Online Maximum Independent Set of Hyperrectangles

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 bounds on the competitive ratio of an optimal online algorithm under the adaptive offline, adaptive online, and oblivious adversary models, for several classes of hyperrectangles and restrictions on the order of the input. We are the first to present results on this problem under the oblivious adversary model. We prove bounds on the competitive ratio for unit hypercubes, $\sigma$-bounded hypercubes, unit-volume hypercubes, arbitrary hypercubes, and arbitrary hyperrectangles, in both arbitrary and non-dominated order. We are also the first to present results under the adaptive offline and adaptive online adversary models with input in non-dominated order, proving bounds on the competitive ratio for the same classes of hyperrectangles; for input in arbitrary order, we present the first results on $\sigma$-bounded hypercubes, unit-volume hyperrectangles, arbitrary hypercubes, and arbitrary hyperrectangles. For input in dominating order, we show that the performance of the naive greedy algorithm matches the performance of an optimal offline algorithm in all cases. We also give lower bounds on the competitive ratio of a probabilistic greedy algorithm under the oblivious adversary model. We conclude by discussing several promising directions for future work.Comment: 27 pages, 12 figure

Two Decades of Maude

This paper is a tribute to José Meseguer, from the rest of us in the Maude team, reviewing the past, the present, and the future of the language and system with which we have been working for around two decades under his leadership. After reviewing the origins and the language's main features, we present the latest additions to the language and some features currently under development. This paper is not an introduction to Maude, and some familiarity with it and with rewriting logic are indeed assumed.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech