On the Two-Dimensional Knapsack Problem for Convex Polygons

Minimum Coverage by Convex Polygons: The CG:SHOP Challenge 2023

We give an overview of the 2023 Computational Geometry Challenge targeting the problem Minimum Coverage by Convex Polygons, which consists of covering a given polygonal region (possibly with holes) by a minimum number of convex subsets, a problem with a long-standing tradition in Computational Geometry.

Constrained episodic reinforcement learning in concave-convex and knapsack settings

We propose an algorithm for tabular episodic reinforcement learning with constraints. We provide a modular analysis with strong theoretical guarantees for settings with concave rewards and convex constraints, and for settings with hard constraints (knapsacks). Most of the previous work in constrained reinforcement learning is limited to linear constraints, and the remaining work focuses on either the feasibility question or settings with a single episode. Our experiments demonstrate that the proposed algorithm significantly outperforms these approaches in existing constrained episodic environm…

近似具有有界方向数量的凸多边形的最大独立集

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

针对边方向数最多为 $d$ 的凸多边形最大独立集问题，提出了一个运行时间为 $O((nd)^{O(d4^d)})$ 的 $8d/3$-近似算法，推广了轴平行矩形情形的常数因子近似结果。

Polygon Convexity: Another O(n) Test

An n-gon is defined as a sequence ¶=(V_0,...,V_{n-1}) of n points on the plane. An n-gon ¶is said to be convex if the boundary of the convex hull of the set {V_0,...,V_{n-1}} of the vertices of ¶coincides with the union of the edges [V_0,V_1],...,[V_{n-1},V_0]; if at that no three vertices of ¶are collinear then ¶is called strictly convex. We prove that an n-gon ¶with n\ge3 is strictly convex if and only if a cyclic shift of the sequence (\al_0,...,\al_{n-1})\in[0,2π)^n of the angles between the x-axis and the vectors V_1-V_0,...,V_0-V_{n-1} is strictly monotone. A ``non-strict'' version of th…

通过近凸Min-Plus卷积加速0-1背包问题

Faster 0-1-Knapsack via Near-Convex Min-Plus-Convolution

本文研究了0-1背包问题的伪多项式时间算法，通过提出高效的近凸函数min-plus卷积计算方法，改进了现有算法的时间复杂度，并在特定条件下首次打破了立方时间障碍。

用固定数量的约束覆盖和打包混合整数线性规划：近似和凸包

Covering and packing mixed-integer linear programs with a fixed number of constraints: Approximation and convex hull

本文提出了一种针对固定数量约束的覆盖混合整数线性规划问题的近似算法，并改进了背包覆盖问题的近似算法。