Successfully reported this slideshow.
Your SlideShare is downloading. ×

Hypergraph Cuts with General Splitting Functions (JMM)

Ad
Ad
Ad
Ad
Ad
Ad
Ad
Ad
Ad
Ad
Ad
Loading in …3
×

Check these out next

1 of 28 Ad

More Related Content

More from Austin Benson (20)

Recently uploaded (20)

Advertisement

Hypergraph Cuts with General Splitting Functions (JMM)

  1. 1. 1 Joint work with Nate Veldt & Jon Kleinberg (Cornell) Hypergraph Cuts with General Splitting Functions Austin R. Benson · Cornell University AMS Special Session on Applied Combinatorial Methods Joint Mathematics Meetings · January 9, 2021
  2. 2. Graph minimum s-t cuts are fundamental. 2 minimizeS⇢V cut(S) subject to s 2 S, t /2 S.<latexit sha1_base64="xm7lCa+sznQv4hBLJYdds1WJ/eg=">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</latexit><latexit sha1_base64="xm7lCa+sznQv4hBLJYdds1WJ/eg=">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</latexit><latexit sha1_base64="xm7lCa+sznQv4hBLJYdds1WJ/eg=">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</latexit><latexit sha1_base64="xm7lCa+sznQv4hBLJYdds1WJ/eg=">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</latexit> 1 3 2 4 5 6 7 8 s t • Maximum flow / min s-t cut [Ford,Fulkerson,Dantzig 1950s] • Computer vision [Bokykov-Kolmogorov 01; Kolmogorov-Zabih 04] • Densest subgraph [Goldberg 84; Shang+ 18] • First graph-based semi-supervised learning algorithms [Blum-Chawla 01] • Local graph clustering [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16] Also see any undergraduate algorithms class poly-time algorithms!
  3. 3. Real-world systems have“higher-order”interactions. 3 Physical proximity • nodes are students • People gather in groups linear-algebra discrete-mathematics math-software combinatorics category-theory logic terminology algebraic-graph-theory combinatorial-designs hypergraphs graph-theory cayley-graphs group-theory finite-groups Categorical information • nodes are tags • groups of tags applied to info (same question on mathoverflow.com) Networks beyond pairwise interactions: structure and dynamics. Battiston et al., 2020. The why, how, and when of representations for complex systems. Torres et al., 2020. Commerce • nodes are products • hyperedges are students in the same class
  4. 4. We can model“higher-order”interactions with hypergraphs. 4 H = (V, E), edge e 2 E is a subset of V (e ⇢ V)<latexit sha1_base64="8oqd642c1xU2WvSPMjDvF/Nrfc4=">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</latexit><latexit sha1_base64="8oqd642c1xU2WvSPMjDvF/Nrfc4=">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</latexit><latexit sha1_base64="8oqd642c1xU2WvSPMjDvF/Nrfc4=">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</latexit><latexit sha1_base64="8oqd642c1xU2WvSPMjDvF/Nrfc4=">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</latexit> 1 2 3 4 5 V = {1, 2, 3, 4, 5} E = {{1, 2, 3}, {2, 4, 5}}<latexit sha1_base64="NNfjaoBWw5b5H6dRzAXbaRk4ses=">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</latexit><latexit sha1_base64="NNfjaoBWw5b5H6dRzAXbaRk4ses=">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</latexit><latexit sha1_base64="NNfjaoBWw5b5H6dRzAXbaRk4ses=">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</latexit><latexit sha1_base64="NNfjaoBWw5b5H6dRzAXbaRk4ses=">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</latexit>
  5. 5. 5 1. What is a hypergraph minimum s-t cut? 2. If we know what they are, can we find them efficiently? 3. If we can find them efficiently, what can we use them for? We should have a foundation for hypergraph minimum s-t cuts,but…
  6. 6. What is a hypergraph minimum s-t cut? 6 s t Should we treat the 2/2 split differently from the 1/3 split? Historically, no. [Lawler 73,Ihler+ 93] More recently, yes. [Li-Milenkovic 17,Veldt-Benson-Kleinberg 20] 1 3 2 4 5 6 7 8 s t There is only one way to split an edge (1/1).
  7. 7. We model hypergraph cuts with splitting functions. 7 s t Given a cut defined by S, we incur penalty of at each hyperedge e. Hypergraph minimum s-t cut problem. Cardinality-Based splitting functions. S<latexit sha1_base64="wtJ1SkLACwJOcMcL9/jLEzSB0Ao=">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</latexit><latexit sha1_base64="wtJ1SkLACwJOcMcL9/jLEzSB0Ao=">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</latexit><latexit sha1_base64="wtJ1SkLACwJOcMcL9/jLEzSB0Ao=">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</latexit><latexit sha1_base64="wtJ1SkLACwJOcMcL9/jLEzSB0Ao=">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</latexit> cutH(S) = f (2) + f (1)<latexit sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit><latexit sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit><latexit sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit><latexit sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit> <latexit sha1_base64="6FFH4JtCJ1Fb69WjugRCayyF5vI=">AAAHU3icfVVRb9s2EFazLum8dU23x72wCwx0hezY6bIkAwoYWFGsQItls9MWsIyMok4SYZLSSCq2R+iX7HX7SXvYb9nLjrazWk42AbYo8r7v7r47knEpuLG93l93dj64++Hu3r2PWh9/cv/TB/sPP3tjikozuGCFKPS7mBoQXMGF5VbAu1IDlbGAt/H0O7/+9gq04YUa2UUJE0kzxVPOqMWpy/0H0dXsEh4DiRgtyfCry/2DXre3fMjNQX89OAjWz/nlw90wSgpWSVCWCWrMuN8r7cRRbTkTULeiykBJ2ZRmMMahohLMxC0jr0kbZxKSFhp/ypLlbGsTgjyaLhosztK4ElTPm7NxUUxxxdQN/NgLY1QlY9CQhLoSkGBwIis0t7k8gi3zyqanE8dVWVlQbBVgWgliC+LlIwnXwKxYkGaUlk9/DRVnkGrKQiqNpDYPS+6zCiWdAgMhVvF6U8FjTfXCJ1fMTBgjS6aLSiUmLKm1oJVBlNV8HpqclmDClNuQUcH8d+IxpSispHpq/ou1K8FSXFxqKsC6UZVa+AmS2qESj057j2KBfjctbA6ZBlC1W768zSznFrZsYlFB7fz/hkWrTXJrS/Pt4aGFeddY5IY5y6nKoMsKefhLBcb3nDnsf3N8dnR2aEBybM0YO1F2ZliNjk+iw1UnxgYGvbR7enKwerUiLyPFBvf6tKJMFDEVEX5GHjYAZSoNg6QQ2BoDbG9WJPAs0iDo/BpbYPDN9hqP+hPni+SL3ajo+WhIlRdXg4IZJiCpSlyUUsnFIoGUVsLWLjLp9bjZECb1HVC32pvODFYQkme97lnIJEenmaYCNwM6sHOTeopmksgdKTv3VIMV2JknY9yFx5N6O6nngNtPw3Ah40K8wJTcisXU7ofXr2qnvAvJaydrxzHcaAj2NmOcSLYh8Rqy9uEBwyrGctrKl/R2B9sehi9ee0muHYz6DflcPK+dEe+deOMV2r1ES68BFWVO6/eh/vxyS/UkE8BZ3llpf9sKFtrgwdM8OaSn2ayyHPJMoqdo1VWezkWxdNFqvr7RFvIVHrnJbYj1Qt108SSax1SPsfmiPC7mLrry/+1WlPsTiuTAs9ziuXtyXFrSJqMcCGW2ooIgrBVN8YTodY+OYd4m10+bPMfrgioGJAY7w/3rbQk6I2YpY2vlqt0iZEnQ6XX7INvX6GFeaFSHq4wUimBTEQGpJYYn4BEbeR30639J8Gp4+r8kepnJkgVFqPF+6W/fJjcHb466/eNu78evDwan65vmXvBF8GXwOOgHJ8Eg+D44Dy4CFlTBb8HvwR+7f+7+vbezd3dlunNnjfk8aDx79/8BrZyTJg==</latexit> we(e S) <latexit sha1_base64="QjrhfsKLxaK/82LxnRurhFCzCik=">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</latexit> minimizeS⇢V P e2E we(e S) ⌘ cutH(S) subject to s 2 S, t /2 S. <latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit> Non-negativity we(A) 0. Non-split ignoring we(e) = we(;) = 0. C-B we(A) = f (min(|A|, |Ae|)).
  8. 8. Cardinality-based splitting functions appear throughout the literature. 8 [Lawler 73; Ihler+ 93; Yin+ 17] [Hu-Moerder 85; Heuer+ 18] [Agarwal+ 06; Zhou+ 06; Benson+ 16] [Yaros- Imielinski 13] [Li-Milenkovic 18] <latexit sha1_base64="gX/87S67KKdqKR6T9Qjl8vcDiHo=">AAAJt3icfVZbb9s2FHbTrau9W7s97oVd4qHbZMdOl7UZUCDBimIFWqxd0gsQehklHVmESVElqdoewx/aX7MdynJrO2n1YFHk+b5zP3RcCm7sYPD2ytbVTz699tn1dufzL7786usbN795YVSlE3ieKKH0q5gZELyA55ZbAa9KDUzGAl7Gk9/D+cs3oA1XxYmdlzCSbFzwjCfM4tbZzS1NYxjzwlkWV4Jp74TwHUKOhOgp3SuUzXkxJj+QHfpmega3j34k90kDSVCv8WSAp9TCzDqekZ0jQnlBqIOIUJClnRuw1O94QikZvpNEWtBTbsBTKNKGaYeg0OkvUo7QgMfoENOkhIIJO980QNY6yPnReUTOAQ1iycQIZnJydE6oJzsrTM8qlmp0N/kA2XmAJKmym0SrJA+4SVRVWJJU9sPG/E2ZKHN20abmYGHaqpe9J8RUsVRpCP4mcaZZ4obe7Xny89pXbW6tmArIcOWGUSNABabbEkyDDgvvCBWZUEqTpQkLAWgE8FUf+w0CWLX/HdnHuRoq4tFFzce5bfzt1EleVljn7Mb2oD+oH3JxMWwW263meXp281pEU5VUEgqboEXmdDgo7cgxjVkV4Du0MlCiuWwMp7gsmAQzcnV3eNLFnZRkaFemQgLDbmcVgjyazddYlsbO1ndjpSZ4Yvwa/jQ0nykqGYOGNNKVgBSNE2Oluc3lHmyIVza7N3K8KCsLRbIwMKsEsYqEFiUp15BYMSfrVlo++TcqeAIhTRGTRjKbRyUPXkWSTSCB0LsBEkQFjzXT8+CcmpooZHOssYRTE5XMWtCFQZTVfBaZnJVgoozbKGEiCd9pwJRCWcn0xHyItS/BMjysYyrAupMqs/AXpN5hJG7dG9yKBepdlcDGH2uAwrv6FWSmObewIROLCrwLvysSnS7JrS3Nb7u7OEX6xiI3zJKcFWPoJ0ruvq7AhLlmdoe/7h/sHewakBwrM8b2l70pZqMXnOjxohfjkARdy925u714dWgII8MhGuLToWOhYiawxSwNsEMoTKXhMFUCS+MQR2iiUrhPNQg2W2Lrybae75PhyIUkhWSvZfTpyTErQnA1FDBFByTDPqEZk1zMU8hYJULPmWy5Xi8Ik4UK8J3uqjKDGYT0/qB/ECU4HyxGmwlsBlRgZyYLFOtOIjct7CxQHS7Azvx0il24P/KbTj0AbD8Nx3MZK/EQXXILFuPdn08ee1cEFZJ7J73jaC49BnuZMG6km5C4gTQ6AuC4ijGdtgopvVzBpobjh09CSJYKToZr4XPxzDsj3isJwgu0e4SSIQb1YPPvTf3n0UbU07EAnuS9RewvO8FEGxw865NDBprVLMtjPpaoiS6qKtA5GktHF/v+QlnIx3itp5chmgO/ruInOouZPsXio3msZo6+Cb/dDs3DhCI51BN60L+7X1rSJSc5EJbYigmCsA6d4IQY9Pf2YdYly6cb7kHLigRIDHaK/RtkCSojpg5jZ6Gqi1dbTdAb9Icgu0v0ca40Rif8o1AFwaIi9fVleAoBseLX9tC/I8Gr4c5HSRZ3Tc2CQfB4vww3b5OLixd7/eF+f/Bsb/vwXnPTXG991/q+dbs1bN1tHbb+aD1tPW8lW2+3/rt6/Wq7fdA+a2ftfCG6daXBfNtae9qv/wf+EGQx</latexit> All-or-nothing we(A) = ( 0 if A 2 {e, ;} 1 otherwise Linear penalty we(A) = min{|A|, |eA|} Quadratic penalty we(A) = |A| · |eA| Discount cut we(A) = min{|A|↵ , |eA|↵ } L-M submodular we(A) = 1 2 + 1 2 · min n 1, |A| b↵|e|c , |eA| b↵|e|c o
  9. 9. Cardinality-based splitting functions are easy to specify. 9 minimizeS⇢V P e2E we(e S) ⌘ cutH(S) subject to s 2 S, t /2 S.<latexit sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">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</latexit><latexit sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">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</latexit><latexit sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">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</latexit><latexit sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">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</latexit> s t One extra scaling DOF, so set w1 = 1. Specify w2, ... , wbr/2c.<latexit sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit><latexit sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit><latexit sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit><latexit sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit> cutH(S) = f (2) + f (1) = w2 + 1<latexit sha1_base64="djHeAEhlAe+wWiTssHtiXt8DSlg=">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</latexit><latexit sha1_base64="djHeAEhlAe+wWiTssHtiXt8DSlg=">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</latexit><latexit sha1_base64="djHeAEhlAe+wWiTssHtiXt8DSlg=">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</latexit><latexit sha1_base64="djHeAEhlAe+wWiTssHtiXt8DSlg=">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</latexit> Only need to specify f(1), f(2), …, f(⌊r / 2⌋), where r = max hyperedge size. Just scalars. f(i) = wi. Cardinality-based splitting functions. <latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit> Non-negativity we(A) 0. Non-split ignoring we(e) = we(;) = 0. C-B we(A) = f (min(|A|, |Ae|)).
  10. 10. Cardinality-based splitting functions are easy to specify. 10 Just need to specify w2, ... , wbr/2c and assume w1 = 1.<latexit sha1_base64="OwBovXiRkyHjnkYdEriLKgfnPdk=">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</latexit><latexit sha1_base64="OwBovXiRkyHjnkYdEriLKgfnPdk=">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</latexit><latexit sha1_base64="OwBovXiRkyHjnkYdEriLKgfnPdk=">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</latexit><latexit sha1_base64="OwBovXiRkyHjnkYdEriLKgfnPdk=">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</latexit> r = 2 (graphs) r = 3 (3-uniform hypergraph) “Only one way to split a triangle” [Benson+ 16; Li-Milenkovic 17; Yin+ 17] s t s t s t r = 4 w2 = 0.5 solution w2 = 1.5 solution w3 = 1.5 solution
  11. 11. 1.0 1.25 1.5 1.75 2.0 fusion- systems topological- stacks graph- invariants adjacency- matrix signed- graph gorenstein cohen- macaulay topological- k- theory difference- sets pushforward regular- rings graph- connectivity block- matrices directed- graphs eulerian- path central- extensions group- extensions semidirect- product wreath- product graded- algebras supergeometry geometric- complexity soliton- theory matrix- congruences teichmueller- theory superalgebra string- theory riemann- surfaces group- cohomology dglas celestial- mechanics s- seed = symplectic- linear- algebra t- seed = bernoulli- numbers Different weights lead to different min cuts in practice. 11 1.00 1.25 1.50 1.75 2.00 0.7 0.8 0.9 1.0 JaccardSimilarity
  12. 12. 12 1. What is a hypergraph minimum s-t cut? 2. If we know what they are, can we find them efficiently? 3. If we can find them efficiently, what can we use them for? We should have a foundation for hypergraph minimum s-t cuts,but…
  13. 13. We solve hypergraph cut problems with graph reductions. 13 1/21/2 1/2 1 1 1 1 ∞ ∞ ∞ ∞ ∞∞ Gadgets (expansions) model a hyperedge with a small graph. clique expansion star expansion Lawler gadget [1973]hyperedge In a graph reduction, we first replace all hyperedges with graph gadgets... s t s t s t s t … then solve the (min s-t cut) problem exactly on the graph, and finally convert the solution to a hypergraph solution.
  14. 14. b We made a new gadget for C-B splitting functions. 14 This gadget models min(|A|, |eA|, b). Theorem [Veldt-Benson-Kleinberg 20a]. Nonnegative linear combinations of the C-B gadget can model any submodular cardinality-based splitting function. See also Graph Cuts for Minimizing Robust Higher Order Potentials,Kohli et al.,2008. <latexit sha1_base64="beQz4cdyY+p8N+9L01TDcNAiwcQ=">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</latexit> C-B we(A) = f (min(|A|, |eA|)). (F is submodular on X if F(A B) + F(A [ B)  F(A) + F(B) for any A, B ✓ X.)<latexit sha1_base64="jx6llVBabtrhi3TShW9c6Ptv2Kc=">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</latexit><latexit sha1_base64="jx6llVBabtrhi3TShW9c6Ptv2Kc=">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</latexit><latexit sha1_base64="jx6llVBabtrhi3TShW9c6Ptv2Kc=">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</latexit><latexit sha1_base64="jx6llVBabtrhi3TShW9c6Ptv2Kc=">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</latexit>
  15. 15. 15 Theorem [Veldt-Benson-Kleinberg 20a]. The hypergraph min s-t cut problem with a cardinality-based splitting function is graph-reducible (via gadgets) if and only if the splitting function is submodular. Cardinality-based splitting functions. s t S<latexit sha1_base64="wtJ1SkLACwJOcMcL9/jLEzSB0Ao=">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</latexit><latexit sha1_base64="wtJ1SkLACwJOcMcL9/jLEzSB0Ao=">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</latexit><latexit sha1_base64="wtJ1SkLACwJOcMcL9/jLEzSB0Ao=">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</latexit><latexit sha1_base64="wtJ1SkLACwJOcMcL9/jLEzSB0Ao=">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</latexit> cutH(S) = f (2) + f (1)<latexit sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit><latexit sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit><latexit sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit><latexit sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit> Submodularity is key to efficient algorithms. What happens when the splitting function isn’t submodular? Is there some other efficient algorithm? <latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit> Non-negativity we(A) 0. Non-split ignoring we(e) = we(;) = 0. C-B we(A) = f (min(|A|, |Ae|)).
  16. 16. 16 Unlike graph min s-t cut, hypergraph min s-t cut can be NP-hard. w1 = 1 0 1 2 w2 ?? Reducible/Submodular NP-hard Unknown Hard Reducible w3 3 2.5 2 1.5 1 0.5 1 1.5 2 2.5 w2 0.5 w2 w3 w4 4 3 2 1 0 1 1.5 2 2.5 1 2 3 max hyperedge size 4 or 5 max hyperedge size 6 or 7 max hyperedge size 8 or 9 Theorem [Veldt-Benson-Kleinberg 20]. For C-B splitting functions, Open Question: For 4-uniform hypergraphs, is there an efficient algorithm to find the minimum s-t cut with no 2-2 splits (w1 = 1, w2 = ∞). s t cutH(S) = f (2) + f (1) = w2 + 1<latexit sha1_base64="buuvN8Zq181Nh/WWQIuguiXmNlg=">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</latexit><latexit sha1_base64="buuvN8Zq181Nh/WWQIuguiXmNlg=">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</latexit><latexit sha1_base64="buuvN8Zq181Nh/WWQIuguiXmNlg=">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</latexit><latexit sha1_base64="buuvN8Zq181Nh/WWQIuguiXmNlg=">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</latexit>
  17. 17. 17 1. What is a hypergraph minimum s-t cut? 2. If we know what they are, can we find them efficiently? 3. If we can find them efficiently, what can we use them for? We should have a foundation for hypergraph minimum s-t cuts,but…
  18. 18. G = (V,E) is a graph. R ⊆ V (Reference or seed set). Finds a “good” cluster S “near” R. 18 Background.Local clustering has been studied extensively in graphs,but not much in hypergraphs.
  19. 19. Rewards high overlap with R. Penalizes nodes outside R. R(S) = cut(S) vol(S R) "vol(S ¯R) Max Flow.Quot.Imp.(Lang,Rao,2004) Flow-Improve (Andersen,Lang 2008) Local-Improve (Orecchia,Allen-Zhou 2014) SimpleLocal (Veldt,Gleich,Mahoney 2016) FlowSeed (Veldt,Klymko,Gleich 2019) Great survey paper! (Fountoulakis et al.2020) 19 Background.Flow-based methods minimize a localized variant of conductance. Rewards contained clusters vol(T) = sum of degrees in T. minimize node sets S FAST ALGORITHMS FOR EXACT MINIMIZATION!
  20. 20. s t 2 4 4 7 3 4 7 3 2 1 3 2 6 4 5 7 8 9 10Set R 4 1 3 2 6 4 5 7 8 9 10 s t 2 4 4 7 3 4 7 3 2 1 3 2 6 4 5 7 8 9 10Set R 4 [Andersen-Lang 08,Orecchia-Zhou 14,Veldt+ 16] Construct G’ R(S) < ↵ () min s-t cut of G0 < ↵vol(R) Compute min s-t cut of G’. 20 Connect R to a source node s ; edges weighted with respect to $. Connect VR to a sink node t ; edges weighted with respect to β = $ε. Is R(S) < ↵ for any S? Background.Flow methods repeatedly solve min-cut problems on an auxiliary graph.
  21. 21. We generalize local flow-based techniques to the hypergraph setting. 21 • We introduce localized hypergraph conductance • We can minimize it exactly with our hypergraph min s-t cuts framework • Strongly-local runtime! (Only depends on size of seed set) • Normalized cut improvement guarantees The analysis provides even new guarantees for the graph case!

×