Removing Connected Obstacles in the Plane is FPT

How to navigate through obstacles?

Given a set of obstacles and two points, is there a path between the two points that does not cross more than $k$ different obstacles? This is a fundamental problem that has undergone a tremendous amount of work. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be generalized into the following graph problem: Given a planar graph $G$ whose vertices are colored by color sets, two designated vertices $s, t \in V(G)$, and $k \in \mathbb{N}$, is there an $s$-$t$ path in $G$ that uses at most $k$ colors? If each o…

An FPT algorithm for planar multicuts with sources and sinks on the outer face

Given a list of k source-sink pairs in an edge-weighted graph G, the minimum multicut problem consists in selecting a set of edges of minimum total weight in G, such that removing these edges leaves no path from each source to its corresponding sink. To the best of our knowledge, no non-trivial FPT result for special cases of this problem, which is APX-hard in general graphs for any fixed k>2, is known with respect to k only. When the graph G is planar, this problem is known to be polynomial-time solvable if k=O(1), but cannot be FPT with respect to k under the Exponential Time Hypothesis. In …

Computing the obstacle number of a plane graph

An obstacle representation of a plane graph G is V(G) together with a set of opaque polygonal obstacles such that G is the visibility graph on V(G) determined by the obstacles. We investigate the problem of computing an obstacle representation of a plane graph (ORPG) with a minimum number of obstacles. We call this minimum size the obstacle number of G. First, we show that ORPG is NP-hard by reduction from planar vertex cover, resolving a question posed by [8]. Second, we give a reduction from ORPG to maximum degree 3 planar vertex cover. Since this reduction preserves solution values, it foll…

一个 $O(3.82^k)$ 时间的凸翻转距离 FPT 算法

An $O(3.82^k)$ Time FPT Algorithm for Convex Flip Distance

针对凸多边形三角剖分间的翻转距离问题，提出时间复杂度为 $O(3.82^k)$、多项式空间的 FPT 算法，显著优于此前最佳算法。

彩色半径和问题的 FPT 常数近似算法

FPT Constant Approximation Algorithms for Colorful Sum of Radii

针对彩色半径和问题，本文提出两种 FPT 常数近似算法，分别取得 $(2+\varepsilon)$ 与 $(7+\varepsilon)$ 近似比，后者消除了对离群点参数 $m$ 的指数依赖。

An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an $n$-vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a graph of lin…