Open Problems from FOCS 2009

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 O(n^3/(log^{2}n)) 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 O(n^{2.9}) 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.