[Paper Review] A counterexample to Beck's conjecture on the discrepancy of three permutations

Alantha Newman, Aleksandar Nikolov|arXiv (Cornell University)|Apr 14, 2011

Mathematical Approximation and Integration3 references3 citations

TL;DR

This paper constructs a recursive family of three permutations on $ n = 3^k $ elements, demonstrating that the discrepancy of the set system formed by all intervals in these permutations is $ \Omega(\log n) $, thereby disproving Beck's conjecture that the discrepancy of three permutations is $ O(1) $. The result also refutes a generalization by Spencer, Srinivasan, and Tetali, showing that $ O(\sqrt{\ell}) $ discrepancy for $ \ell $ permutations is not achievable in general.

ABSTRACT

Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. Jozsef Beck conjectured (c. 1987) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer n = 3^k, we exhibit three permutations whose corresponding set system has discrepancy Omega(log(n)). Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This example also disproves a generalization of Beck's conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set system corresponding to l permutations has discrepancy O(sqrt(l)).

Motivation & Objective

• To disprove József Beck's conjecture (c. 1987) that the discrepancy of any three permutations is $ O(1) $.
• To construct a family of three permutations on $ n = 3^k $ elements with discrepancy $ \Omega(\log n) $.
• To refute a generalization by Spencer, Srinivasan, and Tetali that $ \ell $ permutations yield $ O(\sqrt{\ell}) $ discrepancy.
• To establish tightness of existing upper bounds on discrepancy for set systems derived from a constant number of permutations.

Proposed method

• A recursive construction partitions the ground set $[1, 3^k]$ into three equal blocks $ A, B, C $, and permutes them cyclically across the three permutations.
• At each recursive level, each block is subdivided into three sub-blocks of equal size, and the same cyclic permutation pattern is applied recursively.
• The discrepancy is analyzed inductively by tracking the sum of colorings over prefixes and suffixes of the permutations, using a block-wise decomposition of the coloring contributions.
• The proof uses case analysis based on the sum of block values ($ a, b, c $) and applies inductive bounds on subproblems of size $ 3^{k-1} $.
• Lemmas establish lower bounds on the discrepancy of the full system based on the discrepancy of subproblems and the sum of block contributions.
• Symmetry is exploited to handle both positive and negative total coloring sums by considering the negation of colorings.

Experimental results

Research questions

• RQ1 Is the discrepancy of the set system formed by all intervals in three permutations always $ O(1) $, as conjectured by Beck?
• RQ2 Can the generalization by Spencer, Srinivasan, and Tetali—that $ \ell $ permutations yield $ O(\sqrt{\ell}) $ discrepancy—hold for all $ \ell $?
• RQ3 What is the best possible discrepancy lower bound for three permutations, and can it be $ \Omega(\log n) $?
• RQ4 Are the upper bounds of $ O(\ell \log n) $ and $ O(\sqrt{\ell} \log n) $ for $ \ell $ permutations tight?

Key findings

• The paper constructs, for each $ k > 0 $, three permutations on $ n = 3^k $ elements such that the discrepancy of the corresponding set system is at least $ \lceil k/3 + 1 \rceil $.
• Since $ k = \log_3 n $, this yields a discrepancy lower bound of $ \Omega(\log n) $, which contradicts Beck's conjecture of $ O(1) $ discrepancy.
• The recursive construction generates up to $ 2^k $ distinct permutation triples, all of which maintain the same $ \Omega(\log n) $ discrepancy lower bound due to symmetry in the proof.
• The result implies that the upper bounds of $ O(\ell \log n) $ by Bohus and $ O(\sqrt{\ell} \log n) $ by Spencer, Srinivasan, and Tetali are tight up to the $ \log n $ factor when $ \ell $ is constant.
• The counterexample shows that Beck’s conjecture cannot be true, and thus the connection between discrepancy and the additive integrality gap in Bin Packing cannot be used to prove $ OPT_{LP} + O(1) $ for Bin Packing.