Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions

TL;DR

This work addresses the Maximum Independent Set of Convex Polygons when edges lie on at most $d$ fixed directions ($d$-MISP) and achieves an $8d/3$-approximation in time $O((nd)^{O(d4^d)})$. The authors develop a grid-based dynamic-programming framework over structured containers, generalizing constant-factor MISR techniques to the $d$-direction setting. The analysis hinges on a recursive partition argument aided by a charging scheme that guarantees a $3/(8d)$-fraction of the optimum can be captured by the partition, supported by a maximal extension construction and fences to control interactions. This advances the state of the art by extending MISR-style constant-factor guarantees to a broader class of polygons while maintaining tractable complexity for constant $d$, linking to prior $n^{\varepsilon}$/$OPT^{\varepsilon}$-approximations and PTAS results in related settings.

Abstract

In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the problem where the edges of the polygons can take at most $d$ fixed directions. We present an $8d/3$-approximation algorithm for this problem running in time $O((nd)^{O(d4^d)})$. The previous-best polynomial-time approximation (for constant $d$) was a classical $n^\varepsilon$ approximation by Fox and Pach [SODA'11] that has recently been improved to a $OPT^{\varepsilon}$-approximation algorithm by Cslovjecsek, Pilipczuk and Węgrzycki [SODA '24], which also extends to an arbitrary set of convex polygons. Our result builds on, and generalizes the recent constant factor approximation algorithms for the maximum independent set of axis-parallel rectangles problem (which is a special case of our problem with $d=2$) by Mitchell [FOCS'21] and Gálvez, Khan, Mari, Mömke, Reddy, and Wiese [SODA'22].

Figures (6)

• Figure 1: A convex polygon in $4$ directions. The edge $e_3(P)$ is degenerate.
• Figure 2: A container with $\kappa = 5$. An illustration of crossing and non-crossing.
• Figure 3: Illustration of the process of extending a polygon $P$. We extend $P$ by moving the edge $e$ of $P$ until $\mathop{\mathrm{int}}\nolimits(e)$ touches another polygon in $\mathop{\mathrm{OPT}}\nolimits$.
• Figure 4: A black arrow from $P$ to $P'$ indicates that $P$ sees $P'$ with respect to the option $(v_1, t)$, i.e., direction vertical-up and tail. The blue (resp. red) corners represent the tails (resp. head) of all edges with direction vertical-down ($v_5$). Thus, a polygon $P$ sees a polygon $P'$ if the vertical-up edge of $P$ is touching the red corner of $P'$.
• Figure 6: Example of a structured container with $\kappa = 4$. The black arrows represent "seeing", top-fences are green, bottom-fences are blue. The polygons $P_1, P_3, P_4, P_5, P_8$ are protected (only) from the left, $P_{14}, P_{16}$ are protected (only) from the right, $P_9, P_{11}$ are protected both from the left and from the right. Notice that the fences that protect $P_{14}$ (from the right) are not unique since $P_{14}$ sees $P_{15}$ and $P_{16}$ which are cut and touch $s_4$, respectively. Note also that the bottom-fences touching $P_8, P_{11}$ and $P_{11}, P_{13}$ overlap.

Theorems & Definitions (18)

• Theorem 1
• Lemma 2: Running time
• Definition 3
• Lemma 4: galvezSODA
• Proposition 4
• Definition 4
• Definition 5
• Lemma 6
• Lemma 7: Good charging option
• Definition 9: Top and bottom fences