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.

In many scenarios of binary classification, only positive instances are provided in the training data, leaving the rest of the data unlabeled. This setup, known as positive-unlabeled (PU) learning, is addressed here with a network flow-based method which utilizes pairwise similarities between samples. The method we propose here, 2-HNC, leverages Hochbaum's Normalized Cut (HNC) and the set of solutions it provides by solving a parametric minimum cut problem. The set of solutions, that are nested partitions of the samples into two sets, correspond to varying tradeoff values between the two goals: high intra-similarity inside the sets and low inter-similarity between the two sets. This nested sequence is utilized here to deliver a ranking of unlabeled samples by their likelihood of being negative. Building on this insight, our method, 2-HNC, proceeds in two stages. The first stage generates this ranking without assuming any negative labels, using a problem formulation that is constrained only on positive labeled samples. The second stage augments the positive set with likely-negative samples and recomputes the classification. The final label prediction selects among all generated partitions in both stages, the one that delivers a positive class proportion, closest to a prior estimate of this quantity, which is assumed to be given. Extensive experiments across synthetic and real datasets show that 2-HNC yields strong performance and often surpasses existing state-of-the-art algorithms.

We present and discuss seven different open problems in applied combinatorics. The application areas relevant to this compilation include quantum computing, algorithmic differentiation, topological data analysis, iterative methods, hypergraph cut algorithms, and power systems.

We propose topology-aware feature partitioning into $k$ disjoint partitions for given scene features as a method for object-centric representation learning. To this end, we propose to use minimum $s$-$t$ graph cuts as a partitioning method which is represented as a linear program. The method is topologically aware since it explicitly encodes neighborhood relationships in the image graph. To solve the graph cuts our solution relies on an efficient, scalable, and differentiable quadratic programming approximation. Optimizations specific to cut problems allow us to solve the quadratic programs and compute their gradients significantly more efficiently compared with the general quadratic programming approach. Our results show that our approach is scalable and outperforms existing methods on object discovery tasks with textured scenes and objects.

In this paper, we develop a novel weighted Laplacian method, which is partially inspired by the theory of graph Laplacian, to study recent popular graph problems, such as multilevel graph partitioning and balanced minimum cut problem, in a more convenient manner. Since the weighted Laplacian strategy inherits the virtues of spectral methods, graph algorithms designed using weighted Laplacian will necessarily possess more robust theoretical guarantees for algorithmic performances, comparing with those existing algorithms that are heuristically proposed. In order to illustrate its powerful utility both in theory and in practice, we also present two effective applications of our weighted Laplacian method to multilevel graph partitioning and balanced minimum cut problem, respectively. By means of variational methods and theory of partial differential equations (PDEs), we have established the equivalence relations among the weighted cut problem, balanced minimum cut problem and the initial clustering problem that arises in the middle stage of graph partitioning algorithms under a multilevel structure. These equivalence relations can indeed provide solid theoretical support for algorithms based on our proposed weighted Laplacian strategy. Moreover, from the perspective of the application to the balanced minimum cut problem, weighted Laplacian can make it possible for research of numerical solutions of PDEs to be a powerful tool for the algorithmic study of graph problems. Experimental results also indicate that the algorithm embedded with our strategy indeed outperforms other existing graph algorithms, especially in terms of accuracy, thus verifying the efficacy of the proposed weighted Laplacian.

We propose a scalable and efficient algorithm for coclustering a higher-order tensor. Viewing tensors with hypergraphs, we propose formulating the co-clustering of a tensor as a problem of partitioning the corresponding hypergraph. Our algorithm is based on the random sampling technique, which has been successfully applied to graph cut problems. We extend a random sampling algorithm for the graph multiwaycut problem to hypergraphs, and design a co-clustering algorithm based on it. Each iteration of our algorithm runs in polynomial on the size of hypergraphs, and thus it performs well even for higher-order tensors, which are difficult to deal with for state-of-the-art algorithm.

The (constrained) minimization of a ratio of set functions is a problem frequently occurring in clustering and community detection. As these optimization problems are typically NP-hard, one uses convex or spectral relaxations in practice. While these relaxations can be solved globally optimally, they are often too loose and thus lead to results far away from the optimum. In this paper we show that every constrained minimization problem of a ratio of non-negative set functions allows a tight relaxation into an unconstrained continuous optimization problem. This result leads to a flexible framework for solving constrained problems in network analysis. While a globally optimal solution for the resulting non-convex problem cannot be guaranteed, we outperform the loose convex or spectral relaxations by a large margin on constrained local clustering problems.