A fundamental question in search-guided AI: what topology should guide Monte Carlo Tree Search (MCTS) in puzzle solving? Prior work applied topological features to guide MCTS in ARC-style tasks using grid topology -- the Laplacian spectral properties of cell connectivity -- and found no benefit. We identify the root cause: grid topology is constant across all instances. We propose measuring \emph{solution space topology} instead: the structure of valid color assignments constrained by detected pattern rules. We build this via compatibility graphs where nodes are $(cell, color)$ pairs and edges represent compatible assignments under pattern constraints. Our method: (1) detect pattern rules automatically with 100\% accuracy on 5 types, (2) construct compatibility graphs encoding solution space structure, (3) extract topological features (algebraic connectivity, rigidity, color structure) that vary with task difficulty, (4) integrate these features into MCTS node selection via sibling-normalized scores. We provide formal definitions, a rigorous selection formula, and comprehensive ablations showing that algebraic connectivity is the dominant signal. The work demonstrates that topology matters for search -- but only the \emph{right} topology. For puzzle solving, this is solution space structure, not problem space structure.

Artificial intelligence has gained widespread popularity and adoption across diverse industries due to its ability of automatic decision-making processes. In numerous contexts where artificial intelligence permeates various aspects of our lives, from business operations to societal dynamics and policy formulation, ensuring fairness is of greatest importance to meeting environmental, social, and governance standards. While for nearly any problem in the field of artificial intelligence, there can exist multiple measures of individual fairness as well as multiple measures of subgroup fairness. Often, Subgroup fairness involves multiple protected attributes (e.g., race, sex), creating numerous combinations of subgroups and corresponding subgroup fairness measures, all of which deserve consideration. Hence, it becomes essential to take into account the trade-offs among optimising for multiple fairness measures.

An example assignment with a cycle and a path maximizing the number of patients helped. In their work Parameterized Complexity of Kidney Exchange Revisited, presented at IJCAI 2024, Úrsula Hébert-Johnson, Daniel Lokshtanov, Chinmay Sonar and Vaishali Surianarayanan consider the Kidney Exchange Problem. We hear from Úrsula, Chinmay and Vaishali about kidney exchange, and how they went about solving two of the open problems in this field. Kidney disease affects tens of thousands of patients in the United States and hundreds of millions across the world. One way to treat kidney failure is to regularly undergo dialysis.

This letter proposes a method of global localization on a map with semantic object landmarks. One of the most promising approaches for localization on object maps is to use semantic graph matching using landmark descriptors calculated from the distribution of surrounding objects. These descriptors are vulnerable to misclassification and partial observations. Moreover, many existing methods rely on inlier extraction using RANSAC, which is stochastic and sensitive to a high outlier rate. To address the former issue, we augment the correspondence matching using Vision Language Models (VLMs). Landmark discriminability is improved by VLM embeddings, which are independent of surrounding objects. In addition, inliers are estimated deterministically using a graph-theoretic approach. We also incorporate pose calculation using the weighted least squares considering correspondence similarity and observation completeness to improve the robustness. We confirmed improvements in matching and pose estimation accuracy through experiments on ScanNet and TUM datasets.

Optimal allocation of agricultural water in the event of droughts is an important global problem. In addressing this problem, many aspects, including the welfare of farmers, the economy, and the environment, must be considered. Under this backdrop, our work focuses on several resource-matching problems accounting for agents with multi-crop portfolios, geographic constraints, and fairness. First, we address a matching problem where the goal is to maximize a welfare function in two-sided markets where buyers' requirements and sellers' supplies are represented by value functions that assign prices (or costs) to specified volumes of water. For the setting where the value functions satisfy certain monotonicity properties, we present an efficient algorithm that maximizes a social welfare function. When there are minimum water requirement constraints, we present a randomized algorithm which ensures that the constraints are satisfied in expectation. For a single seller--multiple buyers setting with fairness constraints, we design an efficient algorithm that maximizes the minimum level of satisfaction of any buyer. We also present computational complexity results that highlight the limits on the generalizability of our results. We evaluate the algorithms developed in our work with experiments on both real-world and synthetic data sets with respect to drought severity, value functions, and seniority of agents.

State minimization of combinatorial filters is a fundamental problem that arises, for example, in building cheap, resource-efficient robots. But exact minimization is known to be NP-hard. This paper conducts a more nuanced analysis of this hardness than up till now, and uncovers two factors which contribute to this complexity. We show each factor is a distinct source of the problem's hardness and are able, thereby, to shed some light on the role played by (1) structure of the graph that encodes compatibility relationships, and (2) determinism-enforcing constraints. Just as a line of prior work has sought to introduce additional assumptions and identify sub-classes that lead to practical state reduction, we next use this new, sharper understanding to explore special cases for which exact minimization is efficient. We introduce a new algorithm for constraint repair that applies to a large sub-class of filters, subsuming three distinct special cases for which the possibility of optimal minimization in polynomial time was known earlier. While the efficiency in each of these three cases previously appeared to stem from seemingly dissimilar properties, when seen through the lens of the present work, their commonality now becomes clear. We also provide entirely new families of filters that are efficiently reducible.

What is the minimal information that a robot must retain to achieve its task? To design economical robots, the literature dealing with reduction of combinatorial filters approaches this problem algorithmically. As lossless state compression is NP-hard, prior work has examined, along with minimization algorithms, a variety of special cases in which specific properties enable efficient solution. Complementing those findings, this paper refines the present understanding from the perspective of parameterized complexity. We give a fixed-parameter tractable algorithm for the general reduction problem by exploiting a transformation into minimal clique covering. The transformation introduces new constraints that arise from sequential dependencies encoded within the input filter -- some of these constraints can be repaired, others are treated through enumeration. Through this approach, we identify parameters affecting filter reduction that are based upon inter-constraint couplings (expressed as a notion of their height and width), which add to the structural parameters present in the unconstrained problem of minimal clique covering.

This study introduces a novel framework, G3Reg, for fast and robust global registration of LiDAR point clouds. In contrast to conventional complex keypoints and descriptors, we extract fundamental geometric primitives including planes, clusters, and lines (PCL) from the raw point cloud to obtain low-level semantic segments. Each segment is formulated as a unified Gaussian Ellipsoid Model (GEM) by employing a probability ellipsoid to ensure the ground truth centers are encompassed with a certain degree of probability. Utilizing these GEMs, we then present a distrust-and-verify scheme based on a Pyramid Compatibility Graph for Global Registration (PAGOR). Specifically, we establish an upper bound, which can be traversed based on the confidence level for compatibility testing to construct the pyramid graph. Gradually, we solve multiple maximum cliques (MAC) for each level of the graph, generating numerous transformation candidates. In the verification phase, we adopt a precise and efficient metric for point cloud alignment quality, founded on geometric primitives, to identify the optimal candidate. The performance of the algorithm is extensively validated on three publicly available datasets and a self-collected multi-session dataset, without changing any parameter settings in the experimental evaluation. The results exhibit superior robustness and real-time performance of the G3Reg framework compared to state-of-the-art methods. Furthermore, we demonstrate the potential for integrating individual GEM and PAGOR components into other algorithmic frameworks to enhance their efficacy. To advance further research and promote community understanding, we have publicly shared the source code.

Applications such as employees sharing office spaces over a workweek can be modeled as problems where agents are matched to resources over multiple rounds. Agents' requirements limit the set of compatible resources and the rounds in which they want to be matched. Viewing such an application as a multi-round matching problem on a bipartite compatibility graph between agents and resources, we show that a solution (i.e., a set of matchings, with one matching per round) can be found efficiently if one exists. To cope with situations where a solution does not exist, we consider two extensions. In the first extension, a benefit function is defined for each agent and the objective is to find a multi-round matching to maximize the total benefit. For a general class of benefit functions satisfying certain properties (including diminishing returns), we show that this multi-round matching problem is efficiently solvable. This class includes utilitarian and Rawlsian welfare functions. For another benefit function, we show that the maximization problem is NP-hard. In the second extension, the objective is to generate advice to each agent (i.e., a subset of requirements to be relaxed) subject to a budget constraint so that the agent can be matched. We show that this budget-constrained advice generation problem is NP-hard. For this problem, we develop an integer linear programming formulation as well as a heuristic based on local search. We experimentally evaluate our algorithms on synthetic networks and apply them to two real-world situations: shared office spaces and matching courses to classrooms.

Many scenarios where agents with restrictions compete for resources can be cast as maximum matching problems on bipartite graphs. Our focus is on resource allocation problems where agents may have restrictions that make them incompatible with some resources. We assume that a Principle chooses a maximum matching randomly so that each agent is matched to a resource with some probability. Agents would like to improve their chances of being matched by modifying their restrictions within certain limits. The Principle's goal is to advise an unsatisfied agent to relax its restrictions so that the total cost of relaxation is within a budget (chosen by the agent) and the increase in the probability of being assigned a resource is maximized. We establish hardness results for some variants of this budget-constrained maximization problem and present algorithmic results for other variants. We experimentally evaluate our methods on synthetic datasets as well as on two novel real-world datasets: a vacation activities dataset and a classrooms dataset.