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On Saxe's theorems about the complexity of the Distance Geometry Problem

Maël Kupperschmitt, Leo Liberti

Abstract

In 1979, James B.~Saxe published an extended summary on the complexity of the Distance Geometry Problem in the proceedings of the 17th Allerton Conference. Many of the proofs in his paper are sketches, and even the whole proofs do not have all the details. In this paper we provide a commentary to Saxe's results and hopefully more understandable versions thereof.

On Saxe's theorems about the complexity of the Distance Geometry Problem

Abstract

In 1979, James B.~Saxe published an extended summary on the complexity of the Distance Geometry Problem in the proceedings of the 17th Allerton Conference. Many of the proofs in his paper are sketches, and even the whole proofs do not have all the details. In this paper we provide a commentary to Saxe's results and hopefully more understandable versions thereof.
Paper Structure (26 sections, 2 theorems, 8 equations, 8 figures)

This paper contains 26 sections, 2 theorems, 8 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. Models of computation
  4. Computational complexity
  5. Satisfiability
  6. Graphs and their rigidity
  7. Contents of Saxe's paper saxe79
  8. Membership of EDGP${}_1$ in NP
  9. EDGP${}_1$ is strongly NP-hard
  10. EDGP${}_K$ is strongly NP-hard for any $K\ge 1$
  11. Saxe's realized gadgets $R_1,R_2$ only span $\mathbb{R}^2$
  12. Simpler gadgets that span $\mathbb{R}^K$ and their critique
  13. Gadgets that fill $\mathbb{R}^K$ and yield a rigid graph
  14. Critique of the new gadgets
  15. EDGP with real-valued edge weights
  16. ...and 11 more sections

Key Result

Proposition 3.1

There is a nondeterministic polynomial-time TM for solving EDGP${}_1$.

Figures (8)

  • Figure 1: The gadget for literals used in the 3sat$\to$EDGP${}_1$ reduction: the 3sat variables appear in vertices as either $s_j$ or $\bar{s}_j$ for $j\le n$.
  • Figure 2: The gadget for clause $i$ used in the 3sat$\to$EDGP${}_1$ reduction. There are eight additional vertices for each clause $i\le m$, labelled $c_{i1},\ldots,c_{i8}$. The vertices labeled $\mathsf{L}_{ih}$ represent the $h$-th (positive or negative) literal in clause $i$ for each $h\le 3$, since each clause has exactly three literals.
  • Figure 3: The gadget graphs $T_3$ (left) and $T_4$ (right) show that $\mathcal{G}$ is a YES instance of EDGP${}_1$ iff $\mathcal{G}$ modified so that every edge of weight $3$ (resp. $4$) is replaced by the gadget $T_3$ (resp. $T_4$).
  • Figure 4: The gadget graphs $R_1$ (left) and $R_2$ (right) show that $\mathcal{G}_{1}$ is a YES instance of the EDGP${}_{1}$ iff $\mathcal{G}_2$ is a YES instance of the EDGP${}_2$.
  • Figure 5: The gadget graphs $T_5$ (above) and $T_8$ (below) used in proving the NP-hardness of EDGP${}_K$.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Proposition 3.1
  • proof
  • Theorem 4.1
  • proof