1. Weighted counting of k-matchings is #W[1]-hard
Markus Bläser, Radu Curticapean
Saarland University, Computational Complexity Group

2. counting (perfect) matchings
since 1976 and counting.
biadjacency matrix 𝐴 of bipartite 𝐺
𝑝𝑒𝑟𝑚 𝐴 =#𝑃𝑒𝑟𝑓𝑀𝑎𝑡𝑐ℎ(𝐺)
𝒅𝒆𝒕 𝑨 = 𝜎∈ 𝑆 𝑛 𝑠𝑔𝑛 𝜎 1≤𝑖≤𝑛 𝑎 𝑖,𝜎 𝑖
𝒑𝒆𝒓𝒎 𝑨 = 𝜎∈ 𝑆 𝑛 1≤𝑖≤𝑛 𝑎 𝑖,𝜎 𝑖
considered intractable
poly-time computable
1967 #Planar-PerfMatch ≤ 𝑃 𝑑𝑒𝑡 (Fisher, Kasteleyn)
1976 definition of #𝑷, hardness of 𝑝𝑒𝑟𝑚 (Valiant) 1989 FPRAS for permanent (Jerrum, Sinclair) 2004 parameterized counting complexity (Flum, Grohe) 2007 #Match on planar G of Δ=3 is #P-hard. (Xia et al.)

3. parameterized counting
parameterized problems
input (𝑥,𝑘) with parameter 𝑘
typically solvable in time 𝒏 𝑶(𝒌) or time 𝒇(𝒌)𝒏 𝑶(𝟏)
count 𝑘-vertex covers: 𝑂(𝑛 𝑘 )→ 𝑂(2 𝑘 𝑛) →…
count 𝑘-cliques: 𝑂 𝑛 𝑘 →𝑂( 𝑛 𝜔 3 𝑘 ) →𝑭𝑷𝑻?
define class #W[1] as closure of 𝑘−#𝐶𝑙𝑖𝑞𝑢𝑒𝑠 under…
fpt-turing reduction 𝑨 ≤ 𝒇𝒑𝒕 𝑻 𝑩:
solves 𝐴(𝑥,𝑘) in time 𝑓 𝑘 𝑛 𝑂(1) with oracle for 𝐵
queries B(𝑥′,𝑘′) have 𝑘 ′ ≤𝑔(𝑘) 𝐗𝐏
𝐅𝐏𝐓

4. „simple“ substructure
known results
„simple“ graph
„simple“ substructure
in: graph 𝐺, 𝒌∈ℕ
par: 𝒌
out: #VertexCover s 𝒌 𝐺
for MSOL formula 𝜑(𝑆)
in: graph 𝐺
par: treewidth 𝑮 ,
out: #sets 𝑆 with (𝐺,𝑆)⊨𝜑
𝑘−Clique
𝑘−#Clique
𝑘−Cycle
𝑘−#Cycle
𝑘−Path
𝑘−#Path
in: graph 𝐺
par: genus(𝑮)
out: #𝑃𝑀(𝐺)
𝑘−Match
𝑘−#Match

5. our result status of 𝑘−#𝑀𝑎𝑡𝑐ℎ? 𝑘−#𝒘𝑀𝑎𝑡𝑐ℎ is #𝑊 1 -hard.
„hardness of permanent“ in fpt-world
also: 𝑘−𝑀𝑎𝑡𝑐ℎ∈𝑃
in: edge-weighted bipartite G, 𝑘∈ℕ
out: 𝑀∈𝑀𝑎𝑡𝑐 ℎ 𝑘 [𝐺] 𝑒∈𝑀 𝑤(𝑒)
𝑘−#𝒘𝑀𝑎𝑡𝑐ℎ is #𝑊 1 -hard.
proof by series of reductions

6. partial path-cycle covers
k-partial cycle cover
{ C 1 ,…, C t } of cycles
vertex-disjoint
k edges in total
k-partial path-cycle cover
{ C 1 ,…, C t } of paths and cycles …
𝒞 𝑘 [𝐺]
𝒫𝒞 𝑘 [𝐺]

7. 𝟏 𝟐 reduction chain 𝑘−#𝑤𝑀𝑎𝑡𝑐ℎ 𝑘−#𝑤𝑃𝐶𝐶 𝑘−#𝐶𝐶
in: weighted bipartite G, 𝑘∈ℕ
out: 𝑀∈ ℳ 𝑘 [𝐺] 𝑒∈𝑀 𝑤(𝑒)
𝑘−#𝑤𝑃𝐶𝐶
in: weighted digraph G, 𝑘∈ℕ
out: 𝐶∈ 𝒫𝒞 𝑘 [𝐺] 𝑒∈𝐶 𝑤(𝑒)
𝑘−#𝐶𝐶
in: digraph G, 𝑘∈ℕ
out: #𝑘−partial cycle covers of 𝐺

8. matchings ≥ 𝒇𝒑𝒕 𝑻 path-cycle covers
standard reduction G
S(G)
split in
out
(𝑢,𝑣)
{ 𝑢 𝑜𝑢𝑡 , 𝑣 𝑖𝑛 }
𝓟 𝓒 𝒌 𝑮 ≅ 𝓜 𝒌 𝑺 𝑮 implies 𝐶∈𝒫𝒞 𝐺 𝑒∈𝐶 𝑤(𝑒) = 𝑀∈ℳ 𝐺 𝑒∈𝑀 𝑤(𝑒)

9. 𝟏 𝟐 reduction chain 𝑘−#𝑤𝑀𝑎𝑡𝑐ℎ 𝑘−#𝑤𝑃𝐶𝐶 𝑘−#𝐶𝐶
in: weighted bipartite G, 𝑘∈ℕ
out: 𝑀∈ ℳ 𝑘 [𝐺] 𝑒∈𝑀 𝑤(𝑒)
𝑘−#𝑤𝑃𝐶𝐶
in: weighted digraph G, 𝑘∈ℕ
out: 𝐶∈ 𝒫𝒞 𝑘 [𝐺] 𝑒∈𝐶 𝑤(𝑒)
𝑘−#𝐶𝐶
in: digraph G, 𝑘∈ℕ
out: #𝑘−partial cycle covers of 𝐺

10. path-cycle covers ≥ 𝒇𝒑𝒕 𝑻 cycle covers
analysis via path-cycle polynomial 𝛾 𝐺;𝑥 ≔ 𝐶∈𝒫𝒞 𝐺 𝑥 𝐶 𝑤 𝐶
transform 𝐺↦ 𝐺 𝑏 by gadgets b -b b -b b -b b -b -b b
recipe for path-cycle covers in 𝐺 𝑏
path-cycle cover 𝑪 in 𝐺 as core
at path ends: add nothing or
at isolated vts: add nothing or or b -b b -b
𝛾 𝐺 𝑏 ;𝑥 = 𝐶∈𝒫𝒞 𝐺 𝑥 𝐶 (1+𝑏𝑥) #𝑝𝑎𝑡ℎ 𝐶 (1+𝑏𝑥−𝑏𝑥) #𝑖𝑠𝑜𝑙 𝐶
interpolation along 𝑏 allows to track out paths

11. 𝟏 𝟐 reduction chain 𝑘−#𝐶𝐶 𝑘−#𝑡𝑦𝑝𝑈𝐶𝑊 𝑘−#𝐶𝑙𝑖𝑞𝑢𝑒𝑠 in: digraph G, 𝑘∈ℕ
out: #𝑘−partial cycle covers of 𝐺
𝑘−#𝑡𝑦𝑝𝑈𝐶𝑊
in: digraph G, type 𝜃, 𝑘∈ℕ
out: #k−UCWs of G with type 𝜃
𝑘−#𝐶𝑙𝑖𝑞𝑢𝑒𝑠
in: graph G, 𝑘∈ℕ
out: #𝑘−cliques of 𝐺

12. cycle-like structures
for tuple 𝑊, let 𝒮 𝑊 ≔{cyclic shifts of W} CW
𝑊=𝒮(𝑇)
cycle
𝐶=𝒮(𝑇)
no repeating vts.
UCW
𝑊 1 ,…, 𝑊 𝑡
closed walk
𝑣 1 , …, 𝑣 𝑘−1 , 𝑣 1

13. types of UCWs 𝑓 𝑈 𝑣 ≔ #visits of 𝑈 at 𝑣
similar to types of cyclic walks defined by Flum, Grohe
𝑓 𝑈 𝑣 ≔ #visits of 𝑈 at 𝑣 2 3 1
type 𝜃 𝑈 = 𝑓 𝑈 𝑣 1 ,…, 𝑓 𝑈 ( 𝑣 𝑛 ) 1
size 𝑘=8
𝜃= 0,1,1,1,2,3 (omitting 0s is fine) 1
size 𝑘=4
𝜃= 0,0,1,1,1,1
size 𝑘=9000
𝜃= 9000

14. typed UCWs ≥ 𝒇𝒑𝒕 𝑻 cliques
proof adapted from Flum, Grohe
In G, replace all edges by , and add
Want to count induced subgraphs isomorphic to 𝐾≔ 𝐾 𝑘 .
Consider set 𝒰 of UCWs of type 𝜃 𝑘,𝑙 := 𝑙,…,𝑙 in G. k
Partition 𝒰 according to visited vertices 𝑆∈ 𝑉 𝑘 .
Note: If 𝐺 𝑆 ≃𝐺 𝑆 ′ , then „same“ UCWs in 𝑆 and 𝑆′.
Partition 𝒰 according to isomorphism type of 𝐺 𝑆 , which is some graph 𝐻 on 𝑘 vertices.

15. typed UCWs ≥ 𝒇𝒑𝒕 𝑻 cliques
graph H of order k in host graph G
𝛽 (𝑙) ≔#UCWs of type 𝜃 𝑘,𝑙 in 𝐺
𝛽 𝐻 (𝑙) ≔#UCWs of type 𝜃 𝑘,𝑙 in 𝐻
𝑥 𝐻 ⋅ 𝛽 𝐻 (𝑙) G
𝑥 𝐻 ≔#copies of 𝐻 in 𝐺 H
𝛽 (𝑙) = 𝐻∈ 𝐺𝑟𝑎𝑝ℎ𝑠 𝑘 𝑥 𝐻 ⋅ 𝛽 𝐻 (𝑙)

16. typed UCWs ≥ 𝒇𝒑𝒕 𝑻 cliques
𝛽 (𝑙) ≔#UCWs of type 𝜃 𝑘,𝑙 in 𝐺 oracle call
𝛽 𝐻 (𝑙) ≔#UCWs of type 𝜃 𝑘,𝑙 in 𝐻 oracle call
𝑥 𝐻 ≔#isomorphic copies of 𝐻 in 𝐺 𝒙 𝑲 wanted
𝛽 𝐻 1 (1) ⋯ 𝛽 𝐾 (1) ⋮ ⋱ ⋮ 𝛽 𝐻 1 (𝑙) ⋯ 𝛽 𝐾 (𝑙) 𝑥 𝐻 1 ⋮ 𝒙 𝑲 = 𝛽 ⋮ 𝛽 𝑙
𝛽 (𝑙) = 𝐻∈ 𝐺𝑟𝑎𝑝ℎ𝑠 𝑘 𝑥 𝐻 ⋅ 𝛽 𝐻 (𝑙)
prove that this column is lin. indep. for some 𝑙=𝑓(𝑘)

17. reduction chain in: graph G, 𝑘∈ℕ in: weighted bipartite G, 𝑘∈ℕ
out: #𝑘−cliques of 𝐺
in: weighted bipartite G, 𝑘∈ℕ
out: 𝑀∈ ℳ 𝑘 [𝐺] 𝑒∈𝑀 𝑤(𝑒)
in: digraph G, type 𝜃, 𝑘∈ℕ
out: #k−UCWs of G with type 𝜃
in: weighted digraph G, 𝑘∈ℕ
out: 𝐶∈ 𝒫𝒞 𝑘 [𝐺] 𝑒∈𝐶 𝑤(𝑒)
in: digraph G, 𝑘∈ℕ
out: #𝑘−partial cycle covers of 𝐺

18. future work in: graph G, 𝑘∈ℕ in: weighted bipartite G, 𝑘∈ℕ
out: #𝑘−cliques of 𝐺
in: weighted bipartite G, 𝑘∈ℕ
out: 𝑀∈ ℳ 𝑘 [𝐺] 𝑒∈𝑀 𝑤(𝑒)
in: digraph G, type 𝜃, 𝑘∈ℕ
out: #k−UCWs of G with type 𝜃
in: weighted digraph G, 𝑘∈ℕ
out: 𝐶∈ 𝒫𝒞 𝑘 [𝐺] 𝑒∈𝐶 𝑤(𝑒)
in: digraph G, 𝑘∈ℕ
out: #𝑘−partial cycle covers of 𝐺