Constructive Algorithms for Discrepancy Minimization (1002.2259v4)

Abstract: Given a set system (V,S), V={1,...,n} and S={S1,...,Sm}, the minimum discrepancy problem is to find a 2-coloring of V, such that each set is colored as evenly as possible. In this paper we give the first polynomial time algorithms for discrepancy minimization that achieve bounds similar to those known existentially using the so-called Entropy Method. We also give a first approximation-like result for discrepancy. The main idea in our algorithms is to produce a coloring over time by letting the color of the elements perform a random walk (with tiny increments) starting from 0 until they reach $-1$ or $+1$. At each time step the random hops for various elements are correlated using the solution to a semidefinite program, where this program is determined by the current state and the entropy method.

• The paper presents the first polynomial-time algorithms for discrepancy minimization problems, achieving bounds like O(n^1/2) for general set systems that match non-constructive results.
• Algorithms are provided for bounded degree set systems (O(t^1/2 log n)) and a pseudo-approximation for hereditary discrepancy, converting existential results into constructive methods.
• The methodologies combine semidefinite programming and random walk techniques, offering practical implications for areas like load balancing and resource allocation.

Constructive Algorithms for Discrepancy Minimization

The research paper by Nikhil Bansal addresses the Minimum Discrepancy Problem within set systems, delivering the first polynomial-time algorithms that approximate discrepancy bounds previously known only through existential proofs, particularly using the Entropy Method. The critical objective is to find a 2-coloring of elements in a set system that minimizes the maximum deviation from an equal distribution of colors across subsets. The implications of this work are significant as discrepancy theory branches into numerous domains of mathematics and computer science, including probabilistic algorithms, computational geometry, and more.

The paper presents three main algorithmic contributions:

1. For General Set Systems when $m = O(n)$: The paper devises a randomized algorithm that achieves an $O(n^{1/2})$ discrepancy for general set systems, matching Spencer's existential result up to constant factors. Previously, only random colorings were available for such results, which had a discrepancy bound of $O((n \log n)^{1/2})$.
2. Bounded Degree Set Systems: The algorithm achieves a discrepancy result of $O(t^{1/2} \log n)$ for systems where each element appears in at most $t$ sets. This matches the non-constructive results by Srinivasan and offers a significant advancement as it is achieved constructively.
3. Approximation of Hereditary Discrepancy: The paper introduces a pseudo-approximation result where the hereditary discrepancy is bound by $O(\lambda \log(mn))$ for hereditary discrepancy $\lambda$, providing a partial answer to a previous open question in discrepancy theory.

The methodologies presented are grounded in sophisticated applications of semidefinite programming combined with random walk techniques where the color assignment for each element evolves over time. The process incorporates constraints derived from the Entropy Method to ensure that changes are systematically correlated across elements to minimize overall discrepancy.

Strong Numerical Results and Claims

A major claim in the paper is that for any set system with $n$ elements and $n$ sets, the proposed algorithm guarantees a $O(n^{1/2})$ discrepancy with high probability in polynomial time. This aligns with the bounds achieved by existential methods but previously thought non-constructive. Additionally, the algorithm presents a dependence on $(m/n)$ that, while somewhat worse than the optimal non-constructive bounds, nonetheless constitutes a substantial step forward in algorithmic terms.

Practical and Theoretical Implications

The results of this paper have significant theoretical implications, primarily as they convert previously non-constructive results into actionable algorithmic frameworks. Practically, the methodologies could impact fields requiring efficient distribution methods or resource allocations, such as load balancing in distributed computing networks, sampling methods in computational learning, and more. The paper opens pathways for more practical applications and further research into algorithmic discrepancy minimization.

Future Developments

Going forward, there are intriguing prospects for extending these results. Potential areas of development include improving the dependency on $m$ relative to $n$ to match non-constructive bounds more closely, developing deterministic algorithms that align with these bounds, and extending these techniques to more complex or constrained set systems. Additionally, exploring the inherent bounds and implications of Hereditary Vector Discrepancy could lead to new insights and applications within combinatorial discrepancy problems.

Overall, Bansal's work provides a robust foundation for future explorations in discrepancy theory, linking mathematical concepts with algorithmic implementations effectively and paving the way for practical advancements that build on this theoretical framework.