in non-dominated order.

Theorems & Definitions (48)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Corollary 3.1
• Theorem 4
• proof
• Theorem 5