On a Stackelberg Subset Sum Game

This contribution deals with a two-level discrete decision problem, a so-called Stackelberg strategic game: A Subset Sum setting is addressed with a set $N$ of items with given integer weights. One distinguished player, the leader, may alter the weights of the items in a given subset $L\subset N$, and a second player, the follower, selects a solution $A\subseteq N$ in order to utilize a bounded resource in the best possible way. Finally, the leader receives a payoff from those items of its subset $L$ that were included in the overall solution $A$, chosen by the follower. We assume that the follower applies a publicly known, simple, heuristic algorithm to determine its solution set, which avoids having to solve NP-hard problems. Two variants of the problem are considered, depending on whether the leader is able to control (i.e., change) the weights of its items (i) in the objective function or (ii) in the bounded resource constraint. The leader's objective is the maximization of the overall weight reduction, for the first variant, or the maximization of the weight increase for the latter one. In both variants there is a trade-off for each item between the contribution value to the leader's objective and the chance of being included in the follower's solution set. We analyze the leader's pricing problem for a natural greedy strategy of the follower and discuss the complexity of the corresponding problems. We show that setting the optimal weight values for the leader is, in general, NP-hard. It is even NP-hard to provide a solution within a constant factor of the best possible solution. Exact algorithms, based on dynamic programming and running in pseudopolynomial time, are provided. The additional cases, in which the follower faces a continuous (linear relaxation) version of the above problems, are shown to be straightforward to solve.Comment: 13 pages, 1 figur

Models and algorithms for decomposition problems

This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms. This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced. The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity

The multilevel critical node problem : theoretical intractability and a curriculum learning approach

Évaluer la vulnérabilité des réseaux est un enjeu de plus en plus critique. Dans ce mémoire, nous nous penchons sur une approche étudiant la défense d’infrastructures stratégiques contre des attaques malveillantes au travers de problèmes d'optimisations multiniveaux. Plus particulièrement, nous analysons un jeu séquentiel en trois étapes appelé le « Multilevel Critical Node problem » (MCN). Ce jeu voit deux joueurs s'opposer sur un graphe: un attaquant et un défenseur. Le défenseur commence par empêcher préventivement que certains nœuds soient attaqués durant une phase de vaccination. Ensuite, l’attaquant infecte un sous ensemble des nœuds non vaccinés. Finalement, le défenseur réagit avec une stratégie de protection. Dans ce mémoire, nous fournissons les premiers résultats de complexité pour MCN ainsi que ceux de ses sous-jeux. De plus, en considérant les différents cas de graphes unitaires, pondérés ou orientés, nous clarifions la manière dont la complexité de ces problèmes varie. Nos résultats contribuent à élargir les familles de problèmes connus pour être complets pour les classes NP, $\Sigma_2^p$ et $\Sigma_3^p$. Motivés par l’insolubilité intrinsèque de MCN, nous concevons ensuite une heuristique efficace pour le jeu. Nous nous appuyons sur les approches récentes cherchant à apprendre des heuristiques pour des problèmes d’optimisation combinatoire en utilisant l’apprentissage par renforcement et les réseaux de neurones graphiques. Contrairement aux précédents travaux, nous nous intéressons aux situations dans lesquelles de multiples joueurs prennent des décisions de manière séquentielle. En les inscrivant au sein du formalisme d’apprentissage multiagent, nous concevons un algorithme apprenant à résoudre des problèmes d’optimisation combinatoire multiniveaux budgétés opposant deux joueurs dans un jeu à somme nulle sur un graphe. Notre méthode est basée sur un simple curriculum : si un agent sait estimer la valeur d’une instance du problème ayant un budget au plus B, alors résoudre une instance avec budget B+1 peut être fait en temps polynomial quelque soit la direction d’optimisation en regardant la valeur de tous les prochains états possibles. Ainsi, dans une approche ascendante, nous entraînons notre agent sur des jeux de données d’instances résolues heuristiquement avec des budgets de plus en plus grands. Nous rapportons des résultats quasi optimaux sur des graphes de tailles au plus 100 et un temps de résolution divisé par 185 en moyenne comparé au meilleur solutionneur exact pour le MCN.Evaluating the vulnerability of networks is a problem which has gain momentum in recent decades. In this work, we focus on a Multilevel Programming approach to study the defense of critical infrastructures against malicious attacks. We analyze a three-stage sequential game played in a graph called the Multilevel Critical Node problem (MCN). This game sees two players competing with each other: a defender and an attacker. The defender starts by preventively interdicting nodes from being attacked during what is called a vaccination phase. Then, the attacker infects a subset of non-vaccinated nodes and, finally, the defender reacts with a protection strategy. We provide the first computational complexity results associated with MCN and its subgames. Moreover, by considering unitary, weighted, undirected and directed graphs, we clarify how the theoretical tractability or intractability of those problems vary. Our findings contribute with new NP-complete, $\Sigma_2^p$-complete and $\Sigma_3^p$-complete problems. Motivated by the intrinsic intractability of the MCN, we then design efficient heuristics for the game by building upon the recent approaches seeking to learn heuristics for combinatorial optimization problems through graph neural networks and reinforcement learning. But contrary to previous work, we tackle 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 B, then solving instances with budget B+1 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 100 nodes and a 185 x speedup on average compared to the quickest exact solver known for the MCN

Resource Management in Distributed Camera Systems

The aim of this work is to investigate different methods to solve the problem of allocating the correct amount of resources (network bandwidth and storage space) to video camera systems. Here we explore the intersection between two research areas: automatic control and game theory. Camera systems are a good example of the emergence of the Internet of Things (IoT) and its impact on our daily lives and the environment. We aim to improve today’s systems, shift from resources over-provisioning to allocate dynamically resources where they are needed the most. We optimize the storage and bandwidth allocation of camera systems to limit the impact on the environment as well as provide the best visual quality attainable with the resource limitations. This thesis is written as a collection of papers. It begins by introducing the problem with today’s camera systems, and continues with background information about resource allocation, automatic control and game theory. The third chapter de- scribes the models of the considered systems, their limitations and challenges. It then continues by providing more background on the automatic control and game theory techniques used in the proposed solutions. Finally, the proposed solutions are provided in five papers.Paper I proposes an approach to estimate the amount of data needed by surveillance cameras given camera and scenario parameters. This model is used for calculating the quasi Worst-Case Transmission Times of videos over a network. Papers II and III apply control concepts to camera network storage and bandwidth assignment. They provide simple, yet elegant solutions to the allocation of these resources in distributed camera systems. Paper IV com- bines pricing theory with control techniques to force the video quality of cam- era systems to converge to a common value based solely on the compression parameter of the provided videos. Paper V uses the VCG auction mechanism to solve the storage space allocation problem in competitive camera systems. It allows for a better system-wide visual quality than a simple split allocation given the limited system knowledge, trust and resource constraints