An approximation algorithm for the longest cycle problem in solid grid graphs

单位权重网格上加拿大旅行者问题的紧渐近竞争比

Tight asymptotic competitive ratio for the Canadian Traveller Problem on unit-weighted grids

该研究针对单位权重实心网格图，证明k-加拿大旅行者问题存在竞争比为O(√k)的在线算法，该比值渐近紧，且单位权重与实心性假设对该结果至关重要。

Fast Balanced Partitioning is Hard, Even on Grids and Trees

Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is necessary. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a reduction framework which identifies some necessary conditions on the considered graph class in order to prove the hardness of the problem. We focus …

Decomposition of Map Graphs with Applications

Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or its treewidth is $O(\sqrt{k})$. However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new…