Ankur Nath

Ankur Nath, Alan Kuhnle

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.

Ankur Nath, Alan Kuhnle

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}.

Yixin Chen, Ankur Nath, Chunli Peng et al.

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 $0.377$.

Ankur Nath, Alan Kuhnle

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.