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Federico Della Croce

Publications and source records attributed to Federico Della Croce.

15 recordsLinked to original sources
arXiv2026

Heuristic approaches for solving a bilevel optimistic scheduling problem on parallel machines

This work addresses the uniform parallel machine scheduling problem within an optimistic bilevel optimization framework. The leader seeks to minimize the weighted number of tardy jobs, while the follower aims to minimize the total completion time across a set of uniform machines. The hierarchical decision-making process of the bilevel problem makes designing effective heuristics challenging. To tackle this, we exploit a property of the follower that enables the construction of optimal schedules. From this property, we derive an effective branching scheme that simultaneously accounts for both leader and follower decisions. This branching scheme allows us to design a Recovering Beam Search (RBS), which represents a significant contribution from a bilevel perspective. Then we propose a Multi-Start Local Search (MSLS) algorithm based on an innovative scheme that couples the RBS and a Local Search (LS). To the best of our knowledge, while hybridizing beam search with local search is known in other contexts, our approach of leveraging a bilevel-specific branching scheme to efficiently explore the search space is novel. Moreover, we propose an automated method for determining heuristic parameters via Bayesian optimization. This reduces computational resource requirements and yields better parameters than the usual empirical approach. Finally, computational experiments are presented for instances with up to 500 jobs and 10 machines.

math.OC↗
arXiv2026

Solution of a bilevel optimistic scheduling problem on parallel machines

We consider the uniform parallel machines scheduling problem in the context of optimistic bilevel optimization, where two speed options are considered. In this scenario, the leader aims to minimize the weighted number of tardy jobs, while the follower seeks to minimize the total completion time on a set of uniform machines. This problem has practical applications in Industry 4.0. We show that this problem is NP-hard in the strong sense by providing a reduction from the Numerical 3-Dimensional Matching problem and we provide a moderately exponential-time dynamic programming algorithm. The problem is solved by means of a concise MIP formulation and a branch-and-bound algorithm that embeds a column generation approach for the lower bound computation. Computational experiments are presented for instances with up to 80 jobs and 4 machines while larger problems are out of reach for the proposed approaches.

math.OC↗
arXiv2026

Revisiting Johnson's rule for minimizing makespan in the Two-Machine Flow Shop scheduling problem

We consider Johnson's rule for minimizing the makespan in the two-machine flow shop problem. Although its worst-case time complexity is O(n log n), we show that it is possible to detect in linear time whether a full sorting of jobs can be avoided and an optimal solution can be computed in O(n) time. A probabilistic analysis indicates that linear time complexity holds with high probability under uniformly distributed processing times, a result further supported by extensive computational experimentation.

math.OC↗
arXiv2023

ITERATED INSIDE OUT: a new exact algorithm for the transportation problem

We propose a novel exact algorithm for the transportation problem, one of the paradigmatic network optimization problems. The algorithm, denoted Iterated Inside Out, requires in input a basic feasible solution and is composed by two main phases that are iteratively repeated until an optimal basic feasible solution is reached. In the first "inside" phase, the algorithm progressively improves upon a given basic solution by increasing the value of several non-basic variables with negative reduced cost. This phase typically outputs a non-basic feasible solution interior to the constraints set polytope. The second "out" phase moves in the opposite direction by iteratively setting to zero several variables until a new improved basic feasible solution is reached. Extensive computational tests show that the proposed approach strongly outperforms all versions of network and linear programming algorithms available in the commercial solvers Cplex and Gurobi and other exact algorithms available in the literature.

math.OC↗
arXiv2020

Merging Combinatorial Design and Optimization: the Oberwolfach Problem

The Oberwolfach Problem $OP(F)$ -- posed by Gerhard Ringel in 1967 -- is a paradigmatic Combinatorial Design problem asking whether the complete graph $K_v$ decomposes into edge-disjoint copies of a $2$-regular graph $F$ of order $v$. In this paper, we provide all the necessary equipment to generate solutions to $OP(F)$ for relatively small orders by using the so-called difference methods. From the theoretical standpoint, we present new insights on the combinatorial structures involved in the solution of the problem. Computationally, we provide a full recipe whose base ingredients are advanced optimization models and tailored algorithms. This algorithmic arsenal can solve the $OP(F)$ for all possible orders up to $60$ with the modest computing resources of a personal computer. The new $20$ orders, from $41$ to $60$, encompass $241200$ instances of the Oberwolfach Problem, which is 22 times greater than those solved in previous contributions.

math.CO↗
arXiv2020

An enhanced pinwheel algorithm for the bamboo garden trimming problem

In the Bamboo Garden Trimming Problem (BGT), there is a garden populated by n bamboos b(1), b(2), ... , b(n)$ with daily growth rates h(1) >= h(2) >= ... >= h(n). We assume that the initial heights of bamboos are zero. A gardener is in charge of the bamboos and trims them to height zero according to some schedule. The objective is to design a perpetual schedule of trimming so as to maintain the height of the bamboo garden as low as possible. We consider the so-called discrete BGT variant, where the gardener is allowed to trim only one bamboo at the end of each day. For discrete BGT, the current state-of-the-art approximation algorithm exploits the relationship between BGT and the classical Pinwheel scheduling problem and provides a solution that guarantees a 2-approximation ratio. We propose an alternative Pinwheel scheduling algorithm with approximation ratio converging to 12/7 when sum h(j) > > h(1). Also, we show that the approximation ratio of the proposed algorithm never exceeds 32000/16947 approximately 1.888. This is the first algorithm reaching a ratio strictly inferior to 19/10.

cs.DS↗
arXiv2020

An improved solution approach for the Budget constrained Fuel Treatment Scheduling problem

This paper considers the budget constrained fuel treatment scheduling (BFTS) problem where, in the context of wildfire mitigation, the goal is to inhibit the potential of fire spread in a landscape by proper fuel treatment activities. Given a time horizon represented by consecutive unit periods, the landscape is divided into cells and represented as a grid graph where each cell has a fuel age that increases over time and becomes old if no treatment is applied in the meantime: this induces a potential high fire risk whenever two contiguous cells are old. Cells fuel ages can be reset to zero under appropriate fuel treatments but there is a limited budget for treatment in each period. The problem calls for finding a suitable selection of cells to be treated so as to minimize the presence of old contiguous cells over the whole time horizon. We prove that problem BFTS is strongly NP-complete on paths and thus on grid graphs and show that no polynomial time approximation algorithm exists unless P = NP. We provide an enhanced integer linear programming formulation of the problem with respect to the relevant literature that shows up to be efficiently solved by an ILP solver on reasonably large size instances. Finally, we consider a harder periodic variant of the problem with the aim of finding a cyclic treatment plan with cycles of length T and propose a matheuristic approach capable of efficiently tackling those instances where an ILP solver applied to the ILP formulation runs into difficulties.

cs.DS↗
arXiv2019

Introducing Fairness and Diversification in WTA and ATP Tennis Tournaments Generation

Single-elimination tournaments are the standard paradigm both for the main tennis professional associations. Schedules are generated by allocating first seeded and then unseeded players with seeds prevented from encountering each other early in the competition. Besides, the distribution of pairings in the first round between unseeded players and seeds for a yearly season may be strongly unbalanced. This provides often a great disadvantage to some "unlucky" unseeded players in terms of money prizes. Also, a fair distribution of matches during a season would benefit from limiting in first rounds the presence of Head-to-Head (H2H) matches between players that met in the recent past. We propose a tournament generation approach in order to reduce in the first round "unlucky" pairings and replay of H2H matches. The approach consists in a clustering optimization problem inducing a consequent draw within each cluster. A Non-Linear Mathematical Programming (NLMP) model is proposed for the clustering problem so as to reach a fair schedule. The solution reached by a commercial NLMP solver on the model is compared to the one reached by a faster hybrid algorithm based on multi-start local search. The approach is successfully tested on historical records from the recent Grand Slams tournaments.

physics.soc-ph↗
arXiv2018

A new exact approach for the Bilevel Knapsack with Interdiction Constraints

We consider the Bilevel Knapsack with Interdiction Constraints, an extension of the classic 0-1 knapsack problem formulated as a Stackelberg game with two agents, a leader and a follower, that choose items from a common set and hold their own private knapsacks. First, the leader selects some items to be interdicted for the follower while satisfying a capacity constraint. Then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits. The goal of the leader is to minimize the follower's profits. The presence of two decision levels makes this problem very difficult to solve in practice: the current state-of-the-art algorithms can solve to optimality instances with 50-55 items at most. We derive effective lower bounds and present a new exact approach that exploits the structure of the induced follower's problem. The approach successfully solves all benchmark instances within one second in the worst case and larger instances with up to 500 items within 60 seconds.

cs.GT↗
arXiv2018

Longest Processing Time rule for identical parallel machines revisited

We consider the Pm || Cmax scheduling problem where the goal is to schedule n jobs on m identical parallel machines to minimize makespan. We revisit the famous Longest Processing Time (LPT) rule proposed by Graham in 1969. LPT requires to sort jobs in non-ascending order of processing times and then to assign one job at a time to the machine whose load is smallest so far. We provide new insights on LPT and discuss the approximation ratio of a modification of LPT that improves Graham's bound from 4/3 - 1/(3m) to 4/3 - 1/(3(m-1)) for m >= 3 and from 7/6 to 9/8 for m = 2. We use Linear Programming (LP) to analyze the approximation ratio of our approach. This performance analysis can be seen as a valid alternative to formal proofs based on analytical derivation. Also, we derive from the proposed approach an O(n log n) heuristic. The heuristic splits the sorted jobset in tuples of m consecutive jobs (1,...,m; m+1,...,2m; etc.) and sorts the tuples in non-increasing order of the difference (slack) between largest job and smallest job in the tuple. Then, List Scheduling is applied to the set of sorted tuples. This approach strongly outperforms LPT on benchmark literature instances.

cs.DS↗
arXiv2018

Approximating the Incremental Knapsack Problem

We consider the 0-1 Incremental Knapsack Problem (IKP) where the capacity grows over time periods and if an item is placed in the knapsack in a certain period, it cannot be removed afterwards. The contribution of a packed item in each time period depends on its profit as well as on a time factor which reflects the importance of the period in the objective function. The problem calls for maximizing the weighted sum of the profits over the whole time horizon. In this work, we provide approximation results for IKP and its restricted variants. In some results, we rely on Linear Programming (LP) to derive approximation bounds and show how the proposed LP-based analysis can be seen as a valid alternative to more formal proof systems. We first manage to prove the tightness of some approximation ratios of a general purpose algorithm currently available in the literature and originally applied to a time-invariant version of the problem. We also devise a Polynomial Time Approximation Scheme (PTAS) when the input value indicating the number of periods is considered as a constant. Then, we add the mild and natural assumption that each item can be packed in the first time period. For this variant, we discuss different approximation algorithms suited for any number of time periods and for the special case with two periods.

cs.DS↗
arXiv2017

MILP and Max-Clique based heuristics for the Eternity II puzzle

The present paper considers a hybrid local search approach to the Eternity II puzzle and to unsigned, rectangular, edge matching puzzles in general. Both an original mixed-integer linear programming (MILP) formulation and a novel Max-Clique formulation are presented for this NP-hard problem. Although the presented formulations remain computationally intractable for medium and large sized instances, they can serve as the basis for developing heuristic decompositions and very large scale neighbourhoods. As a side product of the Max-Clique formulation, new hard-to-solve instances are published for the academic research community. Two reasonably well performing MILP-based constructive methods are presented and used for determining the initial solution of a multi-neighbourhood local search approach. Experimental results confirm that this local search can further improve the results obtained by the constructive heuristics and is quite competitive with the state of the art procedures.

cs.DS↗
arXiv2017

No-idle, no-wait: when shop scheduling meets dominoes, eulerian and hamiltonian paths

In shop scheduling, several applications exist where it is required that some components perform consecutively. We refer to no-idle schedules if machines are required to operate with no inserted idle time and no-wait schedules if tasks cannot wait between the end of an operation and the start of the following one. We consider here no-idle/no-wait shop scheduling problems with makespan as performance measure and determine related complexity results. We first analyze the two-machine no-idle/no-wait flow shop problem and show that it is equivalent to a special version of the game of dominoes which is polynomially solvable by tackling an Eulerian path problem on a directed graph. We present for this problem an O(n) exact algorithm. As a byproduct we show that the Hamiltonian Path problem on a digraph G(V,A) with a special structure (where every pair of vertices i,j either has all successors in common or has no common successors) reduces to the two-machine no-idle/no-wait flow shop problem. Correspondingly, we provide a new polynomially solvable special case of the Hamiltonian Path problem. Then, we show that also the corresponding $m$-machine no-idle no-wait flow shop problem is polynomially solvable and provide an O(mn log n) exact algorithm. Finally we prove that the 2-machine no-idle/no-wait job shop problem and the 2-machine no-idle/no-wait open shop problem are NP-Hard in the strong sense.

cs.DM↗
arXiv2017

Dynamic programming algorithms, efficient solution of the LP-relaxation and approximation schemes for the Penalized Knapsack Problem

We consider the 0-1 Penalized Knapsack Problem (PKP). Each item has a profit, a weight and a penalty and the goal is to maximize the sum of the profits minus the greatest penalty value of the items included in a solution. We propose an exact approach relying on a procedure which narrows the relevant range of penalties, on the identification of a core problem and on dynamic programming. The proposed approach turns out to be very effective in solving hard instances of PKP and compares favorably both to commercial solver CPLEX 12.5 applied to the ILP formulation of the problem and to the best available exact algorithm in the literature. Then we present a general inapproximability result and investigate several relevant special cases which permit fully polynomial time approximation schemes (FPTASs).

cs.DS↗