[Paper Review] Hamiltonian increasing paths in random edge orderings

M. I. Lavrov, Po‐Shen Loh|arXiv (Cornell University)|Mar 4, 2014

Stochastic processes and statistical mechanics17 references3 citations

TL;DR

This paper investigates the length of the longest increasing path in a random edge ordering of the complete graph $K_n$, where edges are labeled uniformly at random. It proves that with probability at least $1/e$, a Hamiltonian increasing path (of length $n-1$) exists, and with high probability, an increasing path of length at least $0.85n$ exists, revealing a surprising abundance of near-Hamiltonian paths in the random setting—contrasting sharply with the worst-case bounds of $\sqrt{n-1}$ to $n/2$.

ABSTRACT

If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained an open problem for several decades, with asymptotic bounds between $\sqrt{n}$ (Graham and Kleitman, 1973) and $n/2$ (Calderbank, Chung, and Sturtevant, 1984). We consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, an increasing path of the maximum possible length of $n-1$ exists with probability at least about $1/e$. We also prove that with probability $1-o(1)$, there is an increasing path of length at least $0.85n$, suggesting that this Hamiltonian (or near-Hamiltonian) phenomenon may hold asymptotically almost surely.

Motivation & Objective

• To understand the typical behavior of the longest increasing path in a random edge ordering of $K_n$, shifting focus from worst-case to average-case analysis.
• To resolve the long-standing open problem of characterizing the length of the longest increasing path in random edge orderings, which had seen no progress for decades.
• To establish that Hamiltonian increasing paths are common in random orderings, contrasting with the much weaker worst-case lower bound of $\sqrt{n-1}$.

Proposed method

• Analyzes the greedy algorithm for constructing increasing paths from each vertex, using probabilistic and combinatorial arguments to bound path lengths.
• Applies a variant of the pedestrian argument with modified rules to model edge traversal under increasing labels, adapting it to the random setting.
• Employs generating functions and asymptotic analysis to estimate the number of configurations yielding long increasing paths.
• Uses double summation decomposition into $S_1$, $S_2$, and $S_3$ to bound the expected number of long paths, with careful tail analysis for convergence.
• Applies ratio estimates and exponential decay bounds to show that the total contribution of short and intermediate paths is $o(n^2)$, implying concentration of long paths.
• Leverages concentration inequalities and bounds on binomial and multinomial coefficients to control error terms and establish high-probability results.

Experimental results

Research questions

• RQ1 What is the typical length of the longest increasing path in a uniformly random edge ordering of $K_n$?
• RQ2 How frequently does a Hamiltonian increasing path (of length $n-1$) appear in a random edge ordering?
• RQ3 Does the length of the longest increasing path concentrate near $n-1$ with high probability in the random setting?
• RQ4 Can the worst-case lower bound of $\sqrt{n-1}$ be significantly improved in the average case?
• RQ5 Is there a phase transition in the random setting where long increasing paths become asymptotically almost surely present?

Key findings

• With probability at least $1/e$, a random edge ordering of $K_n$ contains a Hamiltonian increasing path of length $n-1$.
• With high probability (asymptotically almost surely), there exists an increasing path of length at least $0.85n$.
• The total number of configurations yielding long increasing paths is bounded by $o(n^2)$, implying concentration of path lengths near $n-1$.
• The analysis shows that the contribution of paths of intermediate length is negligible, with $S_2 = o(n^2)$ and $S_3 = o(n^2)$.
• The paper resolves a long-standing open problem in extremal combinatorics by showing that Hamiltonian increasing paths are common in the random setting, despite being rare in the worst case.
• The result demonstrates a sharp contrast between worst-case and average-case behavior, where the average case exhibits a phase of near-Hamiltonian behavior.