Happy New Year to everybody. I am starting to a new blog on medium to discuss my entrepreneurial experiences and life. Check out my latest blog post titled ‘Happy New Year‘ discussing my life in 2015 and predictions for 2016. I will continue using this wordpress blog to post math-related posts. Follow me on twitter ( @kintali ) to get updates on my new blog posts on medium.
Today’s post is about the following paper, a joint work with Qiuyi Zhang, one of my advisees. Qiuyi Zhang is now a graduate student in the Mathematics department of Berkeley.
- Shiva Kintali, Qiuyi Zhang. Forbidden Directed Minors and Directed Pathwidth. (Preprint is available on my publications page)
Undirected graphs of pathwidth at most one are characterized by two forbidden minors i.e., (i) the complete graph on three vertices and (ii) the spider graph with three legs of length two each (see the following figure).
Directed pathwidth is a natural generalization of pathwidth to digraphs. We proved that digraphs of directed pathwidth at most one are characterized by a finite number of forbidden directed minors. In particular, we proved that the number of vertices in any forbidden directed minor is at most 8*160000+7. Ahem !!
This paper falls in the “directed minors” part of my research interests. In an earlier theorem, proved in April 2013 (see this earlier post), we proved that partial 1-DAGs are characterized by three forbidden directed minors. In a similar vein, I conjectured that the digraphs with directed pathwidth at most 1 are characterized by a finite number of forbidden directed minors. I assumed that the number of forbidden directed minors is number is around 100. So we started this project in May 2013 and started making a list of carefully constructed forbidden directed minors and tried to extend our techniques from partial 1-DAGs. Here is an initial list of minors we found.
All the forbidden minors we found, looked very cute and we assumed that a proof is nigh. Soon, we realized that the list is growing quickly and none of our earlier techniques are applicable. After almost an year of patient efforts and roller coaster rides, we proved our finiteness theorem in May 2014, two weeks before Qiuyi Zhang’s thesis defense. It took us 10 more months to get the paper to its current status. So this is a two year long adventure.
I am hoping to prove more theorems in the “directed minors” area in the coming years. The current paper taught me that patience and focus are big factors to make consistent progress. There should be a nice balance between `proving new theorems’ and `writing up the existing results’.
Today’s blog post is a quick announcement of “personalization” feature in Polytopix. We added a new feature that allows users to add their (possibly multiple) twitter accounts in Polytopix. The user’s twitter stream is used to personalize and rank the news articles on Polytopix. More importantly, our semantic similarity algorithms will display contextual explanatory articles along-side the news articles in the user’s feed.
On popular demand, we added a new feature called “weekly news digest” in Polytopix. Our ranking algorithm picks the top (at most 10) news articles (along with the contextual explanatory articles) from the last one week and emails them every Friday midnight. I will post the details of our ranking algorithm in a future blog post.
If you want to receive this weekly digest, subscribe on polytopix.com
Here is an example from the week 01/24/2015 – 01/30/2015