{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T07:00:15Z","timestamp":1762326015739,"version":"build-2065373602"},"publisher-location":"California","reference-count":0,"publisher":"International Joint Conferences on Artificial Intelligence Organization","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:p>Model counting is a powerful extension of constraint reasoning that, instead of finding a solution to a constraint system, allows to identify the number of such solutions. Cardinality constraints are used to filter solutions of a certain quality by restricting the number of elements that can be added to the solution. Naturally, one would like to combine both in order to count the number of solutions of good quality. Unfortunately, the two concepts do not get along so well as (1) cardinality constraints may not be parsimonious (due to auxiliary variables, the system\u2019s number of solutions may change in an uncontrolled way) and (2) such constraints may destroy structural properties, which are crucial for the performance of modern solvers. This article provides a systematic study of existing cardinality constraints in the light of model counting, observing that none of them are both, parsimonious and treewidth-preserving. We present structure-aware cardinality constraints that are parsimonious and guaranteed to increase the input\u2019s treewidth only in a controlled way. Detailed experiments reveal that our encodings outperform existing ones.<\/jats:p>","DOI":"10.24963\/kr.2025\/8","type":"proceedings-article","created":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T06:10:44Z","timestamp":1762323044000},"page":"78-88","source":"Crossref","is-referenced-by-count":0,"title":["Counting Solutions Under Cardinality Constraints: Structure Counts in Counting"],"prefix":"10.24963","author":[{"given":"Max","family":"Bannach","sequence":"first","affiliation":[{"name":"European Space Agency, AI and Data Science Section, Noordwijk, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Markus","family":"Hecher","sequence":"additional","affiliation":[{"name":"University of Artois, CNRS, UMR8188, Computer Science Research Center of Lens (CRIL), France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"10584","event":{"name":"22nd International Conference on Principles of Knowledge Representation and Reasoning {KR-2025}","theme":"Artificial Intelligence","location":"Melbourne, Australia","acronym":"KR-2025","number":"22","sponsor":["Artificial Intelligence Journal","Principles of Knowledge Representation and Reasoning Inc.","Academic College of Tel-Aviv","European Association for Artificial Intelligence","National Science Foundation"],"start":{"date-parts":[[2025,11,11]]},"end":{"date-parts":[[2025,11,17]]}},"container-title":["Proceedings of the TwentySecond International Conference on Principles of Knowledge Representation and Reasoning"],"original-title":[],"deposited":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T06:10:51Z","timestamp":1762323051000},"score":1,"resource":{"primary":{"URL":"https:\/\/proceedings.kr.org\/2025\/8"}},"subtitle":[],"proceedings-subject":"Artificial Intelligence Research Articles","short-title":[],"issued":{"date-parts":[[2025,11]]},"references-count":0,"URL":"https:\/\/doi.org\/10.24963\/kr.2025\/8","relation":{},"subject":[],"published":{"date-parts":[[2025,11]]}}}