Minimum-Complexity Graph Simplification under Fr\'echet-Like Distances

TL;DR

This paper investigates the computational complexity of simplifying geometric graphs while maintaining a bounded distance, providing NP-hardness results and an efficient algorithm for tree cases.

Contribution

It introduces new NP-hardness results for general graphs and presents a polynomial-time algorithm for tree simplification under specific conditions.

Findings

Graph simplification is NP-hard in general cases.

An $O(kn^5)$ algorithm is developed for tree simplification.

The complexity depends on input/output graph types and distance measures.

Abstract

Simplifying graphs is a very applicable problem in numerous domains, especially in computational geometry. Given a geometric graph and a threshold, the minimum-complexity graph simplification asks for computing an alternative graph of minimum complexity so that the distance between the two graphs remains at most the threshold. In this paper, we propose several NP-hardness and algorithmic results depending on the type of input and simplified graphs, the vertex placement of the simplified graph, and the distance measures between them (graph and traversal distances [1,2]). In general, we show that for arbitrary input and output graphs, the problem is NP-hard under some specific vertex-placement of the simplified graph. When the input and output are trees, and the graph distance is applied from the simplified tree to the input tree, we give an $O(kn^5)$ time algorithm, where $k$ is the number of the leaves of the two trees that are identical and $n$ is the number of vertices of the input.