Vasilis Livanos

Kristóf Bérczi, Vasilis Livanos, José A. Soto et al.

The Matroid Secretary Problem is a central question in online optimization, modeling sequential decision-making under combinatorial constraints. We introduce a bipartite graph framework that unifies and extends several known formulations, including bipartite matching, matroid intersection, and matroid secretary problems. In this model, agents and items form a bipartite graph, and the objective is to select a matching that satisfies independence constraints on both sides. We first study the free-order setting under edge-arrivals. For $k$-matroid intersection, we leverage a core lemma by [FSZ, 2022] to design an $Ω(1/k^2)$-competitive algorithm, extending known results for single matroids. Building on this, we introduce $k$-growth systems -- a new class of independence systems that lie properly between $k$-matchoids and $k$-extendible systems and may be of independent combinatorial interest. We establish a generalized core lemma for $k$-growth systems, showing that a suitably defined set of critical elements retains a $Ω(1/k^2)$ fraction of the optimal weight. Using this lemma, we extend our $Ω(1/k^2)$-competitive algorithm to $k$-growth systems. We then study the agent-arrival model, which presents unique challenges to our framework. We extend the core lemma to this model and then apply it to obtain an $Ω(β/k^2)$-competitive algorithm for $k$-growth systems, where $β$ denotes the competitiveness of an appropriate type of order-oblivious algorithm for the item-side constraint. Finally, we extend our results to the case of multiple item selection, and obtain constant-competitive algorithms for fundamental cases such as partition matroids and $k$-matching constraints. We also study the closure properties and structural role and of $k$-growth systems within the hierarchy of $k$-systems.

Vasilis Livanos, Ruta Mehta, Aniket Murhekar

We study the problem of finding fair and efficient allocations of a set of indivisible items to a set of agents, where each item may be a good (positively valued) for some agents and a bad (negatively valued) for others, i.e., a mixed manna. As fairness notions, we consider arguably the strongest possible relaxations of envy-freeness and proportionality, namely envy-free up to any item (EFX and EFX$_0$), and proportional up to the maximin good or any bad (PropMX and PropMX$_0$). Our efficiency notion is Pareto-optimality (PO). We study two types of instances: (i) Separable, where the item set can be partitioned into goods and bads, and (ii) Restricted mixed goods (RMG), where for each item $j$, every agent has either a non-positive value for $j$, or values $j$ at the same $v_j>0$. We obtain polynomial-time algorithms for the following: (i) Separable instances: PropMX$_0$ allocation. (ii) RMG instances: Let pure bads be the set of items that everyone values negatively. - PropMX allocation for general pure bads. - EFX+PropMX allocation for identically-ordered pure bads. - EFX+PropMX+PO allocation for identical pure bads. Finally, if the RMG instances are further restricted to binary mixed goods where all the $v_j$'s are the same, we strengthen the results to guarantee EFX$_0$ and PropMX$_0$ respectively.