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

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

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

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

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.

A simple demo: Springs and Nodes

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

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

The Same demo: Springs and Nodes

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

Demo!

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

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…)

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.

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.

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.

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

Thank you!