Constructive Discrepancy Minimization by Walking on The Edges

Abstract: Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer (AMS 1985): In any system of n sets in a universe of size n, there always exists a coloring which achieves discrepancy 6\sqrt{n}. The original proof of Spencer was existential in nature, and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal (FOCS 2010) gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call "Edge-Walk". Our algorithm and its analysis use only basic linear algebra and is "truly" constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

Constructive Discrepancy Minimization by Walking on The Edges

The paper "Constructive Discrepancy Minimization by Walking on The Edges" offers a novel approach to a fundamental problem in combinatorial discrepancy theory, providing a constructive algorithm for discrepancy minimization in set systems. The authors, Lovett and Meka, address the discrepancy minimization problem, which seeks a coloring to minimize discrepancy across set systems. The paper's contributions are both practical, offering an efficient algorithmic solution, and theoretical, providing new insights into longstanding conjectures in discrepancy theory.

Summary of Contributions

1. Algorithm Development: The paper introduces the algorithm "Edge-Walk," a randomized algorithm that achieves Spencer's famous bound for discrepancy in set systems. Unlike previous works, the algorithm is truly constructive, meaning it finds the solution without reliance on non-constructive existential proofs. The algorithm primarily utilizes elementary linear algebra rather than advanced semidefinite programming relaxations.

2. Theoretical Insight: Edge-Walk reveals a new avenue for proving Spencer's theorem constructively. The approach centers around simulating a constrained random walk within a defined polytope until reaching a vertex, effectively yielding a coloring that minimizes discrepancy.

3. Implications for Discrepancy Theory: The new method has widened the understanding of discrepancy minimization by not only matching Spencer's bound but also providing constructive proofs for the partial coloring lemma and extending results to the Beck-Fiala setting. This addresses constraints where elements occur in a bounded number of sets.

Strong Numerical Results

The algorithm achieves a discrepancy bound of (O(\sqrt{n \cdot \log(m/n)})) in time complexity of (\tilde{O}((n+m)^3)), efficiently matching Spencer's non-constructive bounds across different configurations. The algorithm is capable of running in polynomial time, (\tilde{O}((n+m)^3)), highlighting its practical applicability.

Implications and Future Directions

The practical results of this paper have significant implications for computational problems within computer science where minimizing the discrepancy can be critical. Applications range from derandomization techniques to resource allocation problems in systems theory.

From a theoretical perspective, the constructive approach opens pathways to potentially simplifying other discrepancy-based proofs that have traditionally relied on non-constructive methods. There is also scope for the algorithm's methodology to inform improvements in other areas of algorithmic combinatorics.

The paper also highlights a continued challenge in discrepancy theory—the Beck-Fiala conjecture—suggesting avenues where further refinement of this approach could have groundbreaking consequences in proving or disproving long standing conjectures with constructive solutions.

Conclusion

"Constructive Discrepancy Minimization by Walking on The Edges" represents a significant advancement in both algorithmic and theoretical discrepancy minimization. The results not only provide new tools for applied computational problems but also enrich the theoretical dialogue concerning constructive proofs in discrepancy theory. The innovative use of constrained random walks in polyhedral spaces presents opportunities for further research applications and theoretical developments.