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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
Graph minimum s-t cuts are fundamental.
2
minimizeS⇢V cut(S)
subject to s 2 S, t /2 S.<latexit 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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!
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
We can model“higher-order”interactions
with hypergraphs.
4
H = (V, E), edge e 2 E is a subset of V (e ⇢ V)<latexit 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1 2
3
4
5
V = {1, 2, 3, 4, 5}
E = {{1, 2, 3}, {2, 4, 5}}<latexit 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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…
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).
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 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cutH(S) = f (2) + f (1)<latexit 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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=">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</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|)).
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=">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</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
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 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s
t
One extra scaling DOF, so set w1 = 1. Specify w2, ... , wbr/2c.<latexit 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cutH(S) = f (2) + f (1) = w2 + 1<latexit 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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=">AAAIS3icfVXdbts2FLa7Ncm8v2a73A27wEMSyI6dIksyIEC6FsUKtFs2J20B08go6UgiTFIqScX2CD3FbrdH2gPsOXY37GKHtrPYTlYBtijyfN/55wkLwY3tdP6s33vv/ftr6xsfND786ONPPn2w+dkrk5c6gosoF7l+EzIDgiu4sNwKeFNoYDIU8DocPvHnr69AG56rczspYCBZqnjCI2Zx63KzvkZDSLlyTPBU7VYNamFs3fe5ailIUeiK20lFvqKZKVgErtPeP4hkRejV6BK2H+8QmsJb0mkTShegBk23BAlzzVWKcLKKnxPADjmZLynIwk4MWL+1SPik9e27DDghyTaVXG1TgY5a8phQ7RcBudkIWTQ0gpmMwPx0Z6fdoKDia78vH2x12p3pQ24vuvPFVm3+nF1urgU0zqNSgrIRUpt+t1PYgWPa8kgABrI0gBYPWQp9XComwQzcNGMVaeJOTJJc409ZMt1tLEKQR7PJEouzLCwF0+Pl3TDPh3hiqiV83xeEUaUMQUMc6FJAjMaJFDNiM7kPK+KlTY4GjquitKCimYFJKYjNiS8bEnMNkRUTsmyl5cNfAsUjSDSLAiaNZDYLCu69CiQbQgRCzOz1ooKHmumJdy4fmcCnJdV5qWITFMxa0Mogymo+DkzGCjBBwm0QMRH579hjCpFbyfTQ/B9rW4JleDiNqQDrzsvEwk8QVw4j8fCo8zAUqHdRwmaQagBVuenLy4wybmFFJhQlVM7/L0g0miSztjDf7O1htbaNRW4YRxlTKbSjXO69LcH4XjN73a8PjveP9wxIjiUYYnPJ1giz0fJOtLhqhdi4oKdyjw63Zq8G9WFk2Ng+Pg2aijxkguIn9bBTUKbUcBrnAkvjFNs6ymM4oRoEG19jczR+ubz6592B80nyyV7K6Nl5jykfXA0KRuiAZNglNGGSi0kMCSuFrRw1yfV6uSBM4iugajQXlRnMIMQnnfZxEGGnWow2E9gMqMCOTeIplp1Ebqrs2FOdzsDO7PaxCw8G1apTTwHbT0NvIsNcPEOX3IzFVO6Hly8qp7wKySsnK8fRXNoDe5cwbsSrkHAOmevwgF4ZYjpt6VN6t4JVDb1nL31IrhWcd5fC58Jx5Yy4UeKFZ2j3HCV9DJgoMlbdmPrz85Wox6kAHmWtWezvOsFEG7x4lm8O6WkWsyx7PJWoic6qytM5GkpHZ/vVrbKQL3DUxHch5gfVsopdOg6Z7mPx0SzMx45e+f9mg2b+hiIZ8DSzeO8eHhSWNMl5BoRFtmSCIKxBh3hD+AEA4ya5fprkKY5JpiIgIdgR9q+XJaiMmGkYGzNVzQYOHU/Q6rS7IJvX6F6Wa4wOzimSK4JFRQQklhgeg0cs+LXVrf4jwdHw6J0keurJlAWD4OdLd3Wa3F682m93D9qdH/e3To/mk2aj9kXty9p2rVs7rJ3Wvqud1S5qUV3Wf63/Vv99/Y/1v9b/Xv9nJnqvPsd8Xlt6Nu7/C+Rx6YI=</latexit>
Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).
Cardinality-based splitting functions are easy to specify.
10
Just need to specify w2, ... , wbr/2c and assume w1 = 1.<latexit 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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
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
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…
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.
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.
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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 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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 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cutH(S) = f (2) + f (1)<latexit 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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=">AAAIS3icfVXdbts2FLa7Ncm8v2a73A27wEMSyI6dIksyIEC6FsUKtFs2J20B08go6UgiTFIqScX2CD3FbrdH2gPsOXY37GKHtrPYTlYBtijyfN/55wkLwY3tdP6s33vv/ftr6xsfND786ONPPn2w+dkrk5c6gosoF7l+EzIDgiu4sNwKeFNoYDIU8DocPvHnr69AG56rczspYCBZqnjCI2Zx63KzvkZDSLlyTPBU7VYNamFs3fe5ailIUeiK20lFvqKZKVgErtPeP4hkRejV6BK2H+8QmsJb0mkTShegBk23BAlzzVWKcLKKnxPADjmZLynIwk4MWL+1SPik9e27DDghyTaVXG1TgY5a8phQ7RcBudkIWTQ0gpmMwPx0Z6fdoKDia78vH2x12p3pQ24vuvPFVm3+nF1urgU0zqNSgrIRUpt+t1PYgWPa8kgABrI0gBYPWQp9XComwQzcNGMVaeJOTJJc409ZMt1tLEKQR7PJEouzLCwF0+Pl3TDPh3hiqiV83xeEUaUMQUMc6FJAjMaJFDNiM7kPK+KlTY4GjquitKCimYFJKYjNiS8bEnMNkRUTsmyl5cNfAsUjSDSLAiaNZDYLCu69CiQbQgRCzOz1ooKHmumJdy4fmcCnJdV5qWITFMxa0Mogymo+DkzGCjBBwm0QMRH579hjCpFbyfTQ/B9rW4JleDiNqQDrzsvEwk8QVw4j8fCo8zAUqHdRwmaQagBVuenLy4wybmFFJhQlVM7/L0g0miSztjDf7O1htbaNRW4YRxlTKbSjXO69LcH4XjN73a8PjveP9wxIjiUYYnPJ1giz0fJOtLhqhdi4oKdyjw63Zq8G9WFk2Ng+Pg2aijxkguIn9bBTUKbUcBrnAkvjFNs6ymM4oRoEG19jczR+ubz6592B80nyyV7K6Nl5jykfXA0KRuiAZNglNGGSi0kMCSuFrRw1yfV6uSBM4iugajQXlRnMIMQnnfZxEGGnWow2E9gMqMCOTeIplp1Ebqrs2FOdzsDO7PaxCw8G1apTTwHbT0NvIsNcPEOX3IzFVO6Hly8qp7wKySsnK8fRXNoDe5cwbsSrkHAOmevwgF4ZYjpt6VN6t4JVDb1nL31IrhWcd5fC58Jx5Yy4UeKFZ2j3HCV9DJgoMlbdmPrz85Wox6kAHmWtWezvOsFEG7x4lm8O6WkWsyx7PJWoic6qytM5GkpHZ/vVrbKQL3DUxHch5gfVsopdOg6Z7mPx0SzMx45e+f9mg2b+hiIZ8DSzeO8eHhSWNMl5BoRFtmSCIKxBh3hD+AEA4ya5fprkKY5JpiIgIdgR9q+XJaiMmGkYGzNVzQYOHU/Q6rS7IJvX6F6Wa4wOzimSK4JFRQQklhgeg0cs+LXVrf4jwdHw6J0keurJlAWD4OdLd3Wa3F682m93D9qdH/e3To/mk2aj9kXty9p2rVs7rJ3Wvqud1S5qUV3Wf63/Vv99/Y/1v9b/Xv9nJnqvPsd8Xlt6Nu7/C+Rx6YI=</latexit>
Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).
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 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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…
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.
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!
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.
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!
Hypergraph Cuts with General Splitting Functions (JMM)
Hypergraph Cuts with General Splitting Functions (JMM)
Hypergraph Cuts with General Splitting Functions (JMM)
Hypergraph Cuts with General Splitting Functions (JMM)
Hypergraph Cuts with General Splitting Functions (JMM)
Hypergraph Cuts with General Splitting Functions (JMM)
Hypergraph Cuts with General Splitting Functions (JMM)

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Hypergraph Cuts with General Splitting Functions (JMM)

  • 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. Graph minimum s-t cuts are fundamental. 2 minimizeS⇢V cut(S) subject to s 2 S, t /2 S.<latexit 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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. 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. We can model“higher-order”interactions with hypergraphs. 4 H = (V, E), edge e 2 E is a subset of V (e ⇢ V)<latexit 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1 2 3 4 5 V = {1, 2, 3, 4, 5} E = {{1, 2, 3}, {2, 4, 5}}<latexit 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  • 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. 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. 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. 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cutH(S) = f (2) + f (1)<latexit 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<latexit 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we(e S) <latexit 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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. 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=">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</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. 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=">AAAICXicfVXdbhw1FN4UaMrw05RecuMSLQrR7mY3VUgCqhSJULVSKwKbtJXiVfDMnNlx1/ZMbU+yieUn4Gm4Q9zyFNzwLBzPbJrdJOCLGfv4nO/z+bPjUnBj+/2/l+588OFHd5fvfRx98ulnn99fefDFK1NUOoGjpBCFfhMzA4IrOLLcCnhTamAyFvA6nvwQ9l+fgja8UIf2vISRZGPFM54wi6KTlX9oDGOuHLyrasm6j2YSpjU7904IlFiYWie54pJfgD9xQ0JNFRuw5JUnX4eFPHFAKFfkR0/o6dkJrOEyYSUZfkMogvNT0qAklUUAKpnNEybcM+/XggqNHFWFlkxkhbIzXeR4C4kltiA+8JiaYdih3xNUUYUNq15EQaWz4zbzK2dOVlb7vX49yM3JYDZZbc3GwcmDuw9pWiSVBGUTwYw5HvRLO0J4yxMBSFAZKFkyYWM4xqliEszI1YnwpI2SlGSFJrUXtTSaN3l/zDmRZXElmJ4uSuOimOCO8VG0yGmznZHjqqwsqKShzCoRYhTyS1KuMWTinCzyWj656CieQKZZ0mHShAR0Sh7O2bGTi+5YszLvSDaBBIS4EjWnCuaCx5rp8+BCcWY6MSKPdVGp1HRKZi1oZdDeaj7tmJyVYDoZtx1MchLWabApRWEl0xPzX6g9CZbhZh05AdYdVpmFXyD1TkP6aKf/KBbIO69hcxhrAOVd/Qs6Zzm3cE0nFhV4F75zGlGb5NaW5ruNDSy4nrGIDdMkZ2oMvaSQG+8qMKGSzMbg263dzd0NA5JjP8VYX7J7xm3eDU50uerG2HWga73H26vNL6IhoAy7MsQnomNRxExQXNJgtgfKVBr20kJgAexhTyZFCk+oBsGml7YFHn6xiI4PByMXEhcKYCHLB4dDpkJwNSg4Qwckw3agGZNcnKeQsUpY76jJLueLRWKyUBU+as+TGcwgpE/6vd1OgleAxWgzgSWPBHZqsgCx6CRiU2WnAWqvMXZm/Rh7bWvkrzu1D9hkGobnMi7EU3TJNSjGu59evvBOBQrJvZPecTwuHYK9TRkF6XWTeGYy4wgGQ7y08JKsQkpvJ7jOMHz6MoTkkuBwsBA+F0+9M+KKJCg31u65b245Jsqc+auj/vr8WtTTsQCe5N0m9rftYKINXi+L94MMMPNZlkM+lshEm6oKcI7G0tFG7m+UhXyB70R6m8Vswy9SrNNpzPQxFh/N42Lq6Gn4tiOa60oAyYGPc4u36/ZWaUmbHOZAWGIrJgiaRXSCN0S/t7kF0za5HG2yj28cUwmQGOwZ9m/QJUhGTB3GqKFqR4TUAN1+bwCyfWk9zAuN0eFqTApFsKiIgMwSw1MIFnN+rQ78exB8AB7/L4iuPalRfIgCPiOD64/Gzcmrzd4Aj/fz5urezuxBudf6svVVa601aG239lrPWgeto1aytL/0dsks2eXfln9f/mP5z0b1ztLM5mFrYSz/9S/5ANcF</latexit><latexit sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">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</latexit><latexit sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">AAAICXicfVXdbhw1FN4UaMrw05RecuMSLQrR7mY3VUgCqhSJULVSKwKbtJXiVfDMnNlx1/ZMbU+yieUn4Gm4Q9zyFNzwLBzPbJrdJOCLGfv4nO/z+bPjUnBj+/2/l+588OFHd5fvfRx98ulnn99fefDFK1NUOoGjpBCFfhMzA4IrOLLcCnhTamAyFvA6nvwQ9l+fgja8UIf2vISRZGPFM54wi6KTlX9oDGOuHLyrasm6j2YSpjU7904IlFiYWie54pJfgD9xQ0JNFRuw5JUnX4eFPHFAKFfkR0/o6dkJrOEyYSUZfkMogvNT0qAklUUAKpnNEybcM+/XggqNHFWFlkxkhbIzXeR4C4kltiA+8JiaYdih3xNUUYUNq15EQaWz4zbzK2dOVlb7vX49yM3JYDZZbc3GwcmDuw9pWiSVBGUTwYw5HvRLO0J4yxMBSFAZKFkyYWM4xqliEszI1YnwpI2SlGSFJrUXtTSaN3l/zDmRZXElmJ4uSuOimOCO8VG0yGmznZHjqqwsqKShzCoRYhTyS1KuMWTinCzyWj656CieQKZZ0mHShAR0Sh7O2bGTi+5YszLvSDaBBIS4EjWnCuaCx5rp8+BCcWY6MSKPdVGp1HRKZi1oZdDeaj7tmJyVYDoZtx1MchLWabApRWEl0xPzX6g9CZbhZh05AdYdVpmFXyD1TkP6aKf/KBbIO69hcxhrAOVd/Qs6Zzm3cE0nFhV4F75zGlGb5NaW5ruNDSy4nrGIDdMkZ2oMvaSQG+8qMKGSzMbg263dzd0NA5JjP8VYX7J7xm3eDU50uerG2HWga73H26vNL6IhoAy7MsQnomNRxExQXNJgtgfKVBr20kJgAexhTyZFCk+oBsGml7YFHn6xiI4PByMXEhcKYCHLB4dDpkJwNSg4Qwckw3agGZNcnKeQsUpY76jJLueLRWKyUBU+as+TGcwgpE/6vd1OgleAxWgzgSWPBHZqsgCx6CRiU2WnAWqvMXZm/Rh7bWvkrzu1D9hkGobnMi7EU3TJNSjGu59evvBOBQrJvZPecTwuHYK9TRkF6XWTeGYy4wgGQ7y08JKsQkpvJ7jOMHz6MoTkkuBwsBA+F0+9M+KKJCg31u65b245Jsqc+auj/vr8WtTTsQCe5N0m9rftYKINXi+L94MMMPNZlkM+lshEm6oKcI7G0tFG7m+UhXyB70R6m8Vswy9SrNNpzPQxFh/N42Lq6Gn4tiOa60oAyYGPc4u36/ZWaUmbHOZAWGIrJgiaRXSCN0S/t7kF0za5HG2yj28cUwmQGOwZ9m/QJUhGTB3GqKFqR4TUAN1+bwCyfWk9zAuN0eFqTApFsKiIgMwSw1MIFnN+rQ78exB8AB7/L4iuPalRfIgCPiOD64/Gzcmrzd4Aj/fz5urezuxBudf6svVVa601aG239lrPWgeto1aytL/0dsks2eXfln9f/mP5z0b1ztLM5mFrYSz/9S/5ANcF</latexit> s t One extra scaling DOF, so set w1 = 1. Specify w2, ... , wbr/2c.<latexit 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sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">AAAHhXicfVXdbts2FFa7re60v3S93A27xMBQyD9ylyYZEMzAgmAFWiSbnbZYZGSUdGQRpkSNpGq5hN5tr7EX2O32CDu0ncVysvFCIg/P9308h4dkWHCmdL//x737H3z40YPWw4/dTz797PMvdh59+VqJUkZwEQku5NuQKuAshwvNNIe3hQSahRzehLMf7PybdyAVE/lYLwqYZHSas4RFVKPpaueXsxwIVFpSoiKKLFNycnbqESWIAk325lc+OSb+XpeMCohYsrCmgUcCHgutPDK/MgFPuBCSSNIjAxLI5aje617t7Pa7/WUjtzv+urPrrNv51aMHj4NYRGUGuY44VerS7xd6YqjULOJQu0GpoKDRjE7hErs5zUBNzDIJNWmjJSYJLiQRuSZLq7sJQR5JFw0Wo2lYciqrpjUUYoYzqnbdpqZODieG5UWpIY9WkknJiRbE5pbETEKk+YI0dTWbvfdyFkEiaeTRTGVUp17B7Do9PXvfmUpapF5GZxAB5zem1aosnLNQUrmwIYi58kJknkpR5rHyCqo1yFwhXktWeSqlBSgvYdrDDY3sOLaYggudUTlT/8XazUBTnFxmjoM24zLR8DPEtZEQPznsPwk56m566BSmEiCvzfJnfeYp07DlE/ISamO/Gx5um6RaF+q7Xk9D1VUauaGKUppPoRuJrPdbCcoWqer5z/ePBkc9BRnDWg6xdLPOnOm0Y4PosLwTYsWDXPo9O9hd/dzAJpTiibD5cYMpFyHlAQ4DCxtCrkoJw1hwLIAhnodIxHAcSOC0usYKXHyziC7H/sTYjbMF0Njl8/GI5ja5EnKYYwAZzWMTJDRjfBFDQkuuaxOo5LrfLBKV2Kqo3fammD2SEB/3u0delDEUxbLgWPIooCuVWIpmkMgd5LqyVMMV2Kinl3jW9if1dlAngIdMwmiRhYKfYkhmxaJqc/bqZW1yK5Gx2mS1YbjcYAT6Lmc0xNuQcA1Za1jAqAxxO3Vpt/RugW2F0ekrm5JrgbHfSJ8Jq9oofiNinVdo8wI9bQ4oL1Ja3yz11xdbWY+nHFiUdla5v2sGN1rh9dK8HzJLs7nL2YhNM1QKVlVl6UwQZiZY2etbZZG9xDs6vguxnqibEk+DKqTyEosvSENRmeCd/bbdIJUlB5ICm6Yab9eD/UKTNhmnQGikS8oJwtxghjdEvzvYh6pNrlubnOD7QvMISAh6jufX+hIUI2qZRncl1XYJWRJ0+l0fsvY1epQKidmxT4bICRYV4ZBoolgMFrER165f/0uCD8Cz/yWRy0iWLLXNAj4j/vajcbvzetD1cXk/DXaHh+sH5aHzlfO1843jOwfO0PnROXcunMj53fnT+cv5u9VqdVrftp6vXO/fW2MeO43W+v4f50qgzw==</latexit><latexit 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cutH(S) = f (2) + f (1) = w2 + 1<latexit 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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. Cardinality-based splitting functions are easy to specify. 10 Just need to specify w2, ... , wbr/2c and assume w1 = 1.<latexit 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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. 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 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. 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. 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 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  • 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 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cutH(S) = f (2) + f (1)<latexit 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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 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 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  • 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. 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. 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. 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. 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!