On Computing Optimal Linear Diagrams

TL;DR

This paper proves that minimizing line segments in linear diagrams is NP-hard, extends the problem to restricted settings, and introduces TSP-based algorithms that efficiently produce optimal solutions.

Contribution

It establishes the NP-hardness of the problem, extends it to new constrained scenarios, and develops TSP-based algorithms for optimal linear diagram computation.

Findings

NP-hardness of line segment minimization proven

TSP-based algorithms solve instances efficiently

Experimental results show optimal solutions within milliseconds

Abstract

Linear diagrams are an effective way to visualize set-based data by representing elements as columns and sets as rows with one or more horizontal line segments, whose vertical overlaps with other rows indicate set intersections and their contained elements. The efficacy of linear diagrams heavily depends on having few line segments. The underlying minimization problem has already been explored heuristically, but its computational complexity has yet to be classified. In this paper, we show that minimizing line segments in linear diagrams is equivalent to a well-studied NP-hard problem, and extend the NP-hardness to a restricted setting. We develop new algorithms for computing linear diagrams with minimum number of line segments that build on a traveling salesperson (TSP) formulation and allow constraints on the element orders, namely, forcing two sets to be drawn as single line segments, giving weights to sets, and allowing hierarchical constraints via PQ-trees. We conduct an experimental evaluation and compare previous algorithms for minimizing line segments with our TSP formulation, showing that a state-of-the art TSP-solver can solve all considered instances optimally, most of them within few milliseconds.