Lydia Tlilane

A Protocol for Cutting Matroids Like Cakes

We study a problem that generalizes the fair allocation of indivisible goods. The input is a matroid and a set of agents. Each agent has his own utility for every element of the matroid. Our goal is to build a base of the matroid and provide worst case guarantees on the additive utilities of the agents. These utilities are private, an assumption that is commonly made for the fair division of divisible resources, Since the use of an algorithm is not appropriate in this context, we resort to protocols, like in cake cutting problems. Our contribution is a protocol where the agents can interact and build a base of the matroid. If there are up to 8 agents, we show how everyone can ensure that his worst case utility for the resulting base is the same as those given by Markakis and Psomas [18] for the fair allocation of indivisible goods, based on the guarantees of Demko and Hill [8].

Approximate tradeoffs on weighted labeled matroids

We consider problems where a solution is evaluated with a couple. Each coordinate of this couple represents the utility of an agent. Due to the possible conflicts, it is unlikely that one feasible solution is optimal for both agents. Then a natural aim is to find a tradeoff. We investigate tradeoff solutions with worst case guarantees for the agents. The focus is on discrete problems having a matroid structure and the utility of an agent is modeled with a function which is either additive or weighted labeled. We provide polynomial-time deterministic algorithms which achieve several guarantees and we prove that some guarantees are not possible to reach.

Near fairness in matroids

This article deals with the fair allocation of indivisible goods and its generalization to matroids. The notions of fairness under consideration are equitability, proportionality and envy-freeness. It is long known that some instances fail to admit a fair allocation. However, an almost fair solution may exist if an appropriate relaxation of the fairness condition is adopted. This article deals with a matroid problem which comprises the allocation of indivisible goods as a special case. It is to find a base of a matroid and to allocate it to a pool of agents. We first adapt the aforementioned fairness concepts to matroids. Next we propose a relaxed notion of fairness said to be near to fairness. Near fairness respects the fairness up to one element. We show that a nearly fair solution always exists and it can be constructed in polynomial time in the general context of matroids.

Approximate tradeoffs on matroids

We consider problems where a solution is evaluated with a couple. Each coordinate of this couple represents an agents utility. Due to the possible conflicts, it is unlikely that one feasible solution is optimal for both agents. Then, a natural aim is to find tradeoffs. We investigate tradeoff solutions with guarantees for the agents. The focus is on discrete problems having a matroid structure. We provide polynomial-time deterministic algorithms which achieve several guarantees and we prove that some guarantees are not possible to reach.

Subset sum problems with digraph constraints

We introduce and study optimization problems which are related to the well-known Subset Sum problem. In each new problem, a node-weighted digraph is given and one has to select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints called digraph constraints and maximality need to be satisfied. The digraph constraint imposes that a node must belong to the solution if at least one of its predecessors is in the solution. An alternative of this constraint says that a node must belong to the solution if all its predecessors are in the solution. The maximality constraint ensures that no superset of a feasible solution is also feasible. The combination of these constraints provides four problems. We study their complexity and present some approximation results according to the type of input digraph, such as directed acyclic graphs and oriented trees.