Planar Ultrametric Rounding for Image Segmentation

层次聚类中LP舍入的处理与通过超度量拟合距离

Handling LP-Rounding for Hierarchical Clustering and Fitting Distances by Ultrametrics

本文针对层次相关聚类问题提出一种简化LP舍入的新范式，将近似保证显著改进至25.7846，并揭示层次聚类核心问题与平均距离割之间的联系，同时为超度量违规距离问题提供了更简单的替代方法。

Approximation algorithms for node-weighted prize-collecting Steiner tree problems on planar graphs

We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a ($2.88 + ε$)-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest…

Simple and Fast Rounding Algorithms for Directed and Node-weighted Multiway Cut

In Directed Multiway Cut(Dir-MC) the input is an edge-weighted directed graph $G=(V,E)$ and a set of $k$ terminal nodes $\{s_1,s_2,\ldots,s_k\} \subseteq V$; the goal is to find a min-weight subset of edges whose removal ensures that there is no path from $s_i$ to $s_j$ for any $i \neq j$. In Node-weighted Multiway Cut(Node-MC) the input is a node-weighted undirected graph $G$ and a set of $k$ terminal nodes $\{s_1,s_2,\ldots,s_k\} \subseteq V$; the goal is to remove a min-weight subset of nodes to disconnect each pair of terminals. Dir-MC admits a $2$-approximation [Naor, Zosin '97] and Node-…

Approximating maximum integral multiflows on bounded genus graphs

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.

Non-Uniform Robust Network Design in Planar Graphs

Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on \textbf{bulk-robustness}, a model recently introduced~\cite{bulk} for addressing the need to model non-uniform failure patterns in systems. We significantly extend the techniq…

平衡分区问题的难度与近似算法

Hardness and Approximation Algorithms for Balanced Districting Problems

针对政治划区驱动的平衡分区问题，证明了多类图上的近似难度，提出基于改进Whack-a-Mole算法的近似方案，引入平衡散射分隔器实现紧近似比。