1. Distance Geometry: NP-Hard, NP-Hard to approximate
Saxe JB et al.
Presented by Jonathan D. Jou

2. NP-Hardness Recap: reduction BUILD: NP-Complete Problem Oracle
GIVEN: Distance Geometry Oracle

3. Terminology: K-embeddability 3SAT –
A graph G(V,E,W) is said to be k-embeddable when there exists a function f such that for any edge e with vertices vi,vj, |f(vi)-f(vj)|=W(e).
3SAT –
Clause
Variable
Expression

4. The Proof: Part 1
1-embeddability of integer graphs is weakly NP-Complete

5. Partition “1-embeddability” proof:
Any set of numbers can be converted into a cycle of edges; any cycle with a 1-embedding function also solves the Partition problem for that set.
Note that any cycle projected onto one dimension must by definition travel the same distance in the positive direction as the negative, because it always returns to the position of the first node.

6. A simple demo: Springs and Nodes

7. The Hard Part: Proving it’s really hard
Claim: 1-embeddability of {1,2} graphs is at least as hard as 3SAT.
The graphs:

8. Disjunct clause:
There is a viable,
1-embeddable graph
only when the
equivalent disjunct
evaluates to be true.

9. The Same demo: Springs and Nodes

10. Wait, that’s not a {1,2} graph!
Well, how about if we turn edges with weights 3 and 4 into these?

11. Demo!

12. Adding Dimensions
There are graphs which cannot be put into one dimension, no matter how you try:
Remember, the idea is to show that an NP-Hard problem is just as hard in additional dimensions, not that all k-dimensional embeddings are attainable from the graph

13. It is possible to replace every edge weighted 1 with R1 and 2 with R2
It is possible to replace every edge weighted 1 with R1 and 2 with R2. This adds an additional dimension.
Result: k-dimensional embeddings reduce to 1-dimensional embeddings, and k-embeddability is NP-hard for all integers k>0.
(but…doesn’t it only add to the third dimension? R2 is 3D, yes, but replacing 3D edges with 3D edges doesn’t make it 4D…)

14. The Fun Part: Proving it’s hard to approximate
Definition:
ε-Approximate k-embeddability:
Given a real number ε, and set {V,E,W} of vertices, edges, and weights, the set is said to be approximately embeddable if there exists a function f such that it maps every vertex in V into k-dimensional space, and for any edge e with vertices = {vi,vj}, |f(vi)-f(vj)| <W(e)+ε.

15. Note: f is still defined to map vertices to integral values.
This means that, since
the graph relies on
cycles of max length 16
any thing smaller than
1/8 will only have one
possible integral solution.

16. Extending it to all :
Using the same reasoning for changing dimensions in the exact problem, we can say that there exists some real number ε such that ε-approximate k-embeddability is NP-Hard.

17. Does this still apply to what we’re interested in?
Can atoms overlap?
Electron cloud repulsion shouldn’t allow it.
Do “edges” intersect?
This depends. Do NOEs get useful data when a lot of hydrogens are close together?
Aren’t edge values real, instead of integral?
Intuitively, this makes the problem harder: more possibilities  greater complexity

18. Thank you!