Here are some open problems (that interest me) from FOCS 2009. If you want to share an open problem, please leave a comment.

Starting with my paper…….

**1) Reducibility Among Fractional Stability Problems **[pdf]

*Shiva Kintali, Laura Poplawski, Rajmohan Rajaraman, Ravi Sundaram and Shang-Hua Teng.*

**Summary : **We resolve the computational complexity of a number of outstanding open problems with practical applications and introduce a simple PPAD-complete game (the preference game). Here is the list of problems we show to be PPAD-complete, along with the domains of practical significance: 1) Fractional Stable Paths Problem (FSPP) – Internet routing; 2) Core of Balanced Games – Economics and Game theory; 3) Scarf’s Lemma – Combinatorics; 4) Hypergraph Matching – Social Choice and Preference Systems; 5) Fractional Bounded Budget Connection Games (FBBC) – Social networks; and 6) Strong Fractional Kernel – Graph Theory.

** Open Problems : **The complexity of Discrete Ham Sandwich Problem and Necklace Splitting Problem are open. These are shown to be in PPAD by Papadimitiou in 1994.

**2) Linear systems over composite moduli** [pdf]

*Arkadev Chattopadhyay and Avi Wigderson.*

** Summary and Open Problems : **Dick Lipton already summarized their result along with open problems.

**3) Regularity Lemmas and Combinatorial Algorithms** [pdf]

*Nikhil Bansal, Ryan Williams*

** Summary : **This paper presents new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. The authors give the first asymptotic improvements on “combinatorial” algorithms for dense BMM in many years, improving on the Four Russians bound. Their use of Triangle removal lemma and Regularity lemma are particularly interesting.

**Open Problems :** Ryan mentioned the following open problems in his talk : (1) Can one construct a Weak Regular partition in time, deterministically ? (2) Can their techniques be applied to All Pairs Shortest Paths Problems ? (3) is there a Regularity concept that is better suited for Matrix Multiplication ?

**4) Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP **[pdf]

*Aaron Archer, MohammadHossein Bateni, MohammadTaghi Hajiaghayi, Howard Karloff*

** Summary : **They give a factor 1.990283 approximation algorithm for Prize-collecting TSP. This is the first result to break the barrier of 2 improving the primal-dual algorithm of Goemans and Williamson. Recently Goemans, presented a 1.91457-approximation algorithm for the prize-collecting travelling salesman problem.

**Open Problem :** Obvious open problem is to improve this approximation factor.

**5) Blackbox Polynomial Identity Testing for Depth 3 Circuits** [pdf]

* Neeraj Kayal and Shubhangi Saraf. *

**Open Problems :** This paper has a well-written open problems section with specific open problems. If you are interested in polynomial identity testing, I encourage you to read them.

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