Undecidability of linear inequalities in graph homomorphism densities

Details Undecidability of linear inequalities in graph homomorphism densities Undecidability of linear inequalities in graph homomorphism densities

2010-05-13

arxiv.org/abs/1005.2382v3

Mathematics

Combinatorics

17 pages

Scientific paper

The purpose of this article is to show that even the most elementary problems in asymptotic extremal graph theory can be highly non-trivial. We study linear inequalities between graph homomorphism densities. In the language of quantum graphs the validity of such an inequality is equivalent to the positivity of a corresponding quantum graph. Similar to the setting of polynomials, a quantum graph that can be represented as a sum of squares of labeled quantum graphs is necessarily positive. Lov\'asz asks whether the opposite is also true. We answer this question and also a related question of Razborov in the negative by introducing explicit valid inequalities that do not satisfy the required conditions. Our solution to these problems is based on a reduction from real multivariate polynomials and uses the fact that there are positive polynomials that cannot be expressed as sums of squares of polynomials. It is known that the problem of determining whether a multivariate polynomial is positive is decidable. Hence it is very natural to ask "Is the problem of determining the validity of a linear inequality between homomorphism densities decidable?" We give a negative answer to this question which shows that such inequalities are inherently difficult in their full generality. Furthermore we deduce from this fact that the analogue of Artin's solution to Hilbert's seventeenth problem does not hold in the setting of quantum graphs.

Affiliated with

Also associated with

No associations

Rating

Undecidability of linear inequalities in graph homomorphism densities does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Undecidability of linear inequalities in graph homomorphism densities, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Undecidability of linear inequalities in graph homomorphism densities will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-641148