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Constructive Discrepancy Minimization by Walking on The Edges
Shachar Lovett, Raghu Meka
TL;DR
The paper tackles the problem of efficiently finding low-discrepancy colorings after Spencer’s existential proof left construction open. It introduces Edge-Walk, a restricted random walk analyzed with basic linear algebra, and obtains constructive algorithms matching Spencer’s bounds while extending to the Beck–Fiala setting.
Problem
Efficiently constructing the low-discrepancy coloring guaranteed by Spencer’s theorem remained unresolved because his proof was non-constructive.
Method
The paper uses a restricted random walk called Edge-Walk, analyzed with basic linear algebra, to construct colorings and prove the partial coloring lemma.
Results
The randomized algorithm runs in ˜O((n+m)^3) time with probability at least 1/2 and achieves discrepancy below K√(n · log2(m/n)); related techniques extend to Beck–Fiala bounds.
Takeaways & Limitations
The approach provides a new elementary, truly constructive proof of Spencer’s result and extends constructive discrepancy minimization to the Beck–Fiala setting.
Takeaways & Limitations
The method does not strictly produce a proper partial coloring because variables not equal to ±1 may take arbitrary values in (−1,1).
Abstract
from arXiv · showhide
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.
1 Introduction
The paper addresses the longstanding challenge of efficiently constructing low-discrepancy colorings, replacing Spencer’s existential proof with randomized polynomial-time algorithms based on elementary techniques. Its bounds match Spencer’s result across parameter ranges and extend constructively to the Beck–Fiala setting.
- Problem: Discrepancy minimization seeks a ±1 coloring of a set system that minimizes the maximum absolute signed sum over its sets.The paper denotes the minimum achievable discrepancy by disc(S).
- Prior result: Spencer’s theorem guarantees discrepancy below K√(n · log2(m/n)), with K=6 when m=n.For m=O(n), the bound is O(√(n log n)).
- Open problem: Spencer’s pigeonhole-principle proof was non-constructive, leaving no efficient way to find a good coloring without enumerating colorings.This non-constructivity created a longstanding open algorithmic problem.
- Main result: The paper gives a new randomized algorithm running in time ˜O((n+m)^3) that finds a coloring with discrepancy below K√(n · log2(m/n)) with probability at least 1/2.The result applies to arbitrary set systems and uses a universal constant K.
- Main result: For m=n, the paper takes K=13 and matches Spencer’s bound across all ranges of m and n, unlike Bansal’s additional Ω(log(m/n)) loss.The comparison concerns the stated discrepancy bound.
- Extension: The techniques also yield a randomized ˜O((n+m)^5)-time algorithm in the Beck–Fiala setting with discrepancy below K√(t · log n).Here each element belongs to at most t sets, and the result matches the best known constructive bounds.
2 Outline of Algorithm
The algorithm recursively constructs a fractional partial coloring using Edge-Walk, a constrained randomized walk inside a polytope. Its analysis shows that the walk fixes many coordinates while preserving discrepancy constraints, yielding an efficient randomized construction.
- Partial coloring: The method first seeks a partial coloring χ: V → [−1,1] that fixes at least cn coordinates at ±1, then recursively colors the remaining variables.The resulting full coloring follows from a geometrically decreasing discrepancy analysis.
- Partial coloring: The partial coloring lemma is formulated for arbitrary vectors v_1,...,v_m and a starting point x_0 ∈ [−1,1]^n.Set discrepancy is represented by inner products ⟨χ,v_i⟩.
- Implementation: The implemented algorithm stops after a prescribed time and approximates the continuous walk with many small discrete steps.The partial-coloring procedure succeeds with probability at least 0.1 and runs in O((m+n)^3 · δ^-2 · log(nm/δ)) time.
- Edge-Walk: Edge-Walk performs a constrained random walk in the polytope P defined by variable bounds and discrepancy constraints.When a constraint becomes tight, the walk remains in the corresponding face rather than crossing the polytope boundary.
- Edge-Walk: The walk hits many variable constraints with good probability when the aggregate exponential threshold expression is sufficiently small relative to n.The proof uses a Gaussian martingale tail bound and the ℓ2 norm as a potential for coordinates near ±1.
3 Comparison with Entropy Method
The paper compares its partial-coloring result with the Entropy method and argues that the new method is both constructive and more tolerant of stringent discrepancy constraints. Its output is fractional rather than a proper partial coloring, but the authors state that this weakness does not prevent subsequent use.
- Comparison: The paper’s partial-coloring lemma is incomparable with Beck’s Entropy-method lemma, and its existence does not follow from the latter.The comparison concerns the strength and form of the respective partial-coloring guarantees.
- Result: Applying Theorem 4 yields a randomized polynomial-time fractional coloring with at least n/2 coordinates at absolute value 1 and discrepancy at most ∆_S + 1/poly(n).The guarantee holds for every set S in the system.
- Constructivity: The new method is constructive, whereas the Entropy method is non-constructive and Bansal’s constructive algorithms do not yield the full partial-coloring lemma.The result therefore supplies an algorithmic version of the stated partial-coloring guarantee.
- Constraint tolerance: The Entropy method allows only O(n/log n) sets with discrepancy 1/n, while this result allows Ω(n) such sets.The paper presents this as greater flexibility for achieving very small discrepancy for a constant fraction of sets.
- Limitation: A limitation is that non-fixed variables may take arbitrary values in (−1,1), rather than only the proper partial-coloring values {−1,0,1}.The authors state that this does not appear to be a significant drawback for their corollary.
4 Preliminaries
The preliminaries establish notation for vectors, subspaces, Gaussian distributions, and Gaussian tail bounds. These tools support the analysis of the constrained random walk and its martingale behavior.
- Notation: The paper uses [n] for {1,...,n}, e_i for standard basis vectors, capital letters for random variables, and calligraphic letters for distributions.The notation X ∼ D means that X is distributed according to D.
- Gaussian distributions: N(µ,σ^2) denotes a Gaussian distribution with mean µ and variance σ^2; the standard Gaussian has mean 0 and variance 1.These conventions are used throughout the random-walk analysis.
- Gaussian distributions: For a subspace V, G ∼ N(V) is the standard multidimensional Gaussian supported on V and expressed in an orthonormal basis of V.The definition is invariant under the choice of orthonormal basis.
- Gaussian facts: If G ∼ N(V), then every inner product ⟨G,u⟩ is a centered Gaussian with variance at most ∥u∥^2.This follows by expanding G in an orthonormal basis of V.
- Gaussian facts: For Gaussian coordinates of a subspace-supported walk, the sum of coordinate variances equals the subspace dimension.This identity connects Gaussian motion to the dimension-based potential used in the analysis.
- Tail bounds: For a standard Gaussian G, Pr[|G| ≥ λ] ≤ 2 exp(−λ^2/2).The paper also introduces a martingale tail bound for conditionally centered Gaussian steps with variance at most one.
5 Main Partial Coloring Lemma
The Edge-Walk random walk stays within a polytope with high probability while progressively fixing variable constraints and controlling discrepancy constraints. Its win-win analysis yields a partial coloring with at least half the variables nearly fixed, and the algorithm is efficiently implementable.
- Random-walk construction: Edge-Walk updates within the subspace orthogonal to variable and discrepancy constraints that are nearly hit.The walk uses V_t, the linear subspace enforcing zero movement in nearly hit constraints.
- Implementation: Sampling each update direction takes O((n + m)^3) time after identifying the active constraint subspace.The full construction later uses the partial-coloring algorithm repeatedly.
- Random-walk construction: The walk remains in the polytope P with probability at least 1 − 1/(mn)^(C−2) for sufficiently large C.The proof bounds first-exit events using Gaussian tail bounds for every variable and discrepancy constraint.
- Analysis: The expected number of nearly hit discrepancy constraints at time T is small.A martingale tail bound shows that, on average, only a few discrepancy constraints are ever nearly hit.
- Analysis: The walk’s squared norm increases by γ^2 E[dim(V_t)] at each update, while remaining at most n inside P.This supports the win-win argument: if few variable constraints are nearly hit, the dimension remains large and the norm grows significantly.
6 Discrepancy Minimization from Partial Coloring
The partial coloring lemma is iterated on restricted systems until every coordinate is nearly fixed, then randomized rounding produces a proper coloring. This yields discrepancy bounds with total running time ˜O((n + m)^3).
- Iterative partial coloring: Each application of Theorem 4 fixes at least half of the currently unrestricted coordinates while preserving bounded discrepancy constraints.The restricted system is formed by projecting each set vector onto the coordinates not fixed in the previous step.
- Iterative partial coloring: After at most 2 log n iterations, every coordinate satisfies |x_i| ≥ 1 − δ and all discrepancy constraints remain bounded.The resulting vector is assembled by concatenating the partial colorings from the restricted systems.
- Rounding: Randomized rounding sets χ_i to sign(x_i) or −sign(x_i) with probabilities chosen so that E[χ_i] = x_i.For Y = χ − x, the rounding error has bounded coordinates, zero mean, and variance at most δ.
- Rounding: With probability at least 1/2, rounding preserves |⟨Y, v_j⟩| ≤ √n for every discrepancy constraint.The bound follows from a Chernoff inequality and a union bound.
- Efficiency: The total running time is O((n + m)^3 log^5(mn)) = ˜O((n + m)^3).The cost is dominated by O(log^2 n) applications of Theorem 4.
- Specialized bound: For systems with each element appearing in at most t sets, the proof groups set vectors by squared norm to derive the specialized discrepancy bound.The number of vectors in each norm range is bounded using the maximum degree t.