Forbidden Directed Minors and Directed Pathwidth

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) K_3 the complete graph on three vertices and (ii) S_{2,2,2} 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’.

Personalization in Polytopix

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.


Weekly news digest from Polytopix

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

Here is an example from the week  01/24/2015 – 01/30/2015


Starting a Startup: Polytopix

I am inaugurating this year’s blogging with some very exciting news. I am starting a startup called Polytopix. I will be finishing my spring semester teaching responsibilities at Princeton and moving to Bay Area this summer to work full-time on Polytopix.


Meanwhile, I am actively designing, coding and deploying new algorithms, hiring R&D engineers, working on legal aspects and many more related action items. For the first time, I am using a book (instead of post-it’s) to keep track of my to-do list.


I can hear the clock ticking louder and faster than usual, perhaps because I am behind my schedule. According to my original plan (enthusiastically devised during my final semester of PhD), this blog post was supposed to appear in January 2013 !! A bunch of interesting (to say the least) events contributed generously to this delay.


What is Polytopix ?
Polytopix is aimed at adding context to news articles by analyzing the semantics of the news events. Polytopix’s algorithms retrieve news articles, categorize them, analyze their semantics and augment them with contextual explanatory articles.


Why Startup ?
The main idea of ‘semantic analysis of news’ is in my mind since my undergraduate senior thesis defense in 1999. Back then, I developed a summarization engine using lexical cohesion and information retrieval algorithms. See my very first publication here. Starting from the final semester of my PhD (Spring 2011), I started developing a semantic engine to understand and analyze news, especially financial news. I used it as a stock picker to invest my savings. It performed much better than my mutual funds. This is my first realization of the potential of ‘semantic analysis’. Polytopix applies semantic analysis to daily news articles. I am planning to “spin-off” the ‘financial news analysis’ as a separate startup. More on this, in a later blog post.


You probably heard the advice “Do not do a PhD just for the sake of doing it”. The same advice applies (with much more emphasis) to starting a startup. You should only start a startup if you are really passionate to solve a particular problem and you are strongly convinced that starting a company is the best way to solve it. I have all the right reasons to start Polytopix.


How do I feel now ?
Well, I am feeling very excited and energetic. The roadmap looks very challenging.


Review: Algorithms Unplugged, The Power of Algorithms

In my last post, I said my next post will be about one of my papers on directed minors. But that paper is taking more time to write than I expected. Meanwhile, here is my review of two algorithms textbooks: Algorithms Unplugged and The Power of Algorithms, to appear in the SIGACT book review column soon.

Merry Christmas and Happy New Year to everybody !!

book1        book2



Algorithms play an integral part in our daily lives. They are everywhere. They help us travel efficiently, retrieve relevant information from huge data sets, secure money transactions, recommend movies, books, videos, predict stock market etc. Algorithms are essentially simple extensions of our daily rational thinking process. It is very tough to think about a daily task that does not benefit from efficient algorithms. Often the algorithms to solve our daily tasks are very simple, yet their impact is tremendous.

Most of the common books on algorithms start with sorting, searching, graph algorithms and conclude with NP-completeness and perhaps some approximation and online algorithms. The breadth of algorithms cannot be covered by a single book. Algorithms Unplugged and The Power of Algorithms take different approach compared to standard Algorithms textbooks. They are aimed at explaining several basics algorithms (written by multiple authors) in an intuitive manner with real-life examples, without compromising the details. This is how algorithms should be taught.

Algorithms Unplugged

This book is divided into four major parts. Each part has several chapters. Here is an overview of these parts and chapters.

Part I: The first part is about sorting and systematic search i.e., finding things quickly. Chapter 1 introduces binary search, one of the most basic search strategies. A recursive implementation of binary search is explained using an intuitive example to find a CD is a sorted sequence of CD’s. Chapter 2 explains insertion sort, one of the most intuitive comparison-based sorting algorithms and Chapter 3 explains merge sort and quick sort, two sorting algorithms based on the divide and conquer paradigm. Chapter 4 explains bitonic sorting circuit to implement a parallel sorting algorithm. Chapter 5 describes topological sorting and explains how to schedule jobs without violating any dependencies between jobs. Chapter 6 considers string searching problem and explains the Boyer-Moore-Horspool algorithm. Chapter 7 considers the search problem in several real-world applications and explains the depth first search algorithm. Chapter 8 explains how to escape from a dark labyrinth using Pledge’s algorithm. Chapter 9 defines strongly-connected components in directed graphs and explains how to efficiently find directed cycles. Chapter 10 introduces basic principles of search engines, introduces PageRank and explains how to find relevant pages in the World-Wide Web.

Part II: The second part deals with arithmetic problems, number theoretic, cryptographic, compression and coding algorithms. Chapter 11 presents Karatsuba’s method of multiplying long integers that is much more efficient than the basic grade school method. Chapter 12 explains how to compute the greatest common divisor of two numbers using the centuries old Euclidean algorithm. Chapter 13 explains the Sieve of Eratosthenes, a practical algorithm to compute the table of prime numbers. Chapter 14 introduces the basics of one-way functions which play a crucial role in the following chapters. Chapter 15 presents One-Time-Pad, a basic symmetric cryptographic algorithm. Chapter 16 explains Public-Key Cryptography, an asymmetric cryptographic method using different keys for encryption and decryption. Chapter 17 explains how to share a secret in such a way that all participants must meet to decode the secret. Chapter 18 presents a method to play poker by email using cryptographic methods. Chapter 19 and 20 presents fingerprinting and hashing techniques to compress large data sets so that they can be compared using only a few bits. Chapter 21 introduces the basics of coding algorithms to protect data against errors and loss.

Part III: The third part deals with strategic thinking and planning. Chapter 22 discusses broadcasting algorithms to disseminate information quickly. Chapter 23 presents algorithms to convert numbers into english words. Chapter 24 explains majority algorithms with applications and extensions. Chapter 25 explains how to generate random numbers. Chapter 26 discusses winning strategies for a matchstick game. Chapter 27 discusses algorithms to schedule sports leagues. Chapter 28 characterizes eulerian circuits and presents algorithms to find them. Chapter 29 details how to approximately draw circles on a pixelated screen. Chapter 30 explains Gauss-Seidel iterative method. Chapter 31 presents a dynamic programming algorithm to compute distance between two DNA sequences.

Part IV: The final part is about optimization problems. Chapter 32, 33 and 34 describes shortest path, minimum spanning tree and maximum-flow algorithms, three basic optimization problems. Chapter 35 discusses stable marriage problem and presents an algorithm to find a stable matching in a bipartite graph. Chapter 36 explains an algorithm to find the smallest enclosing cycle of a given set of points. Chapter 37 presents online algorithms for the Ski-Rental and Paging problems. Chapter 38 and 39 discusses the Bin-Packing and the Knapsack problems. Chapter 40 discusses the Traveling Salesman Problem, one of the most important optimization problems that challenged mathematicians and computer scientists for decades. Chapter 41 introduces Simulated Annealing method to solve a basic tiling problem and the Traveling Salesman Problem.

At the end of every chapter there are references for further reading. Readers are highly encouraged to go through these references to get better understanding of the corresponding concepts.

The Power of Algorithms

This book is divided into two major parts. Each part has several chapters. Here is an overview of these parts and chapters.

Part I: The first part is divided into three chapters. Chapter 1 gives a historical perspective of algorithms, origin of the word {\em algorithm}, recreational algorithms and reasoning with computers. Chapter 2 aims at explaining how to design algorithms by introducing the basics of graph theory and two algorithms techniques: the backtracking technique and the greedy technique. Chapter 3 quickly introduces the complexity classes P and NP and the million dollar P vs NP problem.

Part II: The second part is aimed at explaining several algorithms of daily life. Chapter 4 explains the Shortest Path problems, one of the basic optimization problems. Chapter 5 discusses the basics of Internet and Web Graphs and explains several related algorithms to Crawl, Index and Search the Internet. Chapter 6 discusses the basics of cryptographic algorithms such as RSA and digital signatures. Chapter 7 discusses biological algorithms. Chapter 8 explains networks algorithms with transmission delays. Chapter 9 discusses algorithms for auctions and games. It presents Prisoner’s dilemma, Coordination games, Randomized strategies, Zero-sum games, Nash’s Theorem, Spurners Lemma, Vickery-Clarke-Groves auctions and competitive equilibria. Chapter 10 explains the power of randomness and its role in complexity theory.

At the end of every chapter there are Bibliographic Notes with several pointers for further reading.


Overall I found these two books very interesting and well-written. There is a nice balance between informal introductions and formal algorithms. It was a joy for me to read these books and I recommend them to anyone (including beginners) who is curios to learn some of the basic algorithms that we use in our daily lives in a rigorous way. I strongly encourage you to read these two books even if you have already read a bunch of algorithms books.

There is no prerequisite to follow these books, except for a reasonable mathematical maturity and perhaps some familiarity with basic constructs of at least one programming language. They can be used as a self-study text by undergraduate and advanced high-school students. In terms of being used in a course, some of the topics in these books can be used in an undergraduate algorithms course. I would definitely suggest that you get them for yourself or your university/department library.