Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method

A well-known theorem of Spencer shows that any set system with n sets over n elements admits a coloring of discrepancy \(O(\sqrt{n})\). While the original proof was non-constructive, recent progress brought polynomial time algorithms by Bansal, Lovett and Meka, and Rothvoss. All those algorithms are randomized, even though Bansal’s algorithm admitted a complicated derandomization.

[1]  Wojciech Banaszczyk,et al.  Balancing vectors and Gaussian measures of n-dimensional convex bodies , 1998, Random Struct. Algorithms.

[2]  M. Ziegler.  Volume 152 of Graduate Texts in Mathematics , 1995 .

[3]  Bernard Chazelle,et al.  The discrepancy method - randomness and complexity , 2000 .

[4]  Aravind Srinivasan,et al.  Improving the discrepancy bound for sparse matrices: better approximations for sparse lattice approximation problems , 1997, SODA '97.

[5]  BanaszczykWojciech.  Balancing vectors and Gaussian measures of n-dimensional convex bodies , 1998 .

[6]  N. Alon,et al.  The Probabilistic Method: Alon/Probabilistic , 2008 .

[7]  József Beck,et al.  "Integer-making" theorems , 1981, Discret. Appl. Math..

[8]  A. Giannopoulos.  On some vector balancing problems , 1997 .

[9]  Aleksandar Nikolov,et al.  Beck's Three Permutations Conjecture: A Counterexample and Some Consequences , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[10]  D. Spielman,et al.  Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem , 2013, 1306.3969.

[11]  Géza Bohus,et al.  On the Discrepancy of 3 Permutations , 1990, Random Struct. Algorithms.

[12]  Joel H. Spencer,et al.  Deterministic Discrepancy Minimization , 2011, Algorithmica.

[13]  J. Pach,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[14]  Nikhil Bansal,et al.  An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound , 2019, SIAM J. Comput..

[15]  Anastasios Zouzias,et al.  A Matrix Hyperbolic Cosine Algorithm and Applications , 2011, ICALP.

[16]  Aleksandar Nikolov,et al.  The Komlos Conjecture Holds for Vector Colorings , 2013, ArXiv.

[17]  Aravind Srinivasan,et al.  The discrepancy of permutation families , 1997 .

[18]  JOEL SPENCER,et al.  Balancing games , 1977, J. Comb. Theory B.

[19]  Nikhil Bansal,et al.  Constructive Algorithms for Discrepancy Minimization , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[20]  E. Gluskin.  EXTREMAL PROPERTIES OF ORTHOGONAL PARALLELEPIPEDS AND THEIR APPLICATIONS TO THE GEOMETRY OF BANACH SPACES , 1989 .

[21]  J. Spencer.  Six standard deviations suffice , 1985 .

[22]  József Beck,et al.  Roth’s estimate of the discrepancy of integer sequences is nearly sharp , 1981, Comb..

[23]  Sanjeev Arora,et al.  The Multiplicative Weights Update Method: a Meta-Algorithm and Applications , 2012, Theory Comput..

[24]  Shachar Lovett,et al.  Constructive Discrepancy Minimization by Walking on the Edges , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[25]  Nikhil Bansal,et al.  An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[26]  E. Chong,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .