{"id":1860,"date":"2019-01-01T12:00:00","date_gmt":"2019-01-01T11:00:00","guid":{"rendered":"https:\/\/kindsonthegenius.com\/blog\/network-flow-introduction-to-cuts-in-a-network\/"},"modified":"2026-07-05T03:20:55","modified_gmt":"2026-07-05T01:20:55","slug":"network-flow-introduction-to-cuts-in-a-network","status":"publish","type":"post","link":"https:\/\/kindsonthegenius.com\/blog\/network-flow-introduction-to-cuts-in-a-network\/","title":{"rendered":"Network Flow \u2013 Introduction to Cuts in a Network"},"content":{"rendered":"<p>The concepts of cuts in a network is a way to verify the Ford-Fulkerson algorithm and proof that the Max-Flow Min-Cut theorem.<\/p>\n<p>You can review these topics:<\/p>\n<ul>\n<li><a href=\"https:\/\/kindsonthegenius.com\/tempsite\/introduction-to-network-flow\/\">The Basics of Flow Networks<\/a><\/li>\n<li><a href=\"https:\/\/kindsonthegenius.com\/tempsite\/network-flow-introduction-to-cuts-in-a-network\/\">Max Flow Problem<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=7E9OWqcMTeI\" target=\"_blank\" rel=\"noopener\">Residual Graphs<\/a><\/li>\n<\/ul>\n<p>Consider partitioning the nodes of a graph into two partitions, A and B such that s\u00a0\u2208 \u00a0A and t\u00a0 \u2208\u00a0 B. This is called a cut. Any such partition places an upper bound on the maximum possible flow value since all the flow must pass from A to B at some point.<\/p>\n<p>We can define an s-t cut as a partition (A, B) of the set of nodes (V), such that s\u00a0\u2208 \u00a0A, and t\u00a0\u2208 \u00a0B.\u00a0 The capacity of a cut is given as c(A, B) which is the sum of all the capacities of all edges out of A. This can be written as:<\/p>\n<p style=\"text-align: center;\">c(A, B) =\u00a0\u2211<sub>e out of A<\/sub> c<sub>e<\/sub><\/p>\n<p>This can be formulated into a formal statement as:<\/p>\n<p>Let f be any s-t flow and (A, B) be any s-t cut. Then v(f) = f<sup>out<\/sup>(A) = f<sup>in<\/sup>(A)<\/p>\n<p>This means that the values of the flow\u00a0 can be found by taking the total amount of flow that leaves A minus the amount that comes back into A.<\/p>\n<p><strong>Proof: <\/strong><\/p>\n<p>We know that v(f) = f<sup>out<\/sup>(s): that is values of the flow is same as value of flow out of the source<\/p>\n<p>Also, fin(s) = 0: since there is no incoming node to s. Therefor we can write<\/p>\n<p>v(f) = f<sub>out<\/sub>(s) &#8211; f<sup>in<\/sup>(s)<\/p>\n<p>Considering the partition A, every node in A is an internal node other than s, we can then write:<\/p>\n<p>f<sup>out<\/sup>(v) &#8211; f<sup>in<\/sup>(v) = 0 for all these internal nodes. The can be written in the form:<\/p>\n<p>v(f) =\u00a0\u2211<sub>v\u2208A<\/sub>f<sup>out<\/sup>(v) &#8211; f<sup>in<\/sup>(v)<\/p>\n<p>&nbsp;<\/p>\n<p>To be continued&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The concepts of cuts in a network is a way to verify the Ford-Fulkerson algorithm and proof that the Max-Flow Min-Cut theorem. You can review &hellip; <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":""},"categories":[35],"tags":[],"class_list":["post-1860","post","type-post","status-publish","format-standard","hentry","category-algorithms"],"_links":{"self":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1860","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/comments?post=1860"}],"version-history":[{"count":1,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1860\/revisions"}],"predecessor-version":[{"id":2029,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1860\/revisions\/2029"}],"wp:attachment":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/media?parent=1860"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/categories?post=1860"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/tags?post=1860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}