[Paper Review] Intermittent Map Matching with the Discrete Fréchet Distance

Tim Wylie, Binhai Zhu|arXiv (Cornell University)|Sep 8, 2014

Data Management and Algorithms17 references3 citations

TL;DR

This paper introduces a formal framework for intermittent map matching using the discrete Fréchet distance, proving that the variant restricting vertex usage to unique nodes (Unique Map Matching, UMM) is NP-complete for planar graphs. It establishes computational hardness for several constrained variations while identifying polynomial-time solvable cases, offering foundational complexity insights for geometric path reconstruction in GPS data with sparse or intermittent sampling.

ABSTRACT

In this paper we focus on the map matching problem where the goal is to find a path through a planar graph such that the path through the vertices closely matches a given polygonal curve. The map matching problem is usually approached with the Fréchet distance matching the edges of the path as well. Here, we formally define the discrete map matching problem based on the discrete Fréchet distance. We then look at the complexities of some variations of the problem which allow for vertices in the graph to be unique or reused, and whether there is a bound on the length of the path or the number of vertices from the graph used in the path. We prove several of these problems to be NP-complete, and then conclude the paper with some open questions.

Motivation & Objective

• To formally define the discrete map matching problem based on the discrete Fréchet distance for intermittent GPS data.
• To analyze the computational complexity of variations of map matching under constraints such as unique vertex usage, bounded path length, or limited vertex count.
• To prove NP-completeness for the unique map matching (UMM) problem in planar graphs.
• To identify polynomial-time solvable cases, particularly when path length is bounded.
• To pose open questions on approximation algorithms and tractable special cases for real-world applications.

Proposed method

• Formal definition of the discrete map matching problem using the discrete Fréchet distance as the similarity measure between a GPS polygonal curve and a path in a planar graph.
• Reduction from planar 3-SAT to prove NP-completeness of the unique map matching (UMM) variant, where each graph vertex can be used at most once.
• Construction of a planar graph and polygonal curve such that a valid path with discrete Fréchet distance ≤ ε exists if and only if the 3-SAT formula is satisfiable.
• Use of variable chains, clause diamonds, and alternating connections to encode Boolean satisfiability in geometric form.
• Dynamic programming approach for the case where only path length is bounded, with recurrence relations provided for efficient computation.
• Proof that the NP-completeness result also holds under the continuous Fréchet distance, though not formally detailed here.

Experimental results

Research questions

• RQ1 Is the discrete map matching problem with unique vertex usage NP-complete for planar graphs?
• RQ2 What is the complexity of map matching when the number of vertices used in the path is bounded?
• RQ3 Can the map matching problem be solved in polynomial time when only the path length is constrained?
• RQ4 Are there approximation algorithms for the optimization versions of discrete map matching?
• RQ5 Are there special cases of the problem that are tractable for real-world GPS data with intermittent sampling?

Key findings

• The unique map matching (UMM) problem, where each vertex in the graph can be used at most once, is NP-complete for planar graphs.
• The problem remains NP-complete when the number of vertices used in the path is bounded, even for regular graphs, and under the Hausdorff distance for planar graphs.
• When only the length of the path is bounded, the problem can be solved in polynomial time using dynamic programming with provided recurrence relations.
• The reduction from planar 3-SAT to the discrete map matching problem establishes that the problem is NP-complete even under the discrete Fréchet distance.
• The NP-completeness result for the discrete Fréchet distance implies the same hardness for the continuous Fréchet distance, as independently confirmed in related work.
• The paper identifies open problems, including the complexity of the non-unique map matching with bounded size (NMMS) and the existence of good approximation algorithms.