Directed Minors III. Directed Linked Decompositions

This is a short post about the following paper in my directed minor series :

  • Shiva Kintali. “Directed Minors III. Directed Linked Decompositions“. Preprint available on my publications page.

Thomas [Tho'90] proved that every undirected graph admits a linked tree decomposition of width equal to its treewidth. This theorem is a key technical tool for proving that every set of bounded treewidth graphs is well-quasi-ordered. An analogous theorem for branch-width was proved by Geelen, Gerards and Whittle [GGW'02]. They used this result to prove that all matroids representable over a fixed finite field and with bounded branch-width are well-quasi-ordered under minors. Kim and Seymour [KS'12] proved that every semi-complete digraph admits a linked directed path decomposition of width equal to its directed pathwidth. They used this result to show that all semi-complete digraphs are well-quasi-ordered under “strong” minors.

In this paper, we generalize Thomas’s theorem to all digraphs.

Theorem : Every digraph G admits a linked directed path decomposition and a linked DAG decomposition of width equal to its directed pathwidth and DAG-width respectively.

The above theorem is crucial to prove well-quasi-ordering of some interesting classes of digraphs. I will release Directed Minors IV soon. Stay tuned !!

Directed Width Parameters and Circumference of Digraphs

This is a short post about the following paper :

  • Shiva Kintali. “Directed Width Parameters and Circumference of Digraphs“. Preprint available on my publications page.

We prove that the directed treewidth, DAG-width and Kelly-width of a digraph are bounded above by its circumference plus one. This generalizes a theorem of Birmele stating that the treewidth of an undirected graph is at most its circumference. 

Theorem : Let G be a digraph of circumference l. Then the directed treewidth, DAG-width and Kelly-width of G are at most l + 1.

The above theorem can be seen as a mini mini mini directed grid minor theorem. I will be using this theorem in future papers to make progress towards a directed grid minor theorem. Stay tuned !!

Open problems for 2014

Wish you all a Very Happy New Year. Here is a list of my 10 favorite open problems for 2014. They belong to several research areas inside discrete mathematics and theoretical computer science. Some of them are baby steps towards resolving much bigger open problems. May this new year shed new light on these open problems.

  • 2. Optimization : Improve the approximation factor for the undirected graphic TSP. The best known bound is 7/5 by Sebo and Vygen.
  • 3. Algorithms : Prove that the tree-width of a planar graph can be computed in polynomial time (or) is NP-complete.
  • 4. Fixed-parameter tractability : Treewidth and Pathwidth are known to be fixed-parameter tractable. Are directed treewidth/DAG-width/Kelly-width (generalizations  of  treewidth) and directed pathwidth (a generalization of pathwidth) fixed-parameter tractable ? This is a very important problem to understand the algorithmic and structural differences between undirected and directed width parameters.
  • 5. Space complexity : Is Planar ST-connectvity in logspace ? This is perhaps the most natural special case of the NL vs L problem. Planar ST-connectivity is known to be in UL \cap coUL. Recently, Imai, Nakagawa, Pavan, Vinodchandran and Watanabe proved that it can be solved simultaneously in polynomial time and approximately O(√n) space.
  • 6. Metric embedding : Is the minor-free embedding conjecture true for partial 3-trees (graphs of treewidth 3) ? Minor-free conjecture states that “every minor-free graph can be embedded in l_1 with constant distortion. The special case of planar graphs also seems very difficult. I think the special case of partial 3-trees is a very interesting baby step.
  • 7. Structural graph theory : Characterize pfaffians of tree-width at most 3 (i.e., partial 3-trees). It is a long-standing open problem to give a nice characterization of pfaffians and design a polynomial time algorithm to decide if an input graph is a pfaffian. The special of partial 3-trees is an interesting baby step.
  • 8. Structural graph theory : Prove that every minimal brick has at least four vertices of degree three. Bricks and braces are defined to better understand pfaffians. The characterization of pfaffian braces is known (more generally characterization of bipartite pfaffians is known). To understand pfaffians, it is important to understand the structure of bricks. Norine,Thomas proved that every minimal brick has at least three vertices of degree three and conjectured that every minimal brick has at least cn vertices of degree three.
  • 9. Communication Complexity : Improve bounds for the log-rank conjecture. The best known bound is O(\sqrt{rank})
  • 10. Approximation algorithms : Improve the approximation factor for the uniform sparsest cut problem. The best known factor is O(\sqrt{logn}).

Here are my conjectures for 2014 :)

  • Weak Conjecture : at least one of the above 10 problems will be resolved in 2014.
  • Conjecture : at least five of the above 10 problems will be resolved in 2014.
  • Strong Conjecture : All of the above 10 problems will be resolved in 2014.

Have fun !!

PolyTopix

In the last couple of years, I developed some (research) interest in recommendation algorithms and speech synthesis. My interests in these areas are geared towards developing an automated personalized news radio.

Almost all of us are interesting in consuming news. In this internet age, there is no dearth of news sources. Often we have too many sources. We tend to “read” news from several sources / news aggregators, spending several hours per week. Most of the time we are simply interested in the top and relevant headlines.

PolyTopix is my way of simplifying the process of consuming top and relevant news. The initial prototype is here. The website “reads” several news tweets (collected from different sources) and ordered based on a machine learning algorithm. Users can login and specify their individual interests (and zip code) to narrow down the news.

logo

Try PolyTopix let me know your feedback. Here are some upcoming features :

  • Automatically collect weather news (and local news) based on your location.
  • Reading more details of most important news.
  • News will be classified as exciting/sad/happy etc., (based on a machine learning algorithm) and read with the corresponding emotional voice.

Essentially PolyTopix is aimed towards a completely automated and personalized news radio, that can “read” news from across the world anytime with one click.

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Forbidden Directed Minors and Kelly-width

Today’s post is about the following paper, a joint work with Qiuyi Zhang, one of my advisees. Qiuyi Zhang is an undergraduate (rising senior) in our mathematics department.

  • Shiva Kintali, Qiuyi Zhang. Forbidden Directed Minors and Kelly-width. (Preprint available on my publications page)

It is well-known that an undirected graph is a partial 1-tree (i.e., a forest) if and only if it has no K_3 minor. We generalized this characterization to partial 1-DAGs. We proved that partial 1-DAGs are characterized by three forbidden directed minors, K_3, N_4 and M_5, shown in the following figure. We named the last two graphs as N_4 and M_5 because their bidirected edges resemble the letters N and M.

minors

Partial k-trees characterize bounded treewidth graphs. Similarly, partial k-DAGs characterize bounded Kelly-width digraphs. Kelly-width is the best known generalization of treewidth to digraphs.

As mentioned in the paper, I have a series of upcoming papers (called Directed Minors) making progress towards a directed graph minor theorem (i.e., all digraphs are well-quasi-ordered by the directed minor relation). For more details of the directed minor relation, read the current paper. I will post about the upcoming results as and when the preprints are ready.

Book Review of “Boosting : Foundations and Algorithms”

Following is my review of Boosting : Foundations and Algorithms (by Robert E. Schapire and Yoav Freund) to appear in the  SIGACT book review column soon.

Boosting

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Book : Boosting : Foundations and Algorithms (by Robert E. Schapire and Yoav Freund)
Reviewer : Shiva Kintali

Introduction

You have k friends, each one earning a small amount of money (say 100 dollars) every month by buying and selling stocks. One fine evening, at a dinner conversation, they told you their individual “strategies” (after all, they are your friends). Is it possible to “combine” these individual strategies and make million dollars in an year, assuming your initial capital is same as your average friend ?

You are managing a group of k “diverse” software engineers each one with only an “above-average” intelligence. Is it possible to build a world-class product using their skills ?

The above scenarios give rise to fundamental theoretical questions in machine learning and form the basis of Boosting. As you may know, the goal of machine learning is to build systems that can adapt to their environments and learn from their experience. In the last five decades, machine learning has impacted almost every aspect of our life, for example, computer vision, speech processing, web-search, information retrieval, biology and so on. In fact, it is very hard to name an area that cannot benefit from the theoretical and practical insights of machine learning.

The answer to the above mentioned questions is Boosting, an elegant method for driving down the error of the combined classifier by combining a number of weak classifiers. In the last two decades, several variants of Boosting are discovered. All these algorithms come with a set of theoretical guarantees and made a deep practical impact on the advances of machine learning, often providing new explanations for existing prediction algorithms.

Boosting : Foundations and Algorithms, written by the inventors of Boosting, deals with variants of AdaBoost, an adaptive boosting method. Here is a quick explanation of the basic version of AdaBoost.

AdaBoost makes iterative calls to the base learner. It maintains a distribution over training examples to choose the training sets provided to the base learner on each round. Each training example is assigned a weight, a measure of importance of correctly classifying an example on the current round. Initially, all weights are set equally. On each round, the weights of incorrectly classified examples are increased so that, “hard” examples get successively higher weight. This forces the base learner to focus its attention on the hard example and drive down the generalization errors.

AdaBoost is fast and easy to implement and the only parameter to tune is the number of rounds. The actual performance of boosting is dependent on the data.

Summary

Chapter 1 provides a quick introduction and overview of Boosting algorithms with practical examples. The rest of the book is divided into four major parts. Each part is divided into 3 to 4 chapters.

Part I studies the properties and effectiveness of AdaBoost and theoretical aspects of minimizing its training and generalization errors. It is proved that AdaBoost drives the training error down very fast (as a function of the error rates of the weak classifiers) and the generalization error arbitrarily close to zero. Basic theoretical bounds on the generalization error show that AdaBoost overfits, however empirical studies show that AdaBoost does not overfit. To explain this paradox, a margin-based analysis is presented to explain the absence of overfitting.
Part II explains several properties of AdaBoost using game-theoretic interpretations. It is shown that the principles of Boosting are very intimately related to the classic min-max theorem of von Neumann. A two-player (the boosting algorithm and the weak learning algorithm) game is considered and it is shown that AdaBoost is a special case of a more general algorithm for playing a repeated game. By reversing the roles of the players, a solution is obtained for the online prediction model thus establishing a connection between Boosting and online learning. Loss minimization is studied and AdaBoost is interpreted as an abstract geometric framework for optimizing a particular objective function. More interestingly, AdaBoost is viewed as a special case of more general methods for optimization of an objective function such as coordinate descent and functional gradient descent.

Part III explains several methods of extending AdaBoost to handle classifiers with more than two output classes. AdaBoost.M1, AdaBoost.MH and AdaBoost.MO are presented along with their theoretical analysis and practical applications. RankBoost, an extension of AdaBoost to study ranking problems is studied. Such an algorithm is very useful, for example, to rank webpages based on their relevance to a given query.

Part IV is dedicated to advanced theoretical topics. Under certain assumptions, it is proved that AdaBoost can handle noisy-data and converge to the best possible classifier. An optimal boost-by-majority algorithm is presented. This algorithm is then modified to be adaptive leading to an algorithm called BrownBoost.

Many examples are given throughout the book to illustrate the empirical performance of the algorithms presented. Every chapter ends with Summary and Bibliography mentioning the related publications. There are well-designed exercises at the end of every chapter. Appendix briefly outlines some required mathematical background.

Opinion

Boosting book is definitely a very good reference text for researchers in the area of machine learning. If you are new to machine learning, I encourage you to read an introductory machine learning book (for example, Machine Learning by Tom M. Mitchell) to better understand and appreciate the concepts. In terms of being used in a course, a graduate-level machine learning course can be designed from the topics covered in this book. The exercises in the book can be readily used for such a course.

Overall this book is a stimulating learning experience. It has provided me new perspectives on theory and practice of several variants of Boosting algorithms. Most of the algorithms in this book are new to me and I had no difficulties following the algorithms and the corresponding theorems. The exercises at the end of every chapter made these topics much more fun to learn.

The authors did a very good job compiling different variants of Boosting algorithms and achieved a nice balance between theoretical analysis and practical examples. I highly recommend this book for anyone interested in machine learning.

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FOCS 2013 Accepted Papers (with pdf files)

FOCS 2013 accepted paper list is here. Following are PDF pointers to online versions.

  • OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings, by Jelani Nelson and Huy L. Nguyen [arXiv]
  • A Polynomial Time Algorithm for Lossy Population Recovery, by Ankur Moitra and Michael Saks [arXiv]
  • Direct products in communication complexity, by Mark Braverman, Anup Rao, Omri Weinstein and Amir Yehudayoff [ECCC]
  • Arithmetic circuits: A chasm at depth three, by Ankit Gupta, Pritish Kamath, Neeraj Kayal and Ramprasad Saptharishi [ECCC]
  • Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs, by Joseph Cheriyan and Laszlo A. Vegh [arXiv]
  • The Parity of Directed Hamiltonian Cycles, by Andreas Björklund and Thore Husfeldt [arXiv]
  • Approximating Minimization Diagrams and Generalized Proximity Search, by Sariel Har-Peled and Nirman Kumar [arXiv]
  • Quantum 3-SAT is QMA1-complete, by David Gosset and Daniel Nagaj [arXiv]
  • The Price of Stability for Undirected Broadcast Network Design with Fair Cost Allocation is Constant, by Vittorio Bilò, Michele Flammini and Luca Moscardelli
  • Strong LTCs with inverse poly-log rate and constant soundness, by Michael Viderman [ECCC]
  • All-or-nothing multicommodity flow problem with bounded fractionality in planar graphs, by Ken-ichi Kawarabayashi and Yusuke Kobayashi
  • Approximating Bin Packing within O(log OPT * log log OPT) bins, by Thomas Rothvoss [arXiv]
  • Common information and unique disjointness, by Gábor Braun, and Sebastian Pokutta [ECCC]
  • Chasing the k-colorability threshold, by Amin Coja-Oghlan and Dan Vilenchik [arXiv]
  • Three-player entangled XOR games are NP-hard to approximate, by Thomas Vidick [arXiv]
  • Fully Dynamic $(1+\epsilon)$-Approximate Matchings, by Manoj Gupta, and Richard Peng [arXiv]
  • Estimating the distance from testable affine-invariant properties, by Hamed Hatami and Shachar Lovett [arXiv]
  • Approximation algorithms for Euler genus and related problems, by Chandra Chekuri and Anastasios Sidiropoulos [arXiv]
  • Polar Codes: Speed of polarization and polynomial gap to capacity, by Venkatesan Guruswami and Patrick Xia [arXiv]
  • An Optimal Randomized Online Algorithm for Reordering Buffer Management, by Noa Avigdor-Elgrabli and Yuval Rabani [arXiv]
  • Approximation Schemes for Maximum Weight Independent Set of Rectangles, by Anna Adamaszek and Andreas Wiese
  • Playing Non-linear Games with Linear Oracles, by Dan Garber, and Elad Hazan
  • Faster Canonical Forms for Strongly Regular Graphs, by Laszlo Babai, Xi Chen, Xiaorui Sun, Shang-Hua Teng and John Wilmes
  • Constant rate PCPs for circuit-SAT with sublinear query complexity, by Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir and Henning Stichtenoth [pdf] [video]
  • How to Approximate A Set Without Knowing Its Size In Advance, by Rasmus Pagh, Gil Segev and Udi Wieder [arXiv]
  • Quasipolynomial-time Identity Testing of Non-Commutative and Read-Once Oblivious Algebraic Branching Programs, by Michael A. Forbes, and Amir Shpilka [arXiv]
  • On Kinetic Delaunay Triangulations: A Near Quadratic Bound for Unit Speed Motions, by Natan Rubin
  • Element Distinctness, Frequency Moments, and Sliding Windows, by Paul Beame, Raphael Clifford and Widad Machmouchi
  • Spatial mixing and approximation algorithms for graphs with bounded connective constant, by Alistair Sinclair, Piyush Srivastava and Yitong Yin
  • Algebraic Algorithms for b-Matching, Shortest Undirected Paths, and f-Factors, by Harold N. Gabow and Piotr Sankowski [arXiv]
  • A linear time approximation scheme for Euclidean TSP, by Yair Bartal and Lee-Ad Gottlieb
  • Interactive Coding, Revisited, by Kai-Min Chung, Rafael Pass and Sidharth Telang [ePrint]
  • Improved approximation for 3-dimensional matching via bounded pathwidth local search, by Marek Cygan [arXiv]
  • Strong Backdoors to Bounded Treewidth SAT, by Serge Gaspers and Stefan Szeider [arXiv]
  • Explicit Subspace Designs, by Venkatesan Guruswami and Swastik Kopparty [ECCC]
  • Dynamic Approximate All-Pairs Shortest Paths: Breaking the $ \tilde O(mn) $ Barrier and Derandomization, Monika Henzinger, Sebastian Krinninger and Danupon Nanongkai
  • On Randomized Memoryless Algorithms for the Weighted $k$-server Problem, by Ashish Chiplunkar and Sundar Vishwanathan [arXiv]
  • Constant-Round Concurrent Zero Knowledge From P-Certificates, by Kai-Min Chung, Huijia Lin and Rafael Pass [ePrint]
  • Learning Sums of Independent Integer Random Variables, by Constantinos Daskalakis, Ilias Diakonikolas, Ryan O’Donnell, Rocco A. Servedio and Li-Yang Tan [pdf]
  • Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses, by Parinya Chalermsook, Bundit Laekhanukit, and Danupon Nanongkai
  • Online Node-weighted Steiner Forest and Extensions via Disk Paintings, by Mohammad Taghi Hajiaghayi, Vahid Liaghat, and Debmalya Panigrahi
  • Klee’s Measure Problem Made Easy, by Timothy M. Chan
  • Extractors for a Constant Number of Independent Sources with Polylogarithmic Min-Entropy, by Xin Li [arXiv]
  • Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back, by Aleksander Madry
  • Fourier sparsity, spectral norm, and the Log-rank conjecture, Hing Yin Tsang, Chung Hoi Wong, Ning Xie and Shengyu Zhang [arXiv]
  • Layered Separators for Queue Layouts, 3D Graph Drawing and Nonrepetitive Coloring, by Vida Dujmović, Pat Morin, and David R. Wood
  • Average Case Lower Bounds for Monotone Switching Networks, by Yuval Filmus, Toniann Pitassi, Robert Robere and Stephen A. Cook [arXiv]
  • PCPs via low-degree long code and hardness for constrained hypergraph coloring, by Irit Dinur and Venkatesan Guruswami
  • The planar directed k-Vertex-Disjoint Paths problem is fixed-parameter tractable, by Marek Cygan, Dániel Marx, Marcin Pilipczuk and Michał Pilipczuk [arXiv]
  • On the communication complexity of sparse set disjointness and exists-equal problems, Mert Sağlam and Gábor Tardos [arXiv]
  • The Simple Economics of Approximately Optimal Auctions, Saeed Alaei, Hu Fu, Nima Haghpanah and Jason Hartline [arXiv]
  • Efficient Accelerated Coordinate Descent Methods and Faster Algorithms for Solving Linear Systems, by Yin Tat Lee and Aaron Sidford [arXiv]
  • Nearly Maximum Flows in Nearly Linear Time, by Jonah Sherman [arXiv]
  • Improved Average-Case Lower Bounds for DeMorgan Formula Size, by Ilan Komargodski, Ran Raz and Avishay Tal [ECCC]
  • Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas, by Vitaly Feldman and Jan Vondrak
  • Towards a better approximation for Sparsest Cut?, by Sanjeev Arora, Rong Ge and Ali Kemal Sinop [arXiv]
  • Bandits with Knapsacks, Ashwinkumar Badanidiyuru, Robert Kleinberg and Aleksandrs Slivkins [arXiv]
  • Approximate Constraint Satisfaction Requires Large LP Relaxations, by Siu On Chan, James R. Lee, Prasad Raghavendra and David Steurer
  • Simple Tabulation, Fast Expanders, Double Tabulation, and High Independence, by Mikkel Thorup
  • An LMP O(log n)-Approximation Algorithm for Node Weighted Prize Collecting Steiner Tree, by Jochen Koenemann, Sina Sadeghian and Laura Sanità [arXiv]
  • A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits, by Russell Impagliazzo, Ramamohan Paturi, and Stefan Schneider [arXiv]
  • The Complexity of Approximating Vertex Expansion, by Anand Louis, Prasad Raghavendra and Santosh Vempala [arXiv]
  • Tight Bounds for Set Disjointness in the Message Passing Model, by Mark Braverman, Faith Ellen, Rotem Oshman, Toni Pitassi and Vinod Vaikuntanathan [arXiv]
  • Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees, Adam Marcus, Daniel A. Spielman and Nikhil Srivastava [arXiv]
  • Candidate Indistinguishability Obfuscation and Functional Encryption for all circuits, by Sanjam Garg, Craig Gentry, Shai Halevi, Mariana Raykova, Amit Sahai, and Brent Waters
  • Iterative Row Sampling, by Mu Li, Gary L. Miller, and Richard Peng [arXiv]
  • Simultaneous Resettability from One-Way Functions, Kai-Min Chung, Rafail Ostrovsky, Rafael Pass and Ivan Visconti
  • Rational Protocol Design: Cryptography Against Incentive-driven Adversaries, by Juan Garay , Jonathan Katz, Ueli Maurer, Bjoern Tackmann and Vassilis Zikas
  • Non-positive curvature, and the planar embedding conjecture, by Anastasios Sidiropoulos [arXiv]
  • Local Privacy and Statistical Minimax Rates, by John C. Duchi, Michael I. Jordan, and Martin J. Wainwright [arXiv]
  • The Moser-Tardos Framework with Partial Resampling, by David G. Harris and Aravind Srinivasan
  • On Clustering Induced Voronoi Diagrams, Danny Z. Chen, Ziyun Huang, Yangwei Liu, and Jinhui Xu
  • A forward-backward single-source shortest paths algorithm, by David Wilson and Uri Zwick
  • An O(c^k) n 5-Approximation Algorithm for Treewidth, by Hans L. Bodlaender, Paal G. Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov and Michal Pulipczuk [arXiv]
  • Understanding Incentives: Mechanism Design becomes Algorithm Design, by Yang Cai, Constantinos Daskalakis, and S. Matthew Weinberg [arXiv]
  • Nondeterministic Direct Product Reductions and the Success Probability of SAT Solvers, by Andrew Drucker
  • From Unprovability to Environementally Friendly Protocols, by Ran Canetti, Huijia Lin and Rafael Pass
  • Adaptive Seeding in Social Networks, by Lior Seeman and Yaron Singer
  • Coupled-Worlds Privacy: Exploiting Adversarial Uncertainty in Statistical Data Privacy, by Raef Bassily, Adam Groce, Jonathan Katz and Adam Smith

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TrueShelf 1.0

One year back (on 6/6/12) I announced a beta version of TrueShelf, a social-network for sharing exercises and puzzles especially in mathematics and computer science. After an year of testing and adding new features, now I can say that TrueShelf is out of beta.

TrueShelf turned out to be a very useful website. When students ask me for practice problems (or books) on a particular topic, I simply point them to trueshelf and tell them the tags related to that topic. When I am advising students on research projects, I first tell them to solve all related problems (in the first couple of weeks) to prepare them to read research papers.

Here are the features in TrueShelf 1.0.

  • Post an exercise (or) multiple-choice question (or) video (or) notes.
  • Solve any multiple-choice question directly on the website.
  • Add topic and tags to any post
  • Add source or level (high-school/undergraduate/graduate/research).
  • Show text-books related to a post
  • Show related posts for every post.
  • View printable version (or) LaTex version of any post.
  • Email / Tweet / share on facebook (or) Google+ any post directly from the post.
  • Add any post to your Favorites
  • Like (a.k.a upvote) any post.

Feel free to explore TrueShelf, contribute new exercises and let me know if you have any feedback (or) new features you want to see. You can also follow TrueShelf on facebooktwitter and google+. Here is a screenshot highlighting the important features.

trueshelf

Graceful Diameter-6 Trees

Graceful Tree Conjecture is one of my favorite open problems (See this earlier post). Trees with diameter 4 and 5 are known to be graceful a decade ago.

One of my advisees, Matt Superdock, made progress towards proving that all diameter 6 trees are graceful. Matt is an undergraduate senior in our Mathematics Department. He proved that an interesting class of diameter 6 trees are graceful. His thesis is available here.

I hope his work motivates more researchers to make progress towards resolving Graceful Tree Conjecture, particularly for diameter 6 trees. Matt’s work really tests the limit of current techniques. Perhaps we need new techniques/insights to prove that all diameter 6 trees are graceful.

 

 

 

Happy Birthday Paul Erdos

Today (March 26 2013) is the 100th Birthday of Paul Erdos. The title of my Blog is inspired by one of his famous sayings “My Brain is Open”. In one of my earlier posts I mentioned a book titled “The Man Who Loved Only Numbers” about his biography.

Paul Erdos published more than 1500 papers. Most of them left a legacy of open problems and conjectures. What is your favorite open problem from Erdos’s papers ? Leave a comment. Hope we can solve some of his open problems during this special year.

Here are some interesting links :

If you know any interesting Erdos links, leave a comment.

Here is a painting of Paul Erdos, I made couple years back.

paul_erdos_painting