36 research outputs found
Knapsack Problems with Side Constraints
The thesis considers a specific class of resource allocation problems in Combinatorial Optimization: the Knapsack Problems. These are paradigmatic NP-hard problems where a set of items with given profits and weights is available. The aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. In the classical 0-1 Knapsack Problem (KP), each item can be picked at most once.
The focus of the thesis is on four generalizations of KP involving side constraints beyond the capacity bound. More precisely, we provide solution approaches and insights for the following problems: The Knapsack Problem with Setups; the Collapsing Knapsack Problem; the Penalized Knapsack Problem; the Incremental Knapsack Problem.
These problems reveal challenging research topics with many real-life applications. The scientific contributions we provide are both from a theoretical and a practical perspective. On the one hand, we give insights into structural elements and properties of the problems and derive a series of approximation results for some of them. On the other hand, we offer valuable solution approaches for direct applications of practical interest or when the problems considered arise as sub-problems in broader contexts
The ZERO Regrets Algorithm: Optimizing over Pure Nash Equilibria via Integer Programming
Designing efficient algorithms to compute Nash equilibria poses considerable
challenges in Algorithmic Game Theory (AGT). We shed new light on the
intersection between Algorithmic Game Theory and Integer Programming. We
introduce ZERO Regrets, a general and efficient cutting plane algorithm to
compute, enumerate, and select Pure Nash Equilibria (PNEs) in Integer
Programming Games, a class of simultaneous and non-cooperative games. We
present a theoretical foundation for our algorithmic reasoning and provide a
polyhedral characterization of the convex hull of the Pure Nash Equilibria. We
introduce the concept of equilibrium inequality, and devise an equilibrium
separation oracle to separate non-equilibrium strategies from PNEs. We evaluate
our algorithmic framework on a wide range of problems from the literature and
provide a solid benchmark against the existing algorithmic approaches
The Baggage Belt Assignment Problem
We consider the problem of assigning flights to baggage belts in the baggage
reclaim area of an airport. The problem is originated by a real-life
application in Copenhagen airport. The objective is to construct a robust
schedule taking passenger and airline preferences into account. We consider a
number of business and fairness constraints, avoiding congestions, and ensuring
a good passenger flow. Robustness of the solutions is achieved by matching the
delivery time with the expected arrival time of passengers, and by adding
buffer time between two flights scheduled on the same belt. We denote this
problem as the Baggage Belt Assignment Problem (BBAP). We first derive a
general Integer Linear Programming (ILP) formulation for the problem. Then, we
propose a Branch-and-Price (B&P) algorithm based on a reformulation of the ILP
model tackled by Column Generation. Our approach relies on an effective dynamic
programming algorithm for handling the pricing problems. We tested the proposed
algorithm on a set of real-life data from Copenhagen airport as well as on a
set of instances inspired by the real data. Our B&P scheme outperforms a
commercial solver launched on the ILP formulation of the problem and is
effective in delivering high quality solutions in limited computational times,
making it possible its use in daily operations in medium-sized and large
airports
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
The 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 (m = 3 and from 7/6 to 9/8 for m = 2. We use linear programming 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) time complexity heuristic. The heuristic splits the sorted job set 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, given this new ordering of the job set, list scheduling is applied. This approach strongly outperforms LPT on benchmark literature instances and is competitive with more involved approaches such as COMBINE and LDM