Danil Sagunov

Yuriy Dementiev, Fedor Pribytkov, Danil Sagunov

We study the problem of computing an envy-free (EF) allocation of $m$ indivisible items among $n$ agents when items come in three distinct types. Each agent holds additive valuations over item types that may be positive (goods), negative (chores), or mixed. We present the first subexponential-time algorithm with running time time $(n \cdot m)^{O(\sqrt{n})}$ that finds an EF allocation whenever one exists, or correctly reports that none exists. Our approach exploits a geometric representation of EF allocations as convex polyhedra in $\mathbb{R}^3$ and applies Miller's planar cycle-separator theorem to recursively decompose the agent set into balanced subgroups. We further extend the algorithm to handle agents whose allocations are fixed in advance, preserving envy-freeness across all agents.

Pål Grønås Drange, Fedor V. Fomin, Petr Golovach et al.

We study a Stackelberg variant of the classical Most Vital Links problem, modeled as a one-round adversarial game between an attacker and a defender. The attacker strategically removes up to $k$ edges from a flow network to maximally disrupt flow between a source $s$ and a sink $t$, after which the defender optimally reroutes the remaining flow. To capture this attacker--defender interaction, we introduce a new mathematical model of discounted cuts, in which the cost of a cut is evaluated by excluding its $k$ most expensive edges. This model generalizes the Most Vital Links problem and uncovers novel algorithmic and complexity-theoretic properties. We develop a unified algorithmic framework for analyzing various forms of discounted cut problems, including minimizing or maximizing the cost of a cut under discount mechanisms that exclude either the $k$ most expensive or the $k$ cheapest edges. While most variants are NP-complete on general graphs, our main result establishes polynomial-time solvability for all discounted cut problems in our framework when the input is restricted to bounded-genus graphs, a relevant class that includes many real-world networks such as transportation and infrastructure networks. With this work, we aim to open collaborative bridges between artificial intelligence, algorithmic game theory, and operations research.

Sergei Khargeliia, Danil Sagunov

We study the weighted case of the \textsc{Maximum Acyclic Subgraph (MAS)} problem, where each edge of a given directed graph has a positive weight assigned, and the task is to find a maximum-weight acyclic edge set. The famous and well-studied random ordering lower bound guarantees the existence of an acyclic set that gives at least the half of the total edge weight. The maximum spanning tree (MaxST) guarantee, which is the weight of a maximum-weight acyclic subgraph of the underlying undirected graph of $G$, is another natural lower bound for the weight of an acyclic subgraph. A solution of this weight dominates the random ordering solution on instances where MaxST spans the most of the total edge weight. Our main contribution are two parameterized algorithms that find acyclic subgraphs of total weight larger than the weight of the MaxST of $G$. Both our algorithms find a solution of total weight at least $MaxST(G)+k$, for a given integer $k\ge 0$, or report that it does not exist, and first of our algorithms runs in time $2^{k^{\mathcal{O}(1)}}\cdot \mathcal{I}^{\mathcal{O}(1)}$ and works when all weights are integers; our second algorithm handles rational weights not less than $1$, and its running time is upper-bounded by $n^{k^{\mathcal{O}(1)}}\cdot \mathcal{I}^{\mathcal{O}(1)}$. This positive result is rather surprising since solving \textsc{MAS} above the random ordering lower bound is \classNP-hard in the same rational weights scenario, when $k=1$. Our findings unravel intricate connections between structure of MaxSTs and directed cycles, use perfect graph theorem to tackle rational weights, and raise graph-theoretic questions that are interesting on their own. Of another importance, this is one of the few examples of positive ``above guarantee'' results for a weighted problem on directed graphs, especially for rational weights.