arXiv subjects

Alex Crane

Publications and source records attributed to Alex Crane.

10 recordsLinked to original sources

arXiv2026

A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation

The algorithmic differentiation (AD) of mathematical functions can be interpreted as a sequence of vertex eliminations in an underlying directed acyclic graph. The problem of determining a minimum-cost elimination ordering, which we call Optimal Vertex Elimination, is NP-complete. Consequently, much effort has been devoted to the design of heuristics. Many of these heuristics are widely believed to perform well in practice, but this hypothesis has so far been difficult to test due to the lack of scalable exact methods. We design and engineer new integer programming formulations for Optimal Vertex Eliminatioin and for a related objective we call Minimum Edge Count. Our implementations scale to graphs one-to-two orders of magnitude larger than existing techniques, enabling the assembly of a corpus of medium-sized graphs for which optimal solutions are known. This corpus facilitates a study of existing heuristics, confirming that on real data popular methods achieve high quality solutions. We also make several theoretical contributions. We give a tight analysis of the forward and reverse modes of AD, and extend our techniques to provide a simple algorithm for Optimal Vertex Elimination with approximation ratio parameterized by the size of a minimum source-sink separator. On the complexity side, we give the first approximation lower bounds for both problems.

cs.DS↗

arXiv2025

Structural Optimal Jacobian Accumulation and Minimum Edge Count are NP-Complete Under Vertex Elimination

We study graph-theoretic formulations of two fundamental problems in algorithmic differentiation. The first (Structural Optimal Jacobian Accumulation) is that of computing a Jacobian while minimizing multiplications. The second (Minimum Edge Count) is to find a minimum-size computational graph. For both problems, we consider the vertex elimination operation. Our main contribution is to show that both problems are NP-complete, thus resolving longstanding open questions. In contrast to prior work, our reduction for Structural Optimal Jacobian Accumulation does not rely on any assumptions about the algebraic relationships between local partial derivatives; we allow these values to be mutually independent. We also provide $O^*(2^n)$-time exact algorithms for both problems, and show that under the exponential time hypothesis these running times are essentially tight. Finally, we provide a data reduction rule for Structural Optimal Jacobian Accumulation by showing that false twins may always be eliminated consecutively.

cs.DS↗

arXiv2025

Equalizing Closeness Centralities via Edge Additions

Graph modification problems with the goal of optimizing some measure of a given node's network position have a rich history in the algorithms literature. Less commonly explored are modification problems with the goal of equalizing positions, though this class of problems is well-motivated from the perspective of equalizing social capital, i.e., algorithmic fairness. In this work, we study how to add edges to make the closeness centralities of a given pair of nodes more equal. We formalize two versions of this problem: Closeness Ratio Improvement, which aims to maximize the ratio of closeness centralities between two specified nodes, and Closeness Gap Minimization, which aims to minimize the absolute difference of centralities. We show that both problems are $\textsf{NP}$-hard, and for Closeness Ratio Improvement we present a quasilinear-time $\frac{6}{11}$-approximation, complemented by a bicriteria inapproximability bound. In contrast, we show that Closeness Gap Minimization admits no multiplicative approximation unless $\textsf{P} = \textsf{NP}$. We conclude with a discussion of open directions for this style of problem, including several natural generalizations.

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arXiv2025

Edge-Colored Clustering in Hypergraphs: Beyond Minimizing Unsatisfied Edges

We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to color) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color nodes to minimize the number of unsatisfied hyperedges -- those containing one or more nodes whose color does not match the hyperedge color. We motivate and present advances for several directions that extend beyond this minimization problem. We first provide new algorithms for maximizing satisfied edges, which is the same at optimality but is much more challenging to approximate, with all prior work restricted to graphs. We develop the first approximation algorithm for hypergraphs, and then refine it to improve the best-known approximation factor for graphs. We then introduce new objective functions that incorporate notions of balance and fairness, and provide new hardness results, approximations, and fixed-parameter tractability results.

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arXiv2024

A Space-Efficient Algebraic Approach to Robotic Motion Planning

We consider efficient route planning for robots in applications such as infrastructure inspection and automated surgical imaging. These tasks can be modeled via the combinatorial problem Graph Inspection. The best known algorithms for this problem are limited in practice by exponential space complexity. In this paper, we develop a memory-efficient approach using algebraic tools related to monomial testing on the polynomials associated with certain arithmetic circuits. Our contributions are two-fold. We first repair a minor flaw in existing work on monomial detection using a new approach we call tree certificates. We further show that, in addition to detection, these tools allow us to efficiently recover monomials of interest from circuits, opening the door for significantly broadened application of related algebraic tools. For Graph Inspection, we design and evaluate a complete algebraic pipeline. Our engineered implementation demonstrates that circuit-based algorithms are indeed memory-efficient in practice, thus encouraging further engineering efforts.

cs.RO↗

arXiv2024

Fast algorithms to improve fair information access in networks

We consider the problem of selecting $k$ seed nodes in a network to maximize the minimum probability of activation under an independent cascade beginning at these seeds. The motivation is to promote fairness by ensuring that even the least advantaged members of the network have good access to information. Our problem can be viewed as a variant of the classic influence maximization objective, but it appears somewhat more difficult to solve: only heuristics are known. Moreover, the scalability of these methods is sharply constrained by the need to repeatedly estimate access probabilities. We design and evaluate a suite of $10$ new scalable algorithms which crucially do not require probability estimation. To facilitate comparison with the state-of-the-art, we make three more contributions which may be of broader interest. We introduce a principled method of selecting a pairwise information transmission parameter used in experimental evaluations, as well as a new performance metric which allows for comparison of algorithms across a range of values for the parameter $k$. Finally, we provide a new benchmark corpus of $174$ networks drawn from $6$ domains. Our algorithms retain most of the performance of the state-of-the-art while reducing running time by orders of magnitude. Specifically, a meta-learner approach is on average only $20\%$ less effective than the state-of-the-art on held-out data, but about $75-130$ times faster. Further, the meta-learner's performance exceeds the state-of the-art on about $20\%$ of networks, and the magnitude of its running time advantage is maintained on much larger networks.

cs.SI↗

arXiv2024

Optimizing Probabilistic Propagation in Graphs by Adding Edges

Probabilistic graphs are an abstraction that allow us to study randomized propagation in graphs. In a probabilistic graph, each edge is "active" with a certain probability, independent of the other edges. For two vertices $u,v$, a classic quantity of interest, that we refer to as the proximity $\mathcal{P}_{G}(u, v)$, is the probability that there exists a path between $u$ and $v$ all of whose edges are active. For a given subset of vertices $V_s$, the reach of $V_s$ is defined as the minimum over pairs $u \in V_s$ and $v \in V$ of the proximity $\mathcal{P}_{G}(u,v)$. This quantity has been studied in the context of multicast in unreliable communication networks and in social network analysis. We study the problem of improving the reach in a probabilistic graph via edge augmentation. Formally, given a budget $k$ of edge additions and a set of source vertices $V_s$, the goal of Reach Improvement is to maximize the reach of $V_s$ by adding at most $k$ new edges to the graph. The problem was introduced in earlier empirical work in the algorithmic fairness community. We provide the first approximation guarantees and hardness results for Reach Improvement. We prove that the existence of a good augmentation implies a cluster structure for the graph. We use this structural result to analyze a novel algorithm that outputs a $k$-edge augmentation with an objective value that is poly($\beta^*$), where $\beta^*$ is the objective value for the optimal augmentation. We also give an algorithm that adds $O(k \log n)$ edges and yields a multiplicative approximation to $\beta^*$. Our arguments rely on new probabilistic tools for analyzing proximity, inspired by techniques in percolation theory; these tools may be of broader interest. Finally, we show that significantly better approximations are unlikely, under known hardness assumptions related to gap variants of the classic Set Cover problem.

cs.DS↗

arXiv2024

Leveraging Fixed-Parameter Tractability for Robot Inspection Planning

Autonomous robotic inspection, where a robot moves through its environment and inspects points of interest, has applications in industrial settings, structural health monitoring, and medicine. Planning the paths for a robot to safely and efficiently perform such an inspection is an extremely difficult algorithmic challenge. In this work we consider an abstraction of the inspection planning problem which we term Graph Inspection. We give two exact algorithms for this problem, using dynamic programming and integer linear programming. We analyze the performance of these methods, and present multiple approaches to achieve scalability. We demonstrate significant improvement both in path weight and inspection coverage over a state-of-the-art approach on two robotics tasks in simulation, a bridge inspection task by a UAV and a surgical inspection task using a medical robot.

cs.RO↗

arXiv2024

Correlation Clustering with Vertex Splitting

We explore Cluster Editing and its generalization Correlation Clustering with a new operation called permissive vertex splitting which addresses finding overlapping clusters in the face of uncertain information. We determine that both problems are NP-hard, yet they exhibit significant differences in parameterized complexity and approximability. For Cluster Editing with Permissive Vertex Splitting, we show a polynomial kernel when parameterized by the solution size and develop a polynomial-time algorithm with approximation factor 7. In the case of Correlation Clustering, we establish para-NP-hardness when parameterized by solution size and demonstrate that computing an $n^{1-\epsilon}$-approximation is NP-hard for any constant $\epsilon > 0$. Additionally, we extend the established link between Correlation Clustering and Multicut to the setting with permissive vertex splitting.

cs.DS↗

arXiv2023

Overlapping and Robust Edge-Colored Clustering in Hypergraphs

A recent trend in data mining has explored (hyper)graph clustering algorithms for data with categorical relationship types. Such algorithms have applications in the analysis of social, co-authorship, and protein interaction networks, to name a few. Many such applications naturally have some overlap between clusters, a nuance which is missing from current combinatorial models. Additionally, existing models lack a mechanism for handling noise in datasets. We address these concerns by generalizing Edge-Colored Clustering, a recent framework for categorical clustering of hypergraphs. Our generalizations allow for a budgeted number of either (a) overlapping cluster assignments or (b) node deletions. For each new model we present a greedy algorithm which approximately minimizes an edge mistake objective, as well as bicriteria approximations where the second approximation factor is on the budget. Additionally, we address the parameterized complexity of each problem, providing FPT algorithms and hardness results.

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