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Valerio Dose

Publications and source records attributed to Valerio Dose.

14 recordsLinked to original sources
arXiv2026

Optimal Macroitem Sequences in the Precedence Constrained Knapsack Problem

The Precedence Constrained Knapsack Problem (PCKP) asks for a maximum-profit subset of items, subject to a knapsack capacity constraint and precedence constraints encoded by a directed acyclic graph. We study the structure of optimal solutions of the Linear Programming (LP) relaxation of the natural Integer Linear Programming formulation of the PCKP. We introduce the notion of macroitem and of feasible sequence of macroitems, which partitions the item set while respecting the precedence structure. We establish that an optimal LP solution is fully characterized by the optimal sequence of macroitems: items are packed in nonincreasing order of the profit-to-weight ratio of their macroitem, with at most one macroitem fractionally included. We further show that the breakpoints of the parametric Lagrangian function of the capacity constraint coincide with the profit-to-weight ratios of the macroitems in the optimal sequence, and provide a complete combinatorial characterization of optimal dual solutions in terms of a feasible flow within each macroitem. Finally, for the special case in which the precedence graph is a forest, we devise an O(n^2) algorithm to compute the optimal sequence, which improves to O(n log n) for in-trees or out-trees, where n denotes the number of items.

math.OC↗
arXiv2025

A Tight 2-Approximation Algorithm for the Bin Packing Problem with Setups

We study approximation algorithms for the Bin Packing Problem with Setups (BPPS), a generalization of the classical Bin Packing Problem (BPP) in which items are partitioned into classes and activating a class in a bin consumes a setup weight and incurs a setup cost. We show that direct adaptations of Next Fit (NF), First Fit (FF), Best Fit (BF), and Worst Fit (WF), as well as their decreasing-order variants, have unbounded absolute worst-case performance ratios, even with unit-weight items and zero setup costs. We then introduce a two-phase algorithm, $\mathrm{TP}_{\mathcal{A}}$ , that packs each class independently with a BPP algorithm and subsequently merges compatible packing patterns. We prove that the solution returned by $\mathrm{TP}_{\mathcal{A}}$ has cost at most twice the optimum under the assumption that produces pairwise merge-maximal solutions, i.e., such that no two packing patterns in the class-wise solution can be feasibly merged. If also runs in polynomial time, this yields a 2-approximation algorithm for the BPPS. The factor is tight: the absolute worst-case performance ratio of $\mathrm{TP}_{\mathcal{A}}$ is exactly 2, even when solves every class-wise BPP instance optimally. Since every Any Fit algorithm returns pairwise merge-maximal solutions, it follows that $\mathrm{TP}_{\mathrm{FF}}$ , $\mathrm{TP}_{\mathrm{BF}}$ , $\mathrm{TP}_{\mathrm{WF}}$ , and their decreasing-order variants all have an absolute worst-case performance ratio exactly 2. If, in addition, is an $\alpha$-approximation algorithm with $\alpha \leq 2$, we obtain a finer, component-wise guarantee with factor 2 for the bin-opening cost and factor $\alpha$ for the setup-cost component.

math.OC↗
arXiv2025

The Bin Packing Problem with Setups: Formulations, Structural Properties and Computational Insights

We introduce the Bin Packing Problem with Setups (BPPS), a generalization of the classical Bin Packing Problem with applications in production planning and logistics. In this problem, the items are partitioned into classes, and packing items of a class in a bin incurs a setup weight and cost. We propose a natural Integer Linear Programming (ILP) formulation for the BPPS and analyze its Linear Programming relaxation. We show that the resulting lower bound can be arbitrarily weak and introduce the Minimum Classes Inequalities (MCIs), which guarantee a worst-case ratio of 1/2 with respect to the optimal objective function value of the BPPS. We also derive the Minimum Bins Inequality (MBI) and an upper bound on the number of bins in any optimal solution, substantially reducing the formulation size. We further develop an arc-flow formulation for the BPPS based on a tailored graph construction and compression procedure. Its LP relaxation dominates that of the natural formulation, and both the MCIs and the MBI are extended to the arc-flow model. Finally, we introduce a benchmark comprising 576 randomly generated instances and 36 real-world instances derived from a vehicle-routing application, and conduct extensive computational experiments. Results show that the natural formulation performs best on instances with small or medium item weights, whereas the arc-flow formulation is more effective for large item weights and on the real-world testbed.

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arXiv2025

Strength of the Upper Bounds for the Edge-Weighted Maximum Clique Problem

We theoretically and computationally compare the strength of the three main upper bounds from the literature on the optimal value of the Edge-Weighted Maximum Clique Problem (EWMCP). We provide a set of instances for which the ratio between any of the three upper bounds and the optimal value of the EWMCP is unbounded, showing that none of them can give a performance guarantee. We further analyze the relative strength among the three upper bounds by determining, for every choice of a ratio between any two of them, the largest values it can attain and providing families of instances for which such values can be reached. Our results show that, for each pair of upper bounds, there exist appropriately chosen instances on which either bound is tighter than the other. Our theoretical analysis is complemented by extensive computational experiments on two benchmark datasets: the standard DIMACS instances and randomly generated instances, providing practical insights into the empirical strength of the upper bounds.

math.OC↗
arXiv2025

Maximal curves over finite fields and a modular isogeny

We prove the existence of curves of genus $7$ and $12$ over the field with $11^5$ elements, reaching the Hasse-Weil-Serre upper bound. These curves are quotients of modular curves and we give explicit equations. We compute the number of points of many quotient modular curves in the same family without providing equations. For various pairs (genus, finite field) we find new records for the largest known number of points. In other instances we find quotient modular curves that are maximal, matching already known results. To perform these computations, we provide a generalization of Chen's isogeny result.

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arXiv2023

Monotonicity of Equilibria in Nonatomic Congestion Games

This paper studies the monotonicity of equilibrium costs and equilibrium loads in nonatomic congestion games, in response to variations of the demands. The main goal is to identify conditions under which a paradoxical non-monotone behavior can be excluded. In contrast to routing games with a single commodity, where the network topology is the sole determinant factor for monotonicity, for general congestion games with multiple commodities the structure of the strategy sets plays a crucial role. We frame our study in the general setting of congestion games, with a special focus on singleton congestion games, for which we establish the monotonicity of equilibrium loads with respect to every demand. We then provide conditions for comonotonicity of the equilibrium loads, i.e., we investigate when they jointly increase or decrease after variations of the demands. We finally extend our study from singleton congestion games to the larger class of constrained series-parallel congestion games, whose structure is reminiscent of the concept of a series-parallel network.

cs.GT↗
arXiv2023

Phase Transitions of the Price-of-Anarchy Function in Multi-Commodity Routing Games

We consider the behavior of the price of anarchy and equilibrium flows in nonatomic multi-commodity routing games as a function of the traffic demand. We analyze their smoothness with a special attention to specific values of the demand at which the support of the Wardrop equilibrium exhibits a phase transition with an abrupt change in the set of optimal routes. Typically, when such a phase transition occurs, the price of anarchy function has a breakpoint, \ie is not differentiable. We prove that, if the demand varies proportionally across all commodities, then, at a breakpoint, the largest left or right derivatives of the price of anarchy and of the social cost at equilibrium, are associated with the smaller equilibrium support. This proves -- under the assumption of proportional demand -- a conjecture of O'Hare et al. (2016), who observed this behavior in simulations. We also provide counterexamples showing that this monotonicity of the one-sided derivatives may fail when the demand does not vary proportionally, even if it moves along a straight line not passing through the origin.

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arXiv2020

High Order Elements in Finite Fields Arising from Recursive Towers

We provide a recipe to construct towers of fields producing high order elements in $\mathrm{GF}(q,2^n)$, for odd $q$, and in $\mathrm{GF}(2,2 \cdot 3^n)$, for $n \ge 1$. These towers are obtained recursively by $x_{n}^2 + x_{n} = v(x_{n - 1})$, for odd $q$, or $x_{n}^3 + x_{n} = v(x_{n - 1})$, for $q=2$, where $v(x)$ is a polynomial of small degree over the prime field $\mathrm{GF}(q,1)$ and $x_n$ belongs to the finite field extension $\mathrm{GF}(q,2^n)$, for $q$ odd, or to $\mathrm{GF}(2,2\cdot 3^n)$. Several examples are carried out and analysed numerically. The lower bounds of the orders of the groups generated by $x_n$, or by the discriminant $\delta_n$ of the polynomial, are similar to the ones obtained in [BCG+09], but we get better numerical results in some cases.

math.NT↗
arXiv2020

Automorphisms of Cartan modular curves of prime and composite level

We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for non-split curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one non-trivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^*(n):=X_0(n)/W$, where $W$ is the group generated by the Atkin-Lehner involutions of $X_0(n)$ and $n$ is a large enough square.

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arXiv2019

The Price of Anarchy in Routing Games as a Function of the Demand

The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone (decreasing or increasing) over the full interval, or it decreases up to a certain minimum point in the interior of the interval and increases afterwards. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite.

cs.GT↗
arXiv2017

Double Covers of Cartan Modular Curves

We present a strategy to obtain explicit equations for the modular double covers associated respectively to both a split and a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_{p})$ with $p$ prime. Then we apply it successfully to the level $13$ case.

math.NT↗
arXiv2016

Modular Curves with many Points over Finite Fields

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients $X_H/W$ for $H$ a subgroup of $\GL_2(\mathbb Z/n\mathbb Z)$ such that for each prime $p$ dividing $n$, the subgroup $H$ at $p$ is either a Borel subroup, a Cartan subgroup, or the normalizer of a Cartan subgroup of $\GL_2(\mathbb Z/p^e\mathbb Z)$, and for $W$ any subgroup of the Atkin-Lehner involutions of $X_H$. We applied our algorithm to more than ten thousands curves of genus up to 50, finding more than one hundred record-breaking curves, namely curves $X/\FF_q$ with genus $g$ that improve the previously known lower bound for the maximum number of points over $\FF_q$ of a curve with genus $g$. As a key technical tool for our computations, we prove the generalization of Chen's isogeny to all the Cartan modular curves of composite level.

math.NT↗
arXiv2015

On the automorphisms of the non-split Cartan modular curves of prime level

We study the automorphisms of the non-split Cartan modular curves $X_{ns}(p)$ of prime level $p$. We prove that if $p\geq 37$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1\text{ mod }12$ and $p\neq 13$, the automorphism group is generated by the modular involution given by the normalizer of a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_p)$. We also prove that for every $p\geq 37$ such that $X_{ns}(p)$ has a CM rational point, the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{ns}^+(p)$ associated to the normalizer of a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_p)$.

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