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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
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions
Hypergraph Cuts with General Splitting Functions

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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