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1
Joint work with
Nate Veldt & Jon Kleinberg (Cornell)
Hypergraph Cuts with General Splitting Functions
Austin R. Benson · Cornell University
Applied and Computational Discrete Algorithms Minisymposium
SIAM Annual · July 6, 2020
Slides. bit.ly/arb-ACDA-AN20
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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1 3
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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 are composed of“higher-order”
interactions that we can model with hypergraphs.
3
Physical proximity
• nodes are students
• hyperedges are students
in the same class
Drug compounds
• nodes are substances
• hyperedges are substances
combined in a drug
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
• hyperedges are groups of tags (e.g.,for the
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.
Real-world systems are composed of“higher-order”
interactions that we can model 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
Non-negativity we(U) 0 for all U ⇢ e.
Symmetry we(U) = we(eU) for all U ⇢ e.
Non-split ignoring we(e) = we(;) = 0.<latexit 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Splitting function for separating edge e into U and U  e.
For each edge e, we have a function we with
minimizeS⇢V
P
e2E we(e  S) ⌘ cutH(S)
subject to s 2 S, t /2 S.<latexit 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Hypergraph minimum s-t cut problem.
1. Anonymity. A node’s identity doesn’t affect the function.
2. Heterogeneity. Same splitting function at each edge.
Cardinality-based splitting functions.
S<latexit 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cutH(S) = f (2) + f (1)<latexit 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we(U) = f (min(|U|, |Ue|))<latexit 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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]
All-or-nothing we(U) =
(
0 if U 2 {e, ;}
1 otherwise
Linear penalty we(U) = min{|U|, |eU|}
Quadratic penalty we(U) = |U| · |eU|
Discount cut we(U) = min{|U|↵ , |eU|↵ }
L-M submodular we(U) = 1
2 + 1
2 · min
n
1, |U|
b↵|e|c , |eU|
b↵|e|c
o
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Cardinality-based splitting functions are easy to specify.
9
Cardinality-based splitting functions.
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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Non-negativity we(U) 0 for all U ⇢ e.
Non-split ignoring we(e) = we(;) = 0.
C-B we(U) = f (min(|U|, |Ue|)).<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 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.
s
t
s
t
s
t
s
t
Existing gadgets model cardinality-based splitting functions.
14
1/21/2
1/2
1
1
1
1
∞
∞ ∞
∞
∞∞
clique expansion star expansion Lawler gadget [1973]hyperedge
Quadratic penalty
wi = i ( k – i )
k = hyperedge size
Linear penalty
wi = i
All-or-nothing
wi = 1
s
t
Existing gadgets model cardinality-based splitting functions.
15
1
∞
∞ ∞
∞
∞∞s
t
1
∞
∞ ∞
∞
∞∞with s
with t
with t
must go
with s
must go
with t
⟶ penalty = 1
1
∞
∞ ∞
∞
∞∞with s
with s
with s
must go
with s
must go
with s
⟶ penalty = 0
Directed min
s-t graph cut
We can encode gadgets as splitting functions.
16
hyperedge e ⟶ graph gadget Ge = (V′, E′ )
Gadget splitting function.
ˆwe(U) = minimum
T✓V0
, Te=U cutGe
(T)<latexit 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1
∞
∞ ∞
∞
∞∞with s
with t
with t
must go
with s
must go
with t
Given a split {U, e  U}, the gadget splitting function “moves” any auxiliary
nodes to yield the smallest penalty, keeping the split {U, eU}.
Theorem [Veldt-Benson-Kleinberg 20].
Gadget splitting functions are submodular (on the ground set e).
Corollary. If a hypergraph min s-t cut problem with a splitting function is
graph-reducible ( ), then the splitting function is submodular.ˆwe(U) = we(U)<latexit 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(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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b
We made a new gadget for C-B splitting functions.
17
ˆwe(U) = min{|U|, |eU|, b}<latexit 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b = 1, 2, 3, …, q =⌊r/2⌋ ⟶ basis of gadgets
(b = 1 ⟶ all-or-nothing penalty)
Theorem [Veldt-Benson-Kleinberg 20]. Nonnegative linear combinations of the
C-B gadget can model any submodular cardinality-based splitting function.
C-B we(U) = f (min(|U|, |Ue|)).<latexit 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(For a similar result, see Graph Cuts for Minimizing
Robust Higher Order Potentials, Kohli et al., 2008.)
18
Theorem [Veldt-Benson-Kleinberg 20]. The hypergraph min s-t cut problem with
a cardinality-based splitting function is graph-reducible 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?
Can we use some other algorithm?
Non-negativity we(U) 0 for all U ⇢ e.
Non-split ignoring we(e) = we(;) = 0.
C-B we(U) = f (min(|U|, |Ue|)).<latexit 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19
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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20
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…
21
Local graph clustering algorithms find a well-connected
clusters nearby a given reference set.
Flow-based Algorithms for Improving Clusters: A
Unifying Framework,Software,and Performance.K.
Fountoulakis,M.Liu, D.F.Gleich,M.W.Mahoney,2020.
some reference nodes
• Used in social community detection, information retrieval, medical
imaging, exploratory data analysis generally.
• Many algorithms based on repeatedly solving graph min s-t cuts
[Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16,Veldt+ 19]
minimize
S⇢V
HLCR,"(S) =
cutH(S)
volH(S  R) "volH(S  ¯R)
cutH(S) = cut from C-B splitting function
volH(X) =
X
i2X
X
e2E
1 = sum of hypergraph degrees in X
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22
We use hypergraph local conductance to
model cluster quality.
Given a reference set R, Not much crosses boundary
Encourage overlap with reference set. Discourage overlap outside reference set
• Generalization of a graph objective [Andersen-Lang 08].
• For a graph with R = V and 𝜀 = 0 ⟶ cut(S) / vol(S) ≈ graph conductance.
23
We can optimize HLC with hypergraph min s-t cut.
s
t
𝜀𝛼dj
𝛼dr
r
j
S
Repeat,starting with S = reference set R.
• S ⟵ hypergraph min s-t cut solution.
• 𝛼 ⟵ HLCR,𝜀(S)
This globally minimizes HLCR,𝜀(S) !
Theorem [Veldt-Benson-Kleinberg 20]. Strong locality. Can make this algorithm run
in time proportional to size of R (does not look at the full hypergraph).
24
Cluster |T| time (s) HyperLocal Baseline1 Baseline2
Amazon Fashion 31 3.5 0.83 0.77 0.6
All Beauty 85 30.8 0.69 0.60 0.28
Appliances 48 9.8 0.82 0.73 0.56
Gift Cards 148 6.5 0.86 0.75 0.71
Magazine Subscriptions 157 14.5 0.87 0.72 0.56
Luxury Beauty 1581 261 0.33 0.31 0.17
Software 802 341 0.74 0.52 0.24
Industrial & Scientific 5334 503 0.55 0.49 0.15
Prime Pantry 4970 406 0.96 0.73 0.36
<latexit 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sha1_base64="ozxPLubxMnSFqTw29mAoiaPKUSk=">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</latexit>
• 2.3M Amazon products (nodes),reviewed by 4.3M users (hyperedges).
• mean hyperedge size > 17,max hyperedge size ~9.3k.
• Product categories provide ground truth cluster labels.
• All-or-nothing penalty (wi = 1).
F1 recovery scores given a handful of nodes from the ground truth cluster T.
25
schem
e
ocam
l
tcl
m
dx
com
m
on- lisp
verilog
lotus- notes
xslt- 1.0
plone
typo3
abap
sitecore
m
arklogic
wolfram
- m
athem
atica
alfresco
axapta
vhdl
sparql
prolog
netsuite
racket
spring- integration
xslt- 2.0
m
ule
wso2
system
- verilog
wso2esb
google- sheets- form
ula
stata
xpages
netlogo
openerp
data.table
google- bigquery
docusignapi
aem
codenam
eone
dax
cypher
julia
sapui5
ibm
- m
obilefirst
office- js jq
apache- nifi
0.2
0.4
0.6
0.8
F1Scores
HyperLocal
TN/BN
FlowSeed
• 15M StackOverflow questions (nodes),answered by 1.1M users (hyperedges).
• mean hyperedge size 23.7,max hyperedge size ~60k.
• Tags provide ground truth cluster labels.
• C-B splitting function wi = min(i,5000).
HyperLocal
ClosestNeighbors
Clique Expansion +
Graph Method
26
Gadget reductions sometimes create dense graphs,
which can make computations expensive.
Theorem [Veldt-Benson-Kleinberg 20].Any submodular C-B splitting function can
be 𝜀-approx with log r / 𝜀 splitting functions (instead of r, r = hyperedge size).
And more…
• Fastest 𝜀-approx min s-t cut solver for certain co-occurrence graphs.
• r = 60k clique expansion only need O(r / √𝜀) instead of O(r2)
0 1 2 3
0
2
4
6
8
10
e0
1 e00
1
e0
2 e00
2
e0
3 e00
3
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
1 1
1 1
1 1
1 1
1 1
1 1
2
4
3
0 1 2 3
0
2
4
6
8
10
e0
e00
5 5
5 5
5 5
5 5
5 5
5 5
9
We can now model and use hypergraph min s-t cuts.
27
1. A model for hypergraph cuts.
C-B splitting functions that depend on # of nodes on small side of the cut
2. Algorithm for min s-t cuts with submodular C-B splitting functions.
Graph-reducible if and only if C-B splitting function is submodular
3. NP-hard and unknown complexity regimes of hypergraph min s-t cuts.
Open questions that we hope others can solve
4. Extensions to multiway cut.
Mostly of theoretical interest right now
5. Applications to local hypergraph clustering.
Strong locality lets us scale to large hypergraphs with large hyperedges
s
t
s
t
s
t
w2 = 0.5 (NP-hard) w2 = 1.5
(polynomial time via graph reduction)
w2 = 2.5 (?)
28
THANKS! Austin R. Benson
Slides. bit.ly/arb-ACDA-AN20
http://cs.cornell.edu/~arb
@austinbenson
arb@cs.cornell.edu
Hypergraph cuts
with general
splitting functions.
Hypergraph Cuts with General Splitting Functions.
Nate Veldt, Austin R. Benson, and Jon Kleinberg.
arXiv:2001.02817, 2020.
Localized Flow-Based Clustering in Hypergraphs.
Nate Veldt, Austin R. Benson, and Jon Kleinberg.
To appear at KDD, 2020.
github.com/nveldt/HypergraphFlowClustering
Augmented Sparsifiers for Generalized Hypergraph Cuts.
Nate Veldt, Austin R. Benson, and Jon Kleinberg.
In preparation.
Has 10 concrete open
algorithmic questions.
29
SIAM Workshop on Network Science 2020 (NS20)
Thursday and Friday, 9am—5pm PT
ns20.cs.cornell.edu
30
We also provide a framework for multiway cuts.
For a partition P of an edge e, a splitting function ze(P) is
s
t
u
Non-negative ze(P) 0
Permutation Invariant ze(P) = ze(P⇡) for any ⇡ 2 Sk
Non-split ignoring ze(P) = 0 if |P| = 1.<latexit 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31
We also provide a framework for multiway cuts.
[Hadley 95]
[Yaros- Imielinski 13]
[Mirzakhani-Vondrak 15]
[Alpert-Kahng 95;
Karypis-Kumar 99;
Chekuri-Ene 11]
All-or-nothing ze(P) =
(
0 if |P| = 1
1 otherwise
Sum of External Degrees ze(P) =
(
0 if |P| = 1
|P| otherwise
K 1 Penalty ze(P) = |P| 1
Rainbow Split ze(P) =
(
1 if |e| = |P|
0 otherwise.
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Theorem [Veldt-Benson-Kleinberg 20]. Minimizing the rainbow split on 3-uniform
hypergraphs is NP-hard to approximate within any constant factor.

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Hypergraph Cuts with General Splitting Functions

  • 1. 1 Joint work with Nate Veldt & Jon Kleinberg (Cornell) Hypergraph Cuts with General Splitting Functions Austin R. Benson · Cornell University Applied and Computational Discrete Algorithms Minisymposium SIAM Annual · July 6, 2020 Slides. bit.ly/arb-ACDA-AN20
  • 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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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 are composed of“higher-order” interactions that we can model with hypergraphs. 3 Physical proximity • nodes are students • hyperedges are students in the same class Drug compounds • nodes are substances • hyperedges are substances combined in a drug 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 • hyperedges are groups of tags (e.g.,for the 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.
  • 4. Real-world systems are composed of“higher-order” interactions that we can model 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 Non-negativity we(U) 0 for all U ⇢ e. Symmetry we(U) = we(eU) for all U ⇢ e. Non-split ignoring we(e) = we(;) = 0.<latexit 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Splitting function for separating edge e into U and U e. For each edge e, we have a function we with minimizeS⇢V P e2E we(e S) ⌘ cutH(S) subject to s 2 S, t /2 S.<latexit 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Hypergraph minimum s-t cut problem. 1. Anonymity. A node’s identity doesn’t affect the function. 2. Heterogeneity. Same splitting function at each edge. Cardinality-based splitting functions. 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cutH(S) = f (2) + f (1)<latexit 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we(U) = f (min(|U|, |Ue|))<latexit 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  • 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] All-or-nothing we(U) = ( 0 if U 2 {e, ;} 1 otherwise Linear penalty we(U) = min{|U|, |eU|} Quadratic penalty we(U) = |U| · |eU| Discount cut we(U) = min{|U|↵ , |eU|↵ } L-M submodular we(U) = 1 2 + 1 2 · min n 1, |U| b↵|e|c , |eU| b↵|e|c o <latexit 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  • 9. Cardinality-based splitting functions are easy to specify. 9 Cardinality-based splitting functions. minimizeS⇢V P e2E we(e S) ⌘ cutH(S) subject to s 2 S, t /2 S.<latexit 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sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">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</latexit> s t One extra scaling DOF, so set w1 = 1. Specify w2, ... , wbr/2c.<latexit 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sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit><latexit sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit> Non-negativity we(U) 0 for all U ⇢ e. Non-split ignoring we(e) = we(;) = 0. C-B we(U) = f (min(|U|, |Ue|)).<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.
  • 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. s t s t s t s t Existing gadgets model cardinality-based splitting functions. 14 1/21/2 1/2 1 1 1 1 ∞ ∞ ∞ ∞ ∞∞ clique expansion star expansion Lawler gadget [1973]hyperedge Quadratic penalty wi = i ( k – i ) k = hyperedge size Linear penalty wi = i All-or-nothing wi = 1
  • 15. s t Existing gadgets model cardinality-based splitting functions. 15 1 ∞ ∞ ∞ ∞ ∞∞s t 1 ∞ ∞ ∞ ∞ ∞∞with s with t with t must go with s must go with t ⟶ penalty = 1 1 ∞ ∞ ∞ ∞ ∞∞with s with s with s must go with s must go with s ⟶ penalty = 0 Directed min s-t graph cut
  • 16. We can encode gadgets as splitting functions. 16 hyperedge e ⟶ graph gadget Ge = (V′, E′ ) Gadget splitting function. ˆwe(U) = minimum T✓V0 , Te=U cutGe (T)<latexit 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1 ∞ ∞ ∞ ∞ ∞∞with s with t with t must go with s must go with t Given a split {U, e U}, the gadget splitting function “moves” any auxiliary nodes to yield the smallest penalty, keeping the split {U, eU}. Theorem [Veldt-Benson-Kleinberg 20]. Gadget splitting functions are submodular (on the ground set e). Corollary. If a hypergraph min s-t cut problem with a splitting function is graph-reducible ( ), then the splitting function is submodular.ˆwe(U) = we(U)<latexit 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(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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  • 17. b We made a new gadget for C-B splitting functions. 17 ˆwe(U) = min{|U|, |eU|, b}<latexit 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b = 1, 2, 3, …, q =⌊r/2⌋ ⟶ basis of gadgets (b = 1 ⟶ all-or-nothing penalty) Theorem [Veldt-Benson-Kleinberg 20]. Nonnegative linear combinations of the C-B gadget can model any submodular cardinality-based splitting function. C-B we(U) = f (min(|U|, |Ue|)).<latexit 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(For a similar result, see Graph Cuts for Minimizing Robust Higher Order Potentials, Kohli et al., 2008.)
  • 18. 18 Theorem [Veldt-Benson-Kleinberg 20]. The hypergraph min s-t cut problem with a cardinality-based splitting function is graph-reducible 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? Can we use some other algorithm? Non-negativity we(U) 0 for all U ⇢ e. Non-split ignoring we(e) = we(;) = 0. C-B we(U) = f (min(|U|, |Ue|)).<latexit 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  • 19. 19 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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  • 20. 20 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…
  • 21. 21 Local graph clustering algorithms find a well-connected clusters nearby a given reference set. Flow-based Algorithms for Improving Clusters: A Unifying Framework,Software,and Performance.K. Fountoulakis,M.Liu, D.F.Gleich,M.W.Mahoney,2020. some reference nodes • Used in social community detection, information retrieval, medical imaging, exploratory data analysis generally. • Many algorithms based on repeatedly solving graph min s-t cuts [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16,Veldt+ 19]
  • 22. minimize S⇢V HLCR,"(S) = cutH(S) volH(S R) "volH(S ¯R) cutH(S) = cut from C-B splitting function volH(X) = X i2X X e2E 1 = sum of hypergraph degrees in X <latexit 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22 We use hypergraph local conductance to model cluster quality. Given a reference set R, Not much crosses boundary Encourage overlap with reference set. Discourage overlap outside reference set • Generalization of a graph objective [Andersen-Lang 08]. • For a graph with R = V and 𝜀 = 0 ⟶ cut(S) / vol(S) ≈ graph conductance.
  • 23. 23 We can optimize HLC with hypergraph min s-t cut. s t 𝜀𝛼dj 𝛼dr r j S Repeat,starting with S = reference set R. • S ⟵ hypergraph min s-t cut solution. • 𝛼 ⟵ HLCR,𝜀(S) This globally minimizes HLCR,𝜀(S) ! Theorem [Veldt-Benson-Kleinberg 20]. Strong locality. Can make this algorithm run in time proportional to size of R (does not look at the full hypergraph).
  • 24. 24 Cluster |T| time (s) HyperLocal Baseline1 Baseline2 Amazon Fashion 31 3.5 0.83 0.77 0.6 All Beauty 85 30.8 0.69 0.60 0.28 Appliances 48 9.8 0.82 0.73 0.56 Gift Cards 148 6.5 0.86 0.75 0.71 Magazine Subscriptions 157 14.5 0.87 0.72 0.56 Luxury Beauty 1581 261 0.33 0.31 0.17 Software 802 341 0.74 0.52 0.24 Industrial & Scientific 5334 503 0.55 0.49 0.15 Prime Pantry 4970 406 0.96 0.73 0.36 <latexit 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• 2.3M Amazon products (nodes),reviewed by 4.3M users (hyperedges). • mean hyperedge size > 17,max hyperedge size ~9.3k. • Product categories provide ground truth cluster labels. • All-or-nothing penalty (wi = 1). F1 recovery scores given a handful of nodes from the ground truth cluster T.
  • 25. 25 schem e ocam l tcl m dx com m on- lisp verilog lotus- notes xslt- 1.0 plone typo3 abap sitecore m arklogic wolfram - m athem atica alfresco axapta vhdl sparql prolog netsuite racket spring- integration xslt- 2.0 m ule wso2 system - verilog wso2esb google- sheets- form ula stata xpages netlogo openerp data.table google- bigquery docusignapi aem codenam eone dax cypher julia sapui5 ibm - m obilefirst office- js jq apache- nifi 0.2 0.4 0.6 0.8 F1Scores HyperLocal TN/BN FlowSeed • 15M StackOverflow questions (nodes),answered by 1.1M users (hyperedges). • mean hyperedge size 23.7,max hyperedge size ~60k. • Tags provide ground truth cluster labels. • C-B splitting function wi = min(i,5000). HyperLocal ClosestNeighbors Clique Expansion + Graph Method
  • 26. 26 Gadget reductions sometimes create dense graphs, which can make computations expensive. Theorem [Veldt-Benson-Kleinberg 20].Any submodular C-B splitting function can be 𝜀-approx with log r / 𝜀 splitting functions (instead of r, r = hyperedge size). And more… • Fastest 𝜀-approx min s-t cut solver for certain co-occurrence graphs. • r = 60k clique expansion only need O(r / √𝜀) instead of O(r2) 0 1 2 3 0 2 4 6 8 10 e0 1 e00 1 e0 2 e00 2 e0 3 e00 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 4 3 0 1 2 3 0 2 4 6 8 10 e0 e00 5 5 5 5 5 5 5 5 5 5 5 5 9
  • 27. We can now model and use hypergraph min s-t cuts. 27 1. A model for hypergraph cuts. C-B splitting functions that depend on # of nodes on small side of the cut 2. Algorithm for min s-t cuts with submodular C-B splitting functions. Graph-reducible if and only if C-B splitting function is submodular 3. NP-hard and unknown complexity regimes of hypergraph min s-t cuts. Open questions that we hope others can solve 4. Extensions to multiway cut. Mostly of theoretical interest right now 5. Applications to local hypergraph clustering. Strong locality lets us scale to large hypergraphs with large hyperedges s t s t s t w2 = 0.5 (NP-hard) w2 = 1.5 (polynomial time via graph reduction) w2 = 2.5 (?)
  • 28. 28 THANKS! Austin R. Benson Slides. bit.ly/arb-ACDA-AN20 http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu Hypergraph cuts with general splitting functions. Hypergraph Cuts with General Splitting Functions. Nate Veldt, Austin R. Benson, and Jon Kleinberg. arXiv:2001.02817, 2020. Localized Flow-Based Clustering in Hypergraphs. Nate Veldt, Austin R. Benson, and Jon Kleinberg. To appear at KDD, 2020. github.com/nveldt/HypergraphFlowClustering Augmented Sparsifiers for Generalized Hypergraph Cuts. Nate Veldt, Austin R. Benson, and Jon Kleinberg. In preparation. Has 10 concrete open algorithmic questions.
  • 29. 29 SIAM Workshop on Network Science 2020 (NS20) Thursday and Friday, 9am—5pm PT ns20.cs.cornell.edu
  • 30. 30 We also provide a framework for multiway cuts. For a partition P of an edge e, a splitting function ze(P) is s t u Non-negative ze(P) 0 Permutation Invariant ze(P) = ze(P⇡) for any ⇡ 2 Sk Non-split ignoring ze(P) = 0 if |P| = 1.<latexit 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  • 31. 31 We also provide a framework for multiway cuts. [Hadley 95] [Yaros- Imielinski 13] [Mirzakhani-Vondrak 15] [Alpert-Kahng 95; Karypis-Kumar 99; Chekuri-Ene 11] All-or-nothing ze(P) = ( 0 if |P| = 1 1 otherwise Sum of External Degrees ze(P) = ( 0 if |P| = 1 |P| otherwise K 1 Penalty ze(P) = |P| 1 Rainbow Split ze(P) = ( 1 if |e| = |P| 0 otherwise. <latexit 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Theorem [Veldt-Benson-Kleinberg 20]. Minimizing the rainbow split on 3-uniform hypergraphs is NP-hard to approximate within any constant factor.