Quantum Algorithms (VQAs) have emerged to obtain possible quantum advantage. In particular, how to effectively incorporate hard constraints in VQAs remains a critical and open question.

Neural Information Processing Systems http://nips.cc/

Graph matching plays a central role in such fields as computer vision, pattern recognition, and bioinformatics. Graph matching problems can be cast as two types of quadratic assignment problems (QAPs): Koopmans-Beckmann's QAP or Lawler's QAP. In our paper, we provide a unifying view for these two problems by introducing new rules for array operations in Hilbert spaces. Consequently, Lawler's QAP can be considered as the Koopmans-Beckmann's alignment between two arrays in reproducing kernel Hilbert spaces (RKHS), making it possible to efficiently solve the problem without computing a huge affinity matrix. Furthermore, we develop the entropy-regularized Frank-Wolfe (EnFW) algorithm for optimizing QAPs, which has the same convergence rate as the original FW algorithm while dramatically reducing the computational burden for each outer iteration. We conduct extensive experiments to evaluate our approach, and show that our algorithm significantly outperforms the state-of-the-art in both matching accuracy and scalability.

Quantum Algorithms (VQAs) have emerged to obtain possible quantum advantage. In particular, how to effectively incorporate hard constraints in VQAs remains a critical and open question.

Neural Information Processing Systems http://nips.cc/

Algorithms are the engine for reproducible problem-solving. We present a framework automating algorithm discovery by conceptualizing them as sequences of operations, represented as tokens. These computational tokens are chained using a grammar, enabling the formation of increasingly sophisticated procedures. Our ensemble Monte Carlo tree search (MCTS) guided by reinforcement learning (RL) explores token chaining and drives the creation of new tokens. This methodology rediscovers, improves, and generates new algorithms that substantially outperform existing methods for strongly NP-hard combinatorial optimization problems and foundational quantum computing approaches such as Grover's and Quantum Approximate Optimization Algorithm. Operating at the computational rather than code-generation level, our framework produces algorithms that can be tailored specifically to problem instances, not merely classes.

Graph matching plays a central role in such fields as computer vision, pattern recognition, and bioinformatics. Graph matching problems can be cast as two types of quadratic assignment problems (QAPs): Koopmans-Beckmann's QAP or Lawler's QAP. In our paper, we provide a unifying view for these two problems by introducing new rules for array operations in Hilbert spaces. Consequently, Lawler's QAP can be considered as the Koopmans-Beckmann's alignment between two arrays in reproducing kernel Hilbert spaces (RKHS), making it possible to efficiently solve the problem without computing a huge affinity matrix. Furthermore, we develop the entropy-regularized Frank-Wolfe (EnFW) algorithm for optimizing QAPs, which has the same convergence rate as the original FW algorithm while dramatically reducing the computational burden for each outer iteration.

Although LLMs have the potential to transform many fields, they still underperform humans in reasoning tasks. Existing methods induce the model to produce step-by-step calculations, but this research explores the question: Does making the LLM analyze the question improve its performance? We propose a novel prompting strategy called Question Analysis Prompting (QAP), in which the model is prompted to explain the question in $n$ words before solving. The value of $n$ influences the length of response generated by the model. QAP is evaluated on GPT 3.5 Turbo and GPT 4 Turbo on arithmetic datasets GSM8K, AQuA, and SAT and commonsense dataset StrategyQA. QAP is compared with other state-of-the-art prompts including Chain-of-Thought (CoT), Plan and Solve Prompting (PS+) and Take A Deep Breath (TADB). QAP outperforms all state-of-the-art prompts on AQuA and SAT datasets on both GPT3.5 and GPT4. QAP consistently ranks among the top-2 prompts on 75\% of the tests. A key factor of QAP performance can be attributed to response length, where detailed responses are beneficial when answering harder questions, but can negatively affect easy questions.

Recently various optimization problems, such as Mixed Integer Linear Programming Problems (MILPs), have undergone comprehensive investigation, leveraging the capabilities of machine learning. This work focuses on learning-based solutions for efficiently solving the Quadratic Assignment Problem (QAPs), which stands as a formidable challenge in combinatorial optimization. While many instances of simpler problems admit fully polynomial-time approximate solution (FPTAS), QAP is shown to be strongly NP-hard. Even finding a FPTAS for QAP is difficult, in the sense that the existence of a FPTAS implies $P = NP$. Current research on QAPs suffer from limited scale and computational inefficiency. To attack the aforementioned issues, we here propose the first solution of its kind for QAP in the learn-to-improve category. This work encodes facility and location nodes separately, instead of forming computationally intensive association graphs prevalent in current approaches. This design choice enables scalability to larger problem sizes. Furthermore, a \textbf{S}olution \textbf{AW}are \textbf{T}ransformer (SAWT) architecture integrates the incumbent solution matrix with the attention score to effectively capture higher-order information of the QAPs. Our model's effectiveness is validated through extensive experiments on self-generated QAP instances of varying sizes and the QAPLIB benchmark.

Quadratic Assignment Problem (QAP) is a practical combinatorial optimization problems that has been studied for several years. Since it is NP-hard, solving large problem instances of QAP is challenging. Although heuristics can find semi-optimal solutions, the execution time significantly increases as the problem size increases. Recently, solving combinatorial optimization problems by deep learning has been attracting attention as a faster solver than heuristics. Even with deep learning, however, solving large QAP is still challenging. In this paper, we propose the deep reinforcement learning model called the two-stage graph pointer network (GPN) for solving QAP. Two-stage GPN relies on GPN, which has been proposed for Euclidean Traveling Salesman Problem (TSP). First, we extend GPN for general TSP, and then we add new algorithms to that model for solving QAP. Our experimental results show that our two-stage GPN provides semi-optimal solutions for benchmark problem instances from TSPlib and QAPLIB.