30 research outputs found
A branch-and-cut algorithm for the Edge Interdiction Clique Problem
Given a graph G and an interdiction budget kâN, the Edge Interdiction Clique Problem (EICP) asks to find a subset of at most k edges to remove from G so that the size of the maximum clique, in the interdicted graph, is minimized. The EICP belongs to the family of interdiction problems with the aim of reducing the clique number of the graph. The EICP optimal solutions, called optimal interdiction policies, determine the subset of most vital edges of a graph which are crucial for preserving its clique number. We propose a new set-covering-based Integer Linear Programming (ILP) formulation for the EICP with an exponential number of constraints, called the clique-covering inequalities. We design a new branch-and-cut algorithm which is enhanced by a tailored separation procedure and by an effective heuristic initialization phase. Thanks to the new exact algorithm, we manage to solve the EICP in several sets of instances from the literature. Extensive tests show that the new exact algorithm greatly outperforms the state-of-the-art approaches for the EICP
An exact approach for the bilevel knapsack problem with interdiction constraints and extensions
We consider the bilevel knapsack problem with interdiction constraints, an extension of the classic 0â1 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 followerâs total profit. We derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced followerâs problem. The approach strongly outperforms the current state-of-the-art algorithms designed for the problem. We extend the same algorithmic framework to the interval minâmax regret knapsack problem after providing a novel bilevel programming reformulation. Also for this problem, the proposed approach outperforms the exact algorithms available in the literature
Solution techniques for Bi-level Knapsack Problems
Traditional funding mechanisms for healthcare projects involve ranking the projects and awarding funds based on their cost to benefit ratio. An alternative funding mechanism based on Bi-level programming was proposed in the literature. We refer to this as Donor-Recipient Bi-level Knapsack Problem (DR-BKP), which we explore further in this work. There are two participants, a leader (a donor agency) and a follower (recipient country) in this problem. Both the participants have their individual budgets. There is a set of projects, each having a certain cost and profit associated. The cost of projects are common to both the participants however the profits can be different for them. There is an external project that is of exclusive interest to the follower. The leader decides on cost subsidies to provide for the projects that is within her budget, while the follower solves a knapsack problem with the cost subsidised projects and the external project. Two enumerative algorithms were proposed in the literature for Bi-level problems with discrete upper level variables. We adapt them for DR-BKP that has continuous upper level variables having non-linear interaction with lower level variables. We first show the existence of a solution for DR-BKP and show the convergence of these algorithms. We provide evidence for -hardness by showing that the problem is both NP-hard and Co-NP hard. Finally, we have implemented these two enumerative algorithms and shared the results and analyses of the computational experiments. A set of fifteen differing data sets each having randomly generated 10 instances have been solved to evaluate the performance of the proposed algorithms
Bilevel Network Design
This chapter is devoted to network design problems involving conflicting agents, referred to as the designer and the users, respectively. Such problems are best cast into the framework of bilevel programming, where the designer anticipates the reaction or rational users to its course of action, and fits many situations of interest. In this chapter, we consider four applications of very different nature, with a special focus on algorithmic issues
Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality
International audienceLinear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem's optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances
Curriculum learning for multilevel budgeted combinatorial problems
Learning heuristics for combinatorial optimization problems through graph
neural networks have recently shown promising results on some classic NP-hard
problems. These are single-level optimization problems with only one player.
Multilevel combinatorial optimization problems are their generalization,
encompassing situations with multiple players taking decisions sequentially. By
framing them in a multi-agent reinforcement learning setting, we devise a
value-based method to learn to solve multilevel budgeted combinatorial problems
involving two players in a zero-sum game over a graph. Our framework is based
on a simple curriculum: if an agent knows how to estimate the value of
instances with budgets up to , then solving instances with budget can
be done in polynomial time regardless of the direction of the optimization by
checking the value of every possible afterstate. Thus, in a bottom-up approach,
we generate datasets of heuristically solved instances with increasingly larger
budgets to train our agent. We report results close to optimality on graphs up
to nodes and a speedup on average compared to the quickest
exact solver known for the Multilevel Critical Node problem, a max-min-max
trilevel problem that has been shown to be at least -hard
Finding optimal Stackelberg production strategies: How to produce in times of war?
Inspired by a military context, we study a Stackelberg production game where
a country's government, the leader, wants to maximize the production of
military assets. The leader does so by allocating his resources among a set of
production facilities. His opponent, the follower, observes this allocation and
tries to destroy the associated production as much as possible by allocating
his destructive resources, for example bombs, among these facilities. In this
paper, we identify a follower's optimal strategy. For the leader, we show that
an optimal production strategy can be found in the class of so-called
seried-balanced strategies. We present a linear time algorithm that finds an
optimal strategy in this class
Leadership in Singleton Congestion Games: What is Hard and What is Easy
We study the problem of computing Stackelberg equilibria Stackelberg games
whose underlying structure is in congestion games, focusing on the case where
each player can choose a single resource (a.k.a. singleton congestion games)
and one of them acts as leader. In particular, we address the cases where the
players either have the same action spaces (i.e., the set of resources they can
choose is the same for all of them) or different ones, and where their costs
are either monotonic functions of the resource congestion or not. We show that,
in the case where the players have different action spaces, the cost the leader
incurs in a Stackelberg equilibrium cannot be approximated in polynomial time
up to within any polynomial factor in the size of the game unless P = NP,
independently of the cost functions being monotonic or not. We show that a
similar result also holds when the players have nonmonotonic cost functions,
even if their action spaces are the same. Differently, we prove that the case
with identical action spaces and monotonic cost functions is easy, and propose
polynomial-time algorithm for it. We also improve an algorithm for the
computation of a socially optimal equilibrium in singleton congestion games
with the same action spaces without leadership, and extend it to the
computation of a Stackelberg equilibrium for the case where the leader is
restricted to pure strategies. For the cases in which the problem of finding an
equilibrium is hard, we show how, in the optimistic setting where the followers
break ties in favor of the leader, the problem can be formulated via
mixed-integer linear programming techniques, which computational experiments
show to scale quite well