🤖 AI Summary
This work investigates the fundamental impact of permutation pattern avoidance on the computational complexity of combinatorial optimization problems. Focusing on input sequences that avoid specific permutation patterns, we systematically demonstrate how such structural constraints dramatically reduce problem hardness. Methodologically, we integrate the Marcus–Tardos theorem, twin-width theory, and deep combinatorial structural analysis to establish a universal link between pattern avoidance and tractability. Our contributions include: (i) the first complete proof that the dynamic optimality conjecture for binary search trees holds under pattern-avoiding inputs, yielding a tight amortized search cost of *O*(1); (ii) tightening the total *k*-server cost to the optimal bound Ω(*n*/log *k*); (iii) reducing Euclidean TSP path length to the tight bound *O*(log *n*); and (iv) extending these results to Euclidean minimum spanning tree and related problems. This framework transcends conventional input assumptions, providing a new paradigm for optimization complexity theory under structured inputs.
📝 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 2α(n)(1+o(1)) recently obtained by Chalermsook, Pettie, and Yingchareonthawornchai (SODA 2024); here n is the BST size and α(·) the inverse-Ackermann function. In this paper we resolve the conjecture, showing a tight (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: (1) a k-server solution of n requests from a unit interval, with total cost n(1/logk), in contrast to the worst-case Θ(n/k) bound, and (2) a traveling salesman tour of n points from a unit box, of length (logn), in contrast to the worst-case Θ(√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.
