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For instance, when analyzing social networks, we may obtain initial communities based on noisy metadata, and we want to improve them by adding influential nodes and removing non-important ones, without making too many changes. However, classic optimization algorithms, which typically find solutions from scratch, potentially return communities that are very dissimilar to the initial one. To mitigate these issues, we introduce the\n OptiRefine framework<\/jats:italic>\n . The framework optimizes initial solutions by making a small number of\n refinements<\/jats:italic>\n , thereby ensuring that the new solution remains close to the initial solution and simultaneously achieving a near-optimal solution for the optimization problem. We apply the OptiRefine framework to two classic graph-optimization problems:\n densest subgraph<\/jats:italic>\n and\n maximum cut<\/jats:italic>\n . For the\n densest-subgraph problem<\/jats:italic>\n , we optimize a given subgraph\u2019s density by adding or removing\n k<\/jats:italic>\n \u00a0nodes. We show that this novel problem is a generalization of\n k<\/jats:italic>\n -densest subgraph, and provide constant-factor approximation algorithms for\n \n $$k=\\Omega (n)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n \u00a0refinements. We also study a version of\n maximum cut<\/jats:italic>\n in which the goal is to improve a given cut. We provide connections to the maximum cut with cardinality constraints and provide an optimal approximation algorithm in most parameter regimes under the Unique Games Conjecture for\n \n $$k=\\Omega (n)$$<\/jats:tex-math>\n <\/jats:inline-formula>\n \u00a0refinements. We evaluate our theoretical methods and scalable heuristics on synthetic and real-world data and show that they are highly effective in practice.\n <\/jats:p>","DOI":"10.1007\/s10618-025-01142-2","type":"journal-article","created":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T11:43:16Z","timestamp":1757590996000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optirefine: densest subgraphs and maximum cuts with k refinements"],"prefix":"10.1007","volume":"39","author":[{"given":"Sijing","family":"Tu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aleksa","family":"Stankovic","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Neumann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aristides","family":"Gionis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,11]]},"reference":[{"key":"1142_CR1","doi-asserted-by":"crossref","unstructured":"Ageev AA, Sviridenko MI (1999) Approximation algorithms for maximum coverage and max cut with given sizes of parts. 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