{"type":"journal_article","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"Consider a convex relaxation f̂ of a pseudo-Boolean function f. We say that the relaxation is totally half-integral if f̂(x) is a polyhedral function with half-integral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form x i=x j, x i=1-x j, and x i=γ where γ∈{0,1,1/2} is a constant. A well-known example is the roof duality relaxation for quadratic pseudo-Boolean functions f. We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-Boolean functions. Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations f̂ by establishing a one-to-one correspondence with bisubmodular functions. Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality. On the conceptual level, our results show that bisubmodular functions provide a natural generalization of the roof duality approach to higher-order terms. This can be viewed as a non-submodular analogue of the fact that submodular functions generalize the s-t minimum cut problem with non-negative weights to higher-order terms."}],"citation":{"apa":"Kolmogorov, V. (2012). Generalized roof duality and bisubmodular functions. <i>Discrete Applied Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.dam.2011.10.026\">https://doi.org/10.1016/j.dam.2011.10.026</a>","chicago":"Kolmogorov, Vladimir. “Generalized Roof Duality and Bisubmodular Functions.” <i>Discrete Applied Mathematics</i>. Elsevier, 2012. <a href=\"https://doi.org/10.1016/j.dam.2011.10.026\">https://doi.org/10.1016/j.dam.2011.10.026</a>.","ama":"Kolmogorov V. Generalized roof duality and bisubmodular functions. <i>Discrete Applied Mathematics</i>. 2012;160(4-5):416-426. doi:<a href=\"https://doi.org/10.1016/j.dam.2011.10.026\">10.1016/j.dam.2011.10.026</a>","ieee":"V. Kolmogorov, “Generalized roof duality and bisubmodular functions,” <i>Discrete Applied Mathematics</i>, vol. 160, no. 4–5. Elsevier, pp. 416–426, 2012.","ista":"Kolmogorov V. 2012. Generalized roof duality and bisubmodular functions. Discrete Applied Mathematics. 160(4–5), 416–426.","mla":"Kolmogorov, Vladimir. “Generalized Roof Duality and Bisubmodular Functions.” <i>Discrete Applied Mathematics</i>, vol. 160, no. 4–5, Elsevier, 2012, pp. 416–26, doi:<a href=\"https://doi.org/10.1016/j.dam.2011.10.026\">10.1016/j.dam.2011.10.026</a>.","short":"V. Kolmogorov, Discrete Applied Mathematics 160 (2012) 416–426."},"publist_id":"3397","oa_version":"Preprint","date_published":"2012-03-01T00:00:00Z","volume":160,"year":"2012","day":"01","status":"public","corr_author":"1","publisher":"Elsevier","publication":"Discrete Applied Mathematics","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"2934"}]},"date_updated":"2024-10-09T20:54:41Z","scopus_import":1,"external_id":{"arxiv":["1005.2305"]},"_id":"3257","author":[{"id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov","first_name":"Vladimir"}],"intvolume":"       160","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1005.2305"}],"department":[{"_id":"VlKo"}],"publication_status":"published","page":"416 - 426","doi":"10.1016/j.dam.2011.10.026","title":"Generalized roof duality and bisubmodular functions","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"03","oa":1,"issue":"4-5","date_created":"2018-12-11T12:02:18Z"}