Guillotine cuts & pattern avoidance in permutations

Abstract

Given a set of axis-parallel rectangles on a plane, can you cut out a constant fraction of them using only Guillotine cuts? We try to answer this question in the project. The general problem of cutting out convex sets was investigated by Pach & Tardos [11]. But there has been a lot of interest in proving better bounds for rectangles. This is because if the answer to the aforementioned question is in the affirmative, it automatically gives us a polytime O(1)-approximation to MISR, as stressed in a paper by Abed, Chalermsook et al. [1], which made progress by showing abound of n=81 for the special case of squares. In our investigation of these structures, a strong connection with permutations showed up. This was mainly due to a combinatorial bijection between Floorplans and Baxter permutations given by Ackerman et al. [2]. Combined with the work of Young et al. [13], it inspired a strategy to prove bounds which work for all rectangle sets: delete just enough rectangles to delete all pinwheels, which we de ne. Moreover, previously unexplored (seemingly) yet natural questions about the eld of Pattern Matching / Avoidance in Permutations presented themselves in our study. We detail numerous results and conjectures in this area.