Mixed-Integer Linear Programming (MILP) lies at the core of many real-world combinatorial optimization (CO) problems, traditionally solved by branch-and-bound (B&B). A key driver influencing B&B solvers efficiency is the variable selection heuristic that guides branching decisions. Looking to move beyond static, hand-crafted heuristics, recent work has explored adapting traditional reinforcement learning (RL) algorithms to the B&B setting, aiming to learn branching strategies tailored to specific MILP distributions. In parallel, RL agents have achieved remarkable success in board games, a very specific type of combinatorial problems, by leveraging environment simulators to plan via Monte Carlo Tree Search (MCTS). Building on these developments, we introduce Plan-and-Branch-and-Bound (PlanB&B), a model-based reinforcement learning (MBRL) agent that leverages a learned internal model of the B&B dynamics to discover improved branching strategies. Computational experiments empirically validate our approach, with our MBRL branching agent outperforming previous state-of-the-art RL methods across four standard MILP benchmarks.

Mixed-Integer Linear Programming (MILP) is a powerful framework used to address a wide range of NP-hard combinatorial optimization problems, often solved by Branch and bound (B&B). A key factor influencing the performance of B&B solvers is the variable selection heuristic governing branching decisions. Recent contributions have sought to adapt reinforcement learning (RL) algorithms to the B&B setting to learn optimal branching policies, through Markov Decision Processes (MDP) inspired formulations, and ad hoc convergence theorems and algorithms. In this work, we introduce BBMDP, a principled vanilla MDP formulation for variable selection in B&B, allowing to leverage a broad range of RL algorithms for the purpose of learning optimal B&B heuristics. Computational experiments validate our model empirically, as our branching agent outperforms prior state-of-the-art RL agents on four standard MILP benchmarks.

Abstract Classic mosaic is an old and durable art form. Generating artificial classic mosaics from digital images is an interesting problem that has attracted attention in recent years. Previous approaches to mosaic generation are largely based on heuristics, and therefore it is harder to analyse, predict and improve their performance. In addition, previous methods have a number of disadvantages, such as requiring that the number of tiles in a mosaic is known a priori, or relying on extensive user interaction, or using heuristics for tile placement that lead to visible artefacts. We propose a classic mosaic generation algorithm that is based on a principled global optimization. Our approach is fully automatic. We design and optimize an objective function that incorporates the desired mosaic properties, such as tile alignment to significant image edges, prohibiting tile overlap, etc. Our optimization method is based on graph cuts, which proved to be a powerful optimization tool in graphics and computer vision. Experimental comparison to previous work demonstrate the advantages of our approach.

Capacity scaling is a hierarchical approach to graph representation that can improve theoretical complexity and practical efficiency of max-flow/min-cut algorithms. Introduced by Edmonds, Karp, and Dinic in 1972, capacity scaling is well known in the combinatorial optimization community. Surprisingly, this major performance improving technique is overlooked in computer vision where graph cut methods typically solve energy minimization problems on huge N-D grids and algorithms' efficiency is a widely studied issue. Unlike some earlier hierarchical methods addressing efficiency of graph cuts in imaging, e.g. (H. Lombaert, 2005), capacity scaling preserves global optimality of the solution. This is the main motivation for our work studying capacity scaling in the context of vision. We show that capacity scaling significantly reduces non-polynomial theoretical time complexity of the max-flow algorithm in (Y. Boykov and V. Kolmorogorov, 2004) to weakly polynomial O(m2n2log(U)) where U is the largest edge weight. While (Y. Boykov and V. Kolmorogorov, 2004) is the fastest method for many applications in vision, capacity scaling gives several folds speed-ups for problems with large number of local minima. The effect is particularly strong in 3D applications with denser neighborhoods.

This paper adds a number of novel concepts into global s/t cut methods improving their efficiency and making them relevant for a wider class of applications in vision where algorithms should ideally run in real-time. Our new Active Cuts (AC) method can effectively use a good approximate solution (initial cut) that is often available in dynamic, hierarchical, and multi-label optimization problems in vision. In many problems AC works faster than the state-of-the-art max-flow methods [2] even if initial cut is far from the optimal one. Moreover, empirical speed improves several folds when initial cut is spatially close to the optima. Before converging to a global minima, Active Cuts outputs a multitude of intermediate solutions (intermediate cuts) that, for example, can be used be accelerate iterative learning-based methods or to improve visual perception of graph cuts realtime performance when large volumetric data is segmented. Finally, it can also be combined with many previous methods for accelerating graph cuts.