1. Radu Curticapean, Holger Dell, Dániel Marx
NEW INSIGHTS INTO
COUNTING SUBGRAPHS
Radu Curticapean, Holger Dell, Dániel Marx
Institute for Computer Science and Control,
Hungarian Academy of Sciences (MTA SZTAKI)
Saarland University,
Cluster of Excellence (MMCI)

2. NOT NECESSARILY INDUCED
𝑘 vertices
pattern 𝐻
𝑛 vertices
host 𝐺
count subgraphs F of G
isomorphic to H
NOT NECESSARILY INDUCED
COUNTING SUBGRAPHS

3. pattern 𝐻 host 𝐺 𝑛 vertices 𝑘 vertices
count subgraphs F of G
isomorphic to H
parameterized problem X: input comes with some parameter 𝑘∈ℕ
X is FPT: problem can be solved in time 𝑓 𝑘 ⋅ 𝑛 𝑂(1)
𝑋 is #W[1]-hard: ∃ FPT-algorithm for #𝑘-clique with 𝑋-oracle and all queries have parameter ≤𝑔(𝑘) for some 𝑔
⇒ X most likely not FPT
any 𝐻
𝑓 𝑘 ⋅𝑛 𝑘
𝑘-clique
𝑘-cycle
𝑘-matching
𝑛 𝜔𝑘/3+𝑐 [NP 85]
𝑛 𝑘/2+𝑐 [BHKK 09]
𝑛 0.9𝑘+𝑐 [BKK 14]
#W[1]-hard
𝑛 𝑜(𝑘) refutes ETH [CCFHJKX 04]
𝑛 0.45𝑘+𝑐 [BKK 14]
#W[1]-hard,
𝑛 𝑜(𝑘/ log 𝑘 ) refutes ETH
#W[1]-hard [FG 04]
[C 13, CM 14, CM 15]
ETH: SAT on 𝑛-variable formulas has no 2 𝑜(𝑛) time algorithm
𝐻 with vtx-cover of size ≤𝑐
𝑛 𝑐+𝑂(1) [VW 09]
𝑘-star
𝒪(𝑛+𝑚)

4. poly-time #W[1]-hard. pattern 𝐻 host 𝐺 𝑛 vertices 𝑘 vertices 𝑘-star
count subgraphs F of G
isomorphic to H
poly-time
#W[1]-hard.
𝑘-star
𝑘-clique
𝑘-matching
𝐻 with vtx-cover of size ≤𝑐
𝑘-cycle

5. classifies when #𝑆𝑢𝑏(𝒞) is poly-time assuming 𝐹𝑃𝑇≠#𝑊 1
for recursively enumerable 𝒞
#𝑺𝒖𝒃(𝓒)
Input: Graphs 𝐻∈𝒞 and 𝐺
Parameter: |𝑉(𝐻)|
Output: #𝑆𝑢𝑏(𝐻→𝐺)
maximum matching in 𝒞
bounded
unbounded
poly-time
#W[1]-hard.
[CM 14]
𝑘-star
classifies when #𝑆𝑢𝑏(𝒞) is poly-time assuming 𝐹𝑃𝑇≠#𝑊 1
despite #𝑃-intermediate cases! [CTW 08]
𝑘-clique
𝑘-matching
𝐻 with vtx-cover of size ≤𝑐
𝑘-cycle

6. poly-time #𝑾[𝟏]-hard 𝐻 𝑇 𝒞 finite count/find induced subgraphs from 𝒞
else
[CTW 08]
count subgraphs from 𝒞
𝒞 has bounded matchings
else
[CM 14]
count homomorphisms from 𝒞
𝒞 has bounded treewidth
else
[G 07]
[DJ 04]
count/find colorful subgraphs from 𝒞
𝒞 has bounded treewidth
else
[GSS 01]
[CM 14]
tree-decomposition of H: tree 𝑇 with bags 𝐵 𝑡 ⊆𝑉(𝐻) at 𝑡∈𝑉 𝑇
Each 𝑣∈𝑉 𝐻 is in some bag
Each 𝑒∈𝐸(𝐻) is in some bag
If 𝑣 appears in bags of 𝑋 𝑣 ={𝑡 1 ,…, 𝑡 𝑠 }, then 𝑋 𝑣 is connected in 𝑇
width of 𝑇: max. bag size treewidth of 𝑯: min. width over all 𝑻
David Eppstein, Wikipedia 𝐻 𝑇

7. 𝒞 has bounded treewidth
poly-time
#𝑾[𝟏]-hard
count homomorphisms from 𝒞
𝒞 has bounded treewidth
else
[G 07]
[DJ 04]
𝑓 𝐻 ⋅ 𝑛 𝑡𝑤 𝐻 +1 time via standard DP 𝐻
𝑓:𝑉 𝐻 →𝑉(𝐺)
COUNTING HOMOMORPHISMS 𝐺

8. COUNTING SUBGRAPH EMBEDDINGS
𝑓:𝑉 𝐻 →𝑉(𝐺) 𝐻 𝐺
COUNTING SUBGRAPH EMBEDDINGS
#𝐸𝑚𝑏(𝐻→𝐺)=#𝐴𝑢𝑡 𝐻 ⋅#𝑆𝑢𝑏(𝐻→𝐺)
COUNTING HOMOMORPHISMS 𝐻 𝐺
𝑓:𝑉 𝐻 →𝑉(𝐺)

9. COUNTING INJECTIVE HOMOMORPHISMS
𝑓:𝑉 𝐻 →𝑉(𝐺) 𝐻 𝐺
COUNTING INJECTIVE HOMOMORPHISMS
COUNTING HOMOMORPHISMS 𝐻 𝐺
𝑓:𝑉 𝐻 →𝑉(𝐺)

10. ≡ COUNTING INJECTIVE HOMOMORPHISMS COUNTING HOMOMORPHISMS 𝐻 𝐺
𝑓:𝑉 𝐻 →𝑉(𝐺) 𝐻 𝐺
COUNTING LINEAR COMBINATIONS OF
COUNTING INJECTIVE HOMOMORPHISMS ≡
COUNTING HOMOMORPHISMS
COUNTING LINEAR COMBINATIONS OF 𝐻 𝐺
𝑓:𝑉 𝐻 →𝑉(𝐺)

11. Spasm 𝐻 ≃ COUNTING INJECTIVE HOMOMORPHISMS
≡ LINEAR COMBINATIONS OF HOMOMORPHISMS
Define 𝑯 𝝆 for partition 𝝆 of 𝑽(𝑯) :
Contract every block of 𝜌 in 𝐻.
Delete multi-edges, keep self-loops.
Spasm 𝐻
≔ 𝐻 𝜌 𝜌 partitions 𝑉 𝐻
1|2|3|4
13|2|4
14|2|3
23|1|4
24|1|3
12|34
13|24
14|23
23|14
24|13
34|12
123|4
142|3
234|1 12 3 4
12|3|4 𝐻 1 2 3 4 1 2 3 4
1234 1 2 34 2
341
34|1|2
341|2

12. ≃ COUNTING INJECTIVE HOMOMORPHISMS
≡ LINEAR COMBINATIONS OF HOMOMORPHISMS
#𝐸𝑚𝑏(𝐻→𝐺)= 𝐹∈Spasm(𝐻) 𝛼 𝐹 ⋅#𝐻𝑜𝑚(𝐹→𝐺)
𝛼 𝐹 = −1 |𝑉 𝐻 |−|𝑉(𝐹)| ⋅ 𝑣∈𝑉(𝐹) 𝜆 𝑣 −1 !
12|34 12 3 4
123|4
12|3|4
13|24 𝐻 2
13|2|4
142|3 1 2 3 4 4
14|23
14|2|3 1 3
1234
23|1|4
23|14
234|1
1|2|3|4 1 2 34
24|1|3 2
341
24|13
34|1|2
341|2
34|12

13. Counting any 𝑘-edge subgraph in time 𝑓 𝑘 ⋅𝑛 0.174𝑘+𝑜(𝑘)
Lovász 1967
#𝐸𝑚𝑏(𝐻→𝐺)= 𝐹∈Spasm(𝐻) 𝛼 𝐹 ⋅#𝐻𝑜𝑚(𝐹→𝐺)
𝛼 𝐹 ≠0
𝑘-paths (before: time 𝑛 0.45𝑘 )
𝑘-matchings (before: time 𝑛 0.9𝑘 )
First application:
Counting any 𝑘-edge subgraph in time 𝑓 𝑘 ⋅𝑛 0.174𝑘+𝑜(𝑘)
best known upper bound on
treewidth of 𝑘-edge graph

14. 𝛼 𝐹 ≠0 #𝐸𝑚𝑏(𝐻→𝐺)= 𝐹∈Spasm(𝐻) 𝛼 𝐹 ⋅#𝐻𝑜𝑚(𝐹→𝐺) poly time
𝑔(𝐻) graphs,
each with ≤𝑘 edges
Scott, Sorkin 2016: Graphs 𝐵 with 𝑘 edges have tw(𝐵)≤ 𝑜 1 𝑘

15. Hardness of counting subgraphs via fine-grained reductions
#𝐸𝑚𝑏(𝐻→𝐺)= 𝐹∈Spasm(𝐻) 𝛼 𝐹 ⋅#𝐻𝑜𝑚(𝐹→𝐺)
𝛼 𝐹 ≠0
Second application:
Hardness of counting subgraphs
via fine-grained reductions

16. 𝛼 𝐹 ≠0 Pick any 𝐵∈Spasm(𝐻). 𝐵 𝐺 #𝐸𝑚𝑏(𝐻→ ⋆)= 𝐹∈Spasm(𝐻) 𝛼 𝐹 ⋅#𝐻𝑜𝑚(𝐹→ ⋆)
Given oracle for this, we can compute this in 𝑓(𝐻) steps.
Pick any 𝐵∈Spasm(𝐻). 𝐵 𝐺
#PartitionedSub(𝐵→ ⋆)

17. 𝐻 has 𝑘-matching ⇒ pick expander 𝐵 with tw B =Ω(𝑘)
𝑛 Ω(𝑣𝑐(𝐻)/ log 𝑣𝑐(𝐻)) under #ETH
even when fixing ℋ and allowing only 𝐻∈ℋ
#𝐸𝑚𝑏(𝐻→ ⋆)= 𝐹∈Spasm(𝐻) 𝛼 𝐹 ⋅#𝐻𝑜𝑚(𝐹→ ⋆)
𝛼 𝐹 ≠0
Pick any 𝐵∈Spasm(𝐻).
Given oracle for this, we can compute this in 𝑓(𝐻) steps.
𝐻 has 𝑘-matching ⇒ pick expander 𝐵 with tw B =Ω(𝑘) 𝐵 𝐺
#PartitionedSub(𝐵→ ⋆)
𝑛 Ω(𝑡𝑤(𝐵)/ log 𝑡𝑤(𝐵)) under #ETH [Marx 2010] even when fixing ℬ=Spasm ℋ and allowing only 𝐵∈ℬ

18. #𝐒𝐮𝐛(𝑯→ ⋆) always in time 𝒇 𝑯 ⋅𝒏 𝒄 𝑯 .
Exact value of 𝑐 𝐻 for fixed 𝐻?
max 𝐹∈𝑆𝑝𝑎𝑠𝑚(𝐻) 𝑐 𝐹 𝑝𝑎𝑟𝑡 ≤
𝑐 𝐻 ≤ max 𝐹∈𝑆𝑝𝑎𝑠𝑚 𝐻 𝑐 𝐹 ℎ𝑜𝑚
exponent in running time for #PartitionedSub(𝐹→ ⋆)
exponent in running time for computing #Hom(𝐹→ ⋆)
Can you beat treewidth: 𝑛 Ω(𝑡𝑤(𝐹)/ log 𝑡𝑤(𝐹))
Clique conjecture: 𝑛 𝜔𝑘/3 if 𝐹= 𝐾 𝑘
#𝐸𝑚𝑏(𝐻→𝐺)= 𝐹∈Spasm(𝐻) 𝛼 𝐹 ⋅#𝐻𝑜𝑚(𝐹→𝐺)
𝛼 𝐹 ≠0
Lovász 1967