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However, it is often computationally hard, in that complexity grows exponentially with problem size. The big\u2010data era urgently demands low\u2010latency and low\u2010energy optimization at the edge, which cannot be handled by digital processors due to their non\u2010(parallel von Neumann architecture. Recent efforts using massively parallel hardware (e.g., memristor crossbars and quantum processors) employing annealing algorithms, while promising, have handled relatively easy and stable problems with sparse or binary representations, such as the max\u2010cut or traveling salesman problems. However, most real\u2010world applications embody three features, which are encoded in the knapsack problem, and cannot be handled by annealing algorithms\u2014dense and nonbinary representations, with destabilizing self\u2010feedback. Herein, a post\u2010digital\u2010hardware\u2010friendly randomized competitive Ising\u2010inspired (RaCI) algorithm performing knapsack optimization, experimentally implemented on a foundry\u2010manufactured complementary metal\u2010oxide\u2010semiconductor\u2010integrated probabilistic analog memristor crossbar, is demonstrated. This solution outperforms digital and quantum approaches by over four orders of magnitude in energy efficiency.<\/jats:p>","DOI":"10.1002\/aisy.202401114","type":"journal-article","created":{"date-parts":[[2025,6,24]],"date-time":"2025-06-24T16:39:02Z","timestamp":1750783142000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Energy\u2010Efficient Knapsack Optimization Using Probabilistic Memristor Crossbars"],"prefix":"10.1002","volume":"7","author":[{"given":"Jinzhan","family":"Li","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering Texas A&amp;M University  College Station TX 77843 USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6772-7250","authenticated-orcid":false,"given":"Suhas","family":"Kumar","sequence":"additional","affiliation":[{"name":"Rain AI  San Francisco CA 94110 USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7957-8953","authenticated-orcid":false,"given":"Su\u2010in","family":"Yi","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering Texas A&amp;M University  College Station TX 77843 USA"}]}],"member":"311","published-online":{"date-parts":[[2025,6,24]]},"reference":[{"key":"e_1_2_11_2_1","first-page":"1","volume-title":"Combinatorial Optimization","author":"Korte B. 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