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)

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

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)

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

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

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

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

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)

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

≡ 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)

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

≃ 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

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

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)≤ 13 75 +𝑜 1 𝑘

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

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→ ⋆)

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

#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