Computing the Grothendieck constant of some graph classes

Mathematics – Combinatorics

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

7 pages

Scientific paper

Given a graph $G=([n],E)$ and $w\in\R^E$, consider the integer program ${\max}_{x\in \{\pm 1\}^n} \sum_{ij \in E} w_{ij}x_ix_j$ and its canonical semidefinite programming relaxation ${\max} \sum_{ij \in E} w_{ij}v_i^Tv_j$, where the maximum is taken over all unit vectors $v_i\in\R^n$. The integrality gap of this relaxation is known as the Grothendieck constant $\ka(G)$ of $G$. We present a closed-form formula for the Grothendieck constant of $K_5$-minor free graphs and derive that it is at most 3/2. Moreover, we show that $\ka(G)\le \ka(K_k)$ if the cut polytope of $G$ is defined by inequalities supported by at most $k$ points. Lastly, since the Grothendieck constant of $K_n$ grows as $\Theta(\log n)$, it is interesting to identify instances with large gap. However this is not the case for the clique-web inequalities, a wide class of valid inequalities for the cut polytope, whose integrality ratio is shown to be bounded by 3.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Computing the Grothendieck constant of some graph classes 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 Computing the Grothendieck constant of some graph classes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Computing the Grothendieck constant of some graph classes will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-414510

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.