On-line Chain Partitions of Up-growing Semi-orders

On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents a partially ordered set, point by point. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of width $w$ is a minimum number $\fVal(w)$ such that Algorithm has a strategy using at most $\fVal(w)$ chains on orders of width at most $w$. We analyze the chain partition game for up-growing semi-orders. Surprisingly, the golden ratio comes into play and the value of the game is $\lfloor\frac{1+\sqrt{5}}{2}; w \rfloor$.

💡 Research Summary

The paper studies the online chain partition game on a restricted class of posets called up‑growing semi‑orders. In this two‑player game, Spoiler presents the elements of a partially ordered set one by one, always adding a maximal element of the current order (the “up‑growing” restriction). After each presentation, Algorithm must irrevocably assign the new element to one of the chains of a chain partition. The performance measure is the number of chains used; the value fₐₗ(w) for width w is the smallest integer n such that there exists an algorithm that never needs more than n chains on any up‑growing semi‑order of width at most w.

Semi‑orders are precisely the posets that admit a unit‑interval representation; equivalently they are (2+2)‑free and (3+1)‑free. For unrestricted semi‑orders the optimal online bound is 2w‑1, while for up‑growing interval orders it is 2w‑1 as well, and for up‑growing general orders it is ⌈(w+1)/2⌉. The authors discover that the up‑growing semi‑order case behaves differently: the exact value is ⌊φ·w⌋, where φ = (1+√5)/2 ≈ 1.618 is the golden ratio.

Lower bound (Spoiler’s strategy).
The authors introduce a system of linear inequalities (Iₖ): x₀ + … + x_{j‑1} + 2x_j – x_{j+1} ≤ w for j = 0,…,k, with integer variables satisfying x₀ > x₁ > … > x_k > x_{k+1}=0.
Given such a solution, Spoiler constructs an up‑growing semi‑order in phases. The construction uses three families of point sets:

• A: w minimal elements (forming w distinct chains initially);
• B_i (i = 1…k+1): points of height 2;
• C_j (j = 0…k): points of height 3.

During phase j Spoiler creates x_j – x_{j+1} “forcing paths”. Each path forces Algorithm either to start a new chain or to place a point on a chain whose current top lies in some B_i. When the latter happens, Spoiler records the top in an auxiliary set D_i, and subsequently forces Algorithm to avoid that top by presenting a higher point that dominates all earlier B‑sets except D_i. This mechanism guarantees that each forcing path either ends a “skip chain” (height‑2 chain whose bottom is in A and top in C_j) or consumes a fresh chain. The total number of chains forced is w + x₀. By choosing the solution of (Iₖ) with x₀ = ⌊(φ−1)·w⌋, the inequality system is satisfied (the construction of the solution uses the recurrence x_{j+1}=⌊(φ−1)(w−Σ_{i=0}^{j}x_i)⌋, which is essentially the Fibonacci recurrence scaled by φ−1). Hence Spoiler can force at least ⌊φ·w⌋ chains, establishing the lower bound.

Upper bound (Algorithm’s strategy).
For the upper bound the authors present a greedy algorithm that always assigns the new element to the chain whose current top is the smallest element larger than the new point (or creates a new chain if none exists). Because semi‑orders are unit‑interval orders, the set of chain tops can be ordered by their interval left endpoints, and the up‑growing condition guarantees that each new point is maximal among the presented ones. The crucial observation is that the height of any up‑growing semi‑order never exceeds three, and the (3+1)‑free property prevents a configuration where many chains would be forced to be simultaneously “active”. By analyzing the evolution of the chain‑top set, the authors show that its size never exceeds ⌊φ·w⌋. The analysis again relies on the Fibonacci numbers: the number of chains of each “type” (bottom in A, top in B_i, etc.) follows a recurrence that mirrors the Fibonacci sequence, whose growth rate is φ. Consequently the greedy algorithm never uses more than ⌊φ·w⌋ chains, establishing the matching upper bound.

Significance.
The result closes the gap for up‑growing semi‑orders, revealing a surprising appearance of the golden ratio in a purely combinatorial online problem. It demonstrates that restricting the presentation order (up‑growing) together with the structural restriction of semi‑orders yields a finer bound than either restriction alone. The use of a linear inequality system tied to the Fibonacci recurrence provides a novel technique that may be applicable to other online coloring or partition problems. Moreover, the exact value ⌊φ·w⌋ bridges the gap between the previously known bounds for related classes (e.g., up‑growing interval orders at 2w‑1 and up‑growing general orders at ⌈(w+1)/2⌉), highlighting how subtle structural properties of the poset class affect online performance.