arXiv subjects

Pedro M. M. de Castro

Publications and source records attributed to Pedro M. M. de Castro.

4 recordsLinked to original sources

arXiv2026

Minimum blockers and extremal graphs for nonnested matchings

We classify all smallest sets of edge deletions that destroy every nonnested perfect matching in a complete ordered graph. The vertices have a fixed linear order. A perfect matching selects edges that use each vertex exactly once; it is nonnested when no selected edge has both endpoints strictly between those of another. On $2k$ vertices, exactly $k$ deletions are necessary, and we describe all $2^k+k-2$ minimum deletion sets for every $k\ge2$. Allowing a matching to leave vertices unused changes the extremal problem. Barát, Freschi and Tóth proposed that an $n$-vertex ordered graph avoiding a nonnested $k$-edge matching can have at most $(k-1)n$ edges. We construct counterexamples for every $k\ge5$ and $n\ge2k+1$. For $n\ge3k$, our constructions exceed the proposed value by a number of edges proportional to $k^2$, matching the order of the known upper bound on this excess. The classification follows from cuts between consecutive intervals; the larger constructions coordinate deletions at the two ends of the vertex order. The exact extremal value remains open in general.

math.CO↗

arXiv2026

Ehrhart $h^*$-polynomials of $(132,213)$-avoiding permutation polytopes: A repair cone and eventual real-rootedness

Let $P_d(132,213)$ be the convex hull of the permutations in $S_d$ that avoid $132$ and $213$, and let $H_d(t)$ be its Ehrhart $h^*$-polynomial, defined by $\sum_{m\ge0}|mP_d(132,213)\cap\mathbb{Z}^d|t^m=\frac{H_d(t)}{(1-t)^d}$. We give an explicit lattice equivalence between this polytope, a path-Laplacian deficit polytope, and twice a translated chain-poset permutahedron. The resulting face description and triangulation give a self-contained proof of Davis and Sagan's cubicality and volume conjecture, with normalized volume $2^{d-1}d^{d-3}$ for $d\ge2$. For $d\ge3$, the coefficient of $t^{d-1}$ in $H_d(t)$ counts strong tournament score sequences. The refined Eulerian expansion of $H_d(t)$ has a negative coordinate already at $d=7$. We construct a compatible cone of adjacent signed differences that accommodates this obstruction. Its largest uniform repair parameter is $2-\sqrt{3}$, and membership reduces to explicit geometric-tail inequalities. An exact dataset and reconstruction algorithm verify these inequalities through $d=1000$. A marked-component identity, discrete smoothing, and Darroch's mode theorem (Ann. Math. Statist. 35 (1964), 1317--1321, Theorem 4) prove them for every $d\ge2^{72}$. Thus $H_d(t)$ has only negative real zeros for $3\le d\le1000$ and for $d\ge2^{72}$, while $H_1(t)=H_2(t)=1$. Uniform real-rootedness in the intervening range remains open.

math.CO↗

arXiv2026

Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions

We study the edge-power cost of the labelled tree generated by the $γ$-strategy, a constant-gain rule for sequential Euclidean connections. Starting with $x_0=p_0$, each input point $p_i$ is attached to $x_{i-1}$, and the state is updated by $x_i=γx_{i-1}+(1-γ)p_i$. Retaining $x_i$ subdivides the insertion segment into a spine edge and a leaf edge. The memory parameter $γ$ controls how long earlier points influence subsequent attachment points. We minimize the sum of the $α$-powers of the edge lengths under independent uniform input and arbitrary input sequences. For uniform points in the unit ball, the stationary problem has a transition at $α=1$. Its continuous extension is minimized at the boundary for $0<α\leq1$, while every global minimizer is interior for $α>1$. We determine the finite optimizer in the joint window $α_N=1+\varepsilon_N$, $\varepsilon_N\log N\toλ$. Below an explicit threshold it lies on the $N^{-1/2}$ scale, at the threshold its scale is $\sqrt{\log N/(N\log\log N)}$, and above the threshold it approaches an explicit stationary root with two computable corrections. A second threshold identifies the governing correction, and differentiated estimates prove eventual uniqueness. At $α=3d+8$, the linear coefficient at the stationary endpoint changes sign and a branch of strict local maxima enters the parameter interval. For arbitrary input sequences, the optimal parameter and asymptotic worst-case edge-power cost per point are explicit for $0<α\leq3$. At high powers, periodic antipodal block inputs give explicit lower bounds which, with a separation argument, show that the optimized cost is asymptotic to $2\log2/\logα$. Exact results for powers two and four, a rational recursion for every even power, and a high-dimensional expansion complete the analysis.

cs.CG↗

arXiv2026

Sequential Euclidean connections with exponential memory: distributional performance and adversarial robustness

Points in the unit ball of $\mathbb R^d$ are processed sequentially. Each new point $p_i$ is connected to a state $x_{i-1}$ that summarizes earlier observations, after which $x_i=γx_{i-1}+(1-γ)p_i$, with $0\leqγ\leq1$. The cost is the sum of the $α$-powers of the connection lengths. This constant-gain rule interpolates between the input-order path and the star centered at the initial point. For independent uniform points, we establish the stationary insertion-length distribution and prove that it decreases in stochastic order as $γ$ increases. If $d+α>2$, or if $(d,α)=(1,1)$, the optimal constant parameter satisfies $1-γ_N^*=Θ(N^{-1/2})$, with an explicit asymptotic constant and closed bounds. For $α=1$, its leading expected tree length equals that of the center star and is eventually smaller than the expected lengths of both endpoint constructions. For $α=2$, the optimizer is unique and characterized exactly. For the same $N$, choosing $1-γ_N$ as a fixed positive multiple of $N^{-1/2}$ gives a sharp two-term expansion of the expected uniform-input cost and a maximal adversarial mean cost of $1+O(N^{-1/2})$. For every fixed $0\leqγ<1$ and $0<α\leq3$, the exact asymptotic adversarial value is $(2/(1+γ))^α$. When $d\geq2$, exponential weighting is within a factor smaller than $1.161^α$ of the best fixed nonnegative weighted rule with the same average look-back, for $0<α\leq3$. Comparison with the running mean highlights its time-homogeneous update, stationary coefficient profile, and fixed effective memory.

cs.CG↗