arXiv subjects

Alan Kuhnle

Publications and source records attributed to Alan Kuhnle.

At least 19 recordsLinked to original sources

arXiv2026

Submodular Welfare under Routing Coupling: A Hierarchical Decomposition with Perturbation Guarantees

This paper studies joint submodular welfare maximization and routing over graphs, where agents select items under diminishing returns and transport them through a network with congestion-dependent costs. Although welfare maximization admits matroid-based approximations and routing reduces to shortest paths under modular costs, their coupling creates supermodular interactions that break separability. We show that, for fixed routing, the objective remains submodular in the allocation variable, enabling a principled decomposition. Building on this property, we propose the Welfare-based Hierarchical Routing Algorithm (WHIRL), which alternates between tractable routing and allocation updates. Routing is initialized through its modular counterpart, while supermodular effects are modeled as bounded perturbations. The method has finite convergence guarantees and approximation bounds that depend explicitly on the deviation from modular routing. Numerical results illustrate the impact of routing-induced coupling and show that WHIRL achieves a favorable tradeoff between solution quality and computational cost.

math.OC↗

arXiv2026

Curvature Beyond Positivity: Greedy Guarantees for Arbitrary Submodular Functions

Submodular functions -- functions exhibiting diminishing returns -- are central to machine learning. When the objective is monotone and non-negative, the greedy algorithm achieves a tight $63\%$ approximation. But many practical objectives incorporate costs that make them negative on some inputs, and all existing multiplicative guarantees require non-negativity. Prior work handles negativity through additive bounds for the special class of decomposable functions and non-monotonicity through partial-monotonicity parameters, but these address each difficulty in isolation and neither extends the classical structural theory. We extend \emph{curvature} -- a parameter measuring how far a function deviates from linearity -- to all submodular functions, handling both non-monotonicity and negativity through a single classical concept. A greedy algorithm with pruning achieves a curvature-controlled multiplicative ratio for \emph{any} submodular function, including those taking negative values -- the first such guarantee beyond monotonicity and non-negativity. In the non-monotone regime $1 \le c_g < 2.2$, the bound strictly beats the best known uniform ratio of $0.401$ (for non-negative $f$), and it recovers the classical $(1-e^{-c_g})/c_g$ guarantee for monotone functions. A multilinear-extension variant extends the framework to general combinatorial constraints via multilinear relaxation. Experiments on cost-penalized experimental design, coverage, feature selection, and a curvature sweep on Multi-News passage selection support the theory.

cs.LG↗

arXiv2026

Submodular Ground-Set Pruning: Monotone Tightness and a Non-Monotone Separation

Large-scale subset selection asks for a small useful set of examples, features, sensors, seed users, or context passages from an enormous ground set. Submodular maximization is a canonical model for such diminishing-returns problems, but rapidly growing datasets make even linear-time algorithms ever costlier. We study \emph{containment pruning}: first reduce the ground set to a smaller core $P$, then require that $P$ contain a near-optimal feasible solution for every downstream budget up to~$k$. Prior work has formulated many heuristics, but the theoretical limits of this preprocessing problem are largely unknown. For monotone submodular objectives, we prove that $1-1/e$ is tight: greedy achieves this containment factor, and no algorithm can beat it even with a larger pruning budget. For non-monotone objectives, we give the first$1/2-\varepsilon$ containment algorithms under cardinality constraints and extend the approach to knapsack constraints. This $1/2$ factor exceeds the best known algorithmic ratio and the known hardness threshold for non-monotone maximization, showing that pruning can be provably easier than optimization. Empirically, pruning lets an exact IP solver run on the reduced MaxCut instance with a ${\approx}620\times$ speedup, and proof-of-concept experiments on LLM context selection demonstrate the utility of non-monotone submodular proxies and our proposed containment algorithms.

cs.DS↗

arXiv2025

Bicriteria Submodular Maximization

Submodular functions and their optimization have found applications in diverse settings ranging from machine learning and data mining to game theory and economics. In this work, we consider the constrained maximization of a submodular function, for which we conduct a principled study of bicriteria approximation algorithms -- algorithms which can violate the constraint, but only up to a bounded factor. Bicrteria optimization allows constrained submodular maximization to capture additional important settings, such as the well-studied submodular cover problem and optimization under soft constraints. We provide results that span both multiple types of constraints (cardinality, knapsack, matroid and convex set) and multiple classes of submodular functions (monotone, symmetric and general). For many of the cases considered, we provide optimal results. In other cases, our results improve over the state-of-the-art, sometimes even over the state-of-the-art for the special case of single-criterion (standard) optimization. Results of the last kind demonstrate that relaxing the feasibility constraint may give a perspective about the problem that is useful even if one only desires feasible solutions.

cs.DS↗

arXiv2025

ResQue Greedy: Rewiring Sequential Greedy for Improved Submodular Maximization

This paper introduces Rewired Sequential Greedy (ResQue Greedy), an enhanced approach for submodular maximization under cardinality constraints. By integrating a novel set curvature metric within a lattice-based framework, ResQue Greedy identifies and corrects suboptimal decisions made by the standard sequential greedy algorithm. Specifically, a curvature-aware rewiring strategy is employed to dynamically redirect the solution path, leading to improved approximation performance over the conventional sequential greedy algorithm without significantly increasing computational complexity. Numerical experiments demonstrate that ResQue Greedy achieves tighter near-optimality bounds compared to the traditional sequential greedy method.

cs.DM↗

arXiv2025

Breaking Barriers: Combinatorial Algorithms for Non-monotone Submodular Maximization with Sublinear Adaptivity and $1/e$ Approximation

With the rapid growth of data in modern applications, parallel algorithms for maximizing non-monotone submodular functions have gained significant attention. In the parallel computation setting, the state-of-the-art approximation ratio of $1/e$ is achieved by a continuous algorithm (Ene & Nguyen, 2020) with adaptivity $ O\left(\log(n)\right)$. In this work, we focus on size constraints and present the first combinatorial algorithm matching this bound -- a randomized parallel approach achieving $1/e-\varepsilon$ approximation ratio. This result bridges the gap between continuous and combinatorial approaches for this problem. As a byproduct, we also develop a simpler $(1/4-\varepsilon)$-approximation algorithm with high probability ($\ge 1-1/n$). Both algorithms achieve $ O\left(\log(n)\log(k)\right)$ adaptivity and $O\left(n\log(n)\log(k)\right)$ query complexity. Empirical results show our algorithms achieve competitive objective values, with the $(1/4-\varepsilon)$-approximation algorithm particularly efficient in queries.

cs.DS↗

arXiv2024

Theoretically Grounded Pruning of Large Ground Sets for Constrained, Discrete Optimization

Modern instances of combinatorial optimization problems often exhibit billion-scale ground sets, which have many uninformative or redundant elements. In this work, we develop light-weight pruning algorithms to quickly discard elements that are unlikely to be part of an optimal solution. Under mild assumptions on the instance, we prove theoretical guarantees on the fraction of the optimal value retained and the size of the resulting pruned ground set. Through extensive experiments on real-world datasets for various applications, we demonstrate that our algorithm, QuickPrune, efficiently prunes over 90% of the ground set and outperforms state-of-the-art classical and machine learning heuristics for pruning.

cs.DS↗

arXiv2024

Scalable Distributed Algorithms for Size-Constrained Submodular Maximization in the MapReduce and Adaptive Complexity Models

Distributed maximization of a submodular function in the MapReduce (MR) model has received much attention, culminating in two frameworks that allow a centralized algorithm to be run in the MR setting without loss of approximation, as long as the centralized algorithm satisfies a certain consistency property -- which had previously only been known to be satisfied by the standard greedy and continous greedy algorithms. A separate line of work has studied parallelizability of submodular maximization in the adaptive complexity model, where each thread may have access to the entire ground set. For the size-constrained maximization of a monotone and submodular function, we show that several sublinearly adaptive (highly parallelizable) algorithms satisfy the consistency property required to work in the MR setting, which yields practical, parallelizable and distributed algorithms. Separately, we develop the first distributed algorithm with linear query complexity for this problem. Finally, we provide a method to increase the maximum cardinality constraint for MR algorithms at the cost of additional MR rounds.

cs.DS↗

arXiv2024

Submodular Maximization in Exactly $n$ Queries

In this work, we study the classical problem of maximizing a submodular function subject to a matroid constraint. We develop deterministic algorithms that are very parsimonious with respect to querying the submodular function, for both the case when the submodular function is monotone and the general submodular case. In particular, we present a 1/4 approximation algorithm for the monotone case that uses exactly one query per element, which gives the same total number of queries n as the number of queries required to compute the maximum singleton. For the general case, we present a constant factor approximation algorithm that requires 2 queries per element, which is the first algorithm for this problem with linear query complexity in the size of the ground set.

cs.DS↗

arXiv2024

Best of Both Worlds: Practical and Theoretically Optimal Submodular Maximization in Parallel

For the problem of maximizing a monotone, submodular function with respect to a cardinality constraint $k$ on a ground set of size $n$, we provide an algorithm that achieves the state-of-the-art in both its empirical performance and its theoretical properties, in terms of adaptive complexity, query complexity, and approximation ratio; that is, it obtains, with high probability, query complexity of $O(n)$ in expectation, adaptivity of $O(\log(n))$, and approximation ratio of nearly $1-1/e$. The main algorithm is assembled from two components which may be of independent interest. The first component of our algorithm, LINEARSEQ, is useful as a preprocessing algorithm to improve the query complexity of many algorithms. Moreover, a variant of LINEARSEQ is shown to have adaptive complexity of $O( \log (n / k) )$ which is smaller than that of any previous algorithm in the literature. The second component is a parallelizable thresholding procedure THRESHOLDSEQ for adding elements with gain above a constant threshold. Finally, we demonstrate that our main algorithm empirically outperforms, in terms of runtime, adaptive rounds, total queries, and objective values, the previous state-of-the-art algorithm FAST in a comprehensive evaluation with six submodular objective functions.

cs.DS↗

arXiv2024

A Benchmark for Maximum Cut: Towards Standardization of the Evaluation of Learned Heuristics for Combinatorial Optimization

Recently, there has been much work on the design of general heuristics for graph-based, combinatorial optimization problems via the incorporation of Graph Neural Networks (GNNs) to learn distribution-specific solution structures.However, there is a lack of consistency in the evaluation of these heuristics, in terms of the baselines and instances chosen, which makes it difficult to assess the relative performance of the algorithms. In this paper, we propose an open-source benchmark suite MaxCut-Bench dedicated to the NP-hard Maximum Cut problem in both its weighted and unweighted variants, based on a careful selection of instances curated from diverse graph datasets. The suite offers a unified interface to various heuristics, both traditional and machine learning-based. Next, we use the benchmark in an attempt to systematically corroborate or reproduce the results of several, popular learning-based approaches, including S2V-DQN [31], ECO-DQN [4], among others, in terms of three dimensions: objective value, generalization, and scalability. Our empirical results show that several of the learned heuristics fail to outperform a naive greedy algorithm, and that only one of them consistently outperforms Tabu Search, a simple, general heuristic based upon local search. Furthermore, we find that the performance of ECO-DQN remains the same or is improved if the GNN is replaced by a simple linear regression on a subset of the features that are related to Tabu Search. Code, data, and pretrained models are available at: \url{https://github.com/ankurnath/MaxCut-Bench}.

cs.AI↗

arXiv2024

Discretely Beyond $1/e$: Guided Combinatorial Algorithms for Submodular Maximization

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: $1/e \approx 0.367$ for size constraint and $0.281$ for the matroid constraint in $\mathcal O (kn)$ queries, where $k$ is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the $1/e$ barrier: we obtain approximation ratio of $0.385$ in $\mathcal O (kn)$ queries to the submodular set function for size constraint, and $0.305$ for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio $0.377$.

cs.DS↗

arXiv2024

Practical and Parallelizable Algorithms for Non-Monotone Submodular Maximization with Size Constraint

We present combinatorial and parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a size constraint. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and nearly optimal query complexity to $0.193 - \varepsilon$. The conference version of this work mistakenly employed a subroutine that does not work for non-monotone, submodular functions. In this version, we propose a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in $O( \log(n) )$ adaptive rounds with high probability. Moreover, we provide two approximation algorithms. The first has approximation ratio $1/6 - \varepsilon$, adaptivity $O( \log (n) )$, and query complexity $O( n \log (k) )$, while the second has approximation ratio $0.193 - \varepsilon$, adaptivity $O( \log^2 (n) )$, and query complexity $O(n \log (k))$. Our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension.

cs.DS↗

arXiv2023

Unveiling the Limits of Learned Local Search Heuristics: Are You the Mightiest of the Meek?

In recent years, combining neural networks with local search heuristics has become popular in the field of combinatorial optimization. Despite its considerable computational demands, this approach has exhibited promising outcomes with minimal manual engineering. However, we have identified three critical limitations in the empirical evaluation of these integration attempts. Firstly, instances with moderate complexity and weak baselines pose a challenge in accurately evaluating the effectiveness of learning-based approaches. Secondly, the absence of an ablation study makes it difficult to quantify and attribute improvements accurately to the deep learning architecture. Lastly, the generalization of learned heuristics across diverse distributions remains underexplored. In this study, we conduct a comprehensive investigation into these identified limitations. Surprisingly, we demonstrate that a simple learned heuristic based on Tabu Search surpasses state-of-the-art (SOTA) learned heuristics in terms of performance and generalizability. Our findings challenge prevailing assumptions and open up exciting avenues for future research and innovation in combinatorial optimization.

cs.AI↗

arXiv2023

Approximation Algorithms for Size-Constrained Non-Monotone Submodular Maximization in Deterministic Linear Time

In this work, we study the problem of finding the maximum value of a non-negative submodular function subject to a limit on the number of items selected, a ubiquitous problem that appears in many applications, such as data summarization and nonlinear regression. We provide the first deterministic, linear-time approximation algorithms for this problem that do not assume the objective is monotone. We present three deterministic, linear-time algorithms: a single-pass streaming algorithm with a ratio of $23.313 + ε$, which is the first linear-time streaming algorithm; a simpler deterministic linear-time algorithm with a ratio of $11.657$; and a $(4 + O(ε))$-approximation algorithm. Finally, we present a deterministic algorithm that obtains ratio of $e + ε$ in $O_ε(n \log(n))$ time, close to the best known expected ratio of $e - 0.121$ in polynomial time.

cs.DS↗

arXiv2023

RELS-DQN: A Robust and Efficient Local Search Framework for Combinatorial Optimization

Combinatorial optimization (CO) aims to efficiently find the best solution to NP-hard problems ranging from statistical physics to social media marketing. A wide range of CO applications can benefit from local search methods because they allow reversible action over greedy policies. Deep Q-learning (DQN) using message-passing neural networks (MPNN) has shown promise in replicating the local search behavior and obtaining comparable results to the local search algorithms. However, the over-smoothing and the information loss during the iterations of message passing limit its robustness across applications, and the large message vectors result in memory inefficiency. Our paper introduces RELS-DQN, a lightweight DQN framework that exhibits the local search behavior while providing practical scalability. Using the RELS-DQN model trained on one application, it can generalize to various applications by providing solution values higher than or equal to both the local search algorithms and the existing DQN models while remaining efficient in runtime and memory.

cs.LG↗

arXiv2023

Learning Strategic Value and Cooperation in Multi-Player Stochastic Games through Side Payments

We study general-sum, multi-player stochastic games with transferable utility, motivated by settings where agents can use side payments to make cooperation individually rational. Building on the Harsanyi--Shapley (HS) value for normal-form games, we introduce two HS-based value notions for stochastic games: HS-S, defined by aggregating dynamic coalition-versus-complement threat powers, and Coco-S, defined as fixed points of a statewise HS Bellman operator. We extend HS-style axioms to the stochastic setting and show that HS-S is the unique mapping satisfying them. We prove that HS-S and Coco-S coincide in all two-player stochastic games, but can disagree when $n>2$, via an explicit three-player counterexample. We prove existence and uniqueness of Coco-S fixed points for all two-player games and for three-player two-state games via topological degree theory, and provide an axiomatic characterization of Coco-S through a new \emph{Markov Consistency} axiom that distinguishes it from HS-S. Finally, we give sampling-based estimators with finite-sample guarantees and empirically compare the induced values, policies, and side payments on multi-player grid-game benchmarks.

cs.GT↗

arXiv2021

Quick Streaming Algorithms for Maximization of Monotone Submodular Functions in Linear Time

We consider the problem of monotone, submodular maximization over a ground set of size $n$ subject to cardinality constraint $k$. For this problem, we introduce the first deterministic algorithms with linear time complexity; these algorithms are streaming algorithms. Our single-pass algorithm obtains a constant ratio in $\lceil n / c \rceil + c$ oracle queries, for any $c \ge 1$. In addition, we propose a deterministic, multi-pass streaming algorithm with a constant number of passes that achieves nearly the optimal ratio with linear query and time complexities. We prove a lower bound that implies no constant-factor approximation exists using $o(n)$ queries, even if queries to infeasible sets are allowed. An empirical analysis demonstrates that our algorithms require fewer queries (often substantially less than $n$) yet still achieve better objective value than the current state-of-the-art algorithms, including single-pass, multi-pass, and non-streaming algorithms.

cs.DS↗