Optimization with pattern-avoiding input

TL;DR

The paper investigates how permutation pattern-avoidance influences optimization problems by leveraging twin-width decompositions and Marcus–Tardos/Stanley–Wilf theory. It provides a tight bound of $O(1)$ amortized cost per access for BST on pattern-avoiding inputs with fixed pattern size, and derives sublinear cost bounds for online problems like $k$-server on the line and Euclidean TSP in pattern-avoiding instances, parameterized by $c_i$ of the avoided pattern. Central to the approach are arborally satisfied supersets for BST and distance-balanced merge sequences for geometry and online optimization, enabling DP-like solutions on structured inputs. The work demonstrates a broad, generalizable phenomenon: pattern-avoidance imposes global sparsity that lowers optimization costs across diverse domains, with implications for dynamic optimality and potential extensions to other online and geometric problems.

Abstract

Permutation pattern-avoidance is a central concept of both enumerative and extremal combinatorics. In this paper we study the effect of permutation pattern-avoidance on the complexity of optimization problems. In the context of the dynamic optimality conjecture (Sleator, Tarjan, STOC 1983), Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak (FOCS 2015) conjectured that the amortized search cost of an optimal binary search tree (BST) is constant whenever the search sequence is pattern-avoiding. The best known bound to date is $2^{α{(n)}(1+o(1))}$ recently obtained by Chalermsook, Pettie, and Yingchareonthawornchai (SODA 2024); here $n$ is the BST size and $α(\cdot)$ the inverse-Ackermann function. In this paper we resolve the conjecture, showing a tight $O(1)$ bound. This indicates a barrier to dynamic optimality: any candidate online BST (e.g., splay trees or greedy trees) must match this optimum, but current analysis techniques only give superconstant bounds. More broadly, we argue that the easiness of pattern-avoiding input is a general phenomenon, not limited to BSTs or even to data structures. To illustrate this, we show that when the input avoids an arbitrary, fixed, a priori unknown pattern, one can efficiently compute a $k$-server solution of $n$ requests from a unit interval, with total cost $n^{O(1/\log k)}$, in contrast to the worst-case $Θ(n/k)$ bound; and a traveling salesman tour of $n$ points from a unit box, of length $O(\log{n})$, in contrast to the worst-case $Θ(\sqrt{n})$ bound; similar results hold for the euclidean minimum spanning tree, Steiner tree, and nearest-neighbor graphs. We show both results to be tight. Our techniques build on the Marcus-Tardos proof of the Stanley-Wilf conjecture, and on the recently emerging concept of twin-width.

Figures (16)

• Figure 1: The permutation 31254 as a matrix (a) and as a point set (b); the highlighted points/ones correspond to an occurrence of the pattern 213 (c). Pattern 321 (d) is avoided by 31254.
• Figure 2: (a) Sum- and (b) skew sum of permutations 213 and 3412; (c) the inflation of 213 by 123, 1, and 21; (d) a $3 \times 4$ grid permutation, and (e) canonical grid.
• Figure 3: A 3-wide merge sequence of 23514, along with its red graphs. Black points are also degenerate rectangles.
• Figure 4: A hierarchy of important permutation classes.
• Figure 5: (a) Accessing node $x$ in a BST with subtree $T'$ re-arranged into $T"$ (shown in grey) at cost 6 and subtrees $T_1, \dots, T_7$ unchanged. (b) A satisfied superset for access sequence $5,2,3,1,4$ (black points denote accesses).

Theorems & Definitions (64)

• Theorem 1.1: informal
• Theorem 1.2: informal
• Theorem 1.3: informal
• Lemma 2.1
• Lemma 2.3
• proof
• Theorem 2.4: Guillemot and Marx GM_PPM
• Theorem 3.1: DHIKP
• Theorem 3.2
• proof