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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 W\k{e}grzycki [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 G\'{a}lvez, Khan, Mari, M\"{o}mke, Reddy, and Wiese [SODA'22]., Comment: To appear at SoCG 2024

We consider an online version of the geometric minimum hitting set problem that can be described as a game between an adversary and an algorithm. For some integers $d$ and $N$, let $P$ be the set of points in $(0, N)^d$ with integral coordinates, and let $\mathcal{O}$ be a family of subsets of $P$, called objects. Both $P$ and $\mathcal{O}$ are known in advance by the algorithm and by the adversary. Then, the adversary gives some objects one by one, and the algorithm has to maintain a valid hitting set for these objects using points from $P$, with an immediate and irrevocable decision. We measure the performance of the algorithm by its competitive ratio, that is the ratio between the number of points used by the algorithm and the offline minimum hitting set for the sub-sequence of objects chosen by the adversary. We present a simple deterministic online algorithm with competitive ratio $((4\alpha+1)^{2d}\log N)$ when objects correspond to a family of $\alpha$-fat objects. Informally, $\alpha$-fatness measures how cube-like is an object. We show that no algorithm can achieve a better ratio when $\alpha$ and $d$ are fixed constants. In particular, our algorithm works for two-dimensional disks and $d$-cubes which answers two open questions from related previous papers in the special case where the set of points corresponds to all the points of integral coordinates with a fixed $d$-cube.

We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell [2021] obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10 and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called corner-clipped rectangles (CCRs), without intersecting certain special horizontal line segments called fences. In this paper, we present a $(2+\epsilon)$-approximation algorithm for MISR which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This allows us to provide an arguably simpler analysis and at the same time already improves the approximation ratio to 6. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to $2+\epsilon$. In particular, we construct a recursive partitioning based on more general fences which can be sequences of up to $O(1/\epsilon)$ line segments each. This partitioning routine and our other new ideas may be useful for future work towards a PTAS for MISR., Comment: 41 pages, updates the previous version which had a 3-approximation algorithm

We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles.