Weighted counting of k-matchings is #W[1]-hard

Markus Bläser, Radu CurticapeanSaarland University, Computational Complexity Group

counting ((perfect) matchings)since 1976 and counting.

𝒅𝒆𝒕 ( 𝑨 )= ∑𝜎 ∈𝑆𝑛

𝑠𝑔𝑛 (𝜎 ) ∏1≤𝑖≤𝑛

𝑎𝑖 ,𝜎 ( 𝑖 )

poly-time computable

𝒑𝒆𝒓𝒎 ( 𝑨 )= ∑𝜎 ∈𝑆𝑛

∏1≤ 𝑖≤𝑛

𝑎𝑖 ,𝜎 ( 𝑖 )

considered intractable

1967 #Planar-PerfMatch (Fisher, Kasteleyn)1976 definition of , hardness of (Valiant)1989 FPRAS for (Jerrum, Sinclair)2004 parameterized counting complexity (Flum, Grohe)2007 on planar G of is #P-hard. (Xia et al.)

for biadjacency matrix of bipartite

parameterized counting

parameterized counting problems on input , decide count something typically solvable in time or time

count -vertex covers count -cliques: ) count -matchings: ) #W[1] can be defined as closure of under fpt-turing reduction

solves in fpt-time with oracle for queries have

known results

„simple“ graph „simple“ substructurein: graph par: treewidthout: sets with

in: graph par: genusout:

in: graph , par: out: 𝑘−Clique𝑘− ¿Cliqu e

𝑘−Path𝑘− ¿Path 𝑘−Cycle

𝑘− ¿Cycle

𝑘−Match𝑘− ¿Match

for any MSOL formula

our result

status of ?• „hardness of permanent“ in fpt-world• decision version

in: edge-weighted bipartite G, out: is -hard.

proof by series of reductions

partial path-cycle covers

k-partial cycle cover • of cycles• vertex-disjoint• edges in total𝒞𝑘[𝐺] 𝒫𝒞𝑘[𝐺 ]

k-partial path-cycle cover • of paths and cycles•…•…

reduction chain

in: weighted bipartite G, out: in: weighted digraph G, out:

𝑘− ¿𝑤 h𝑀𝑎𝑡𝑐

𝑘− ¿𝑤𝑃𝐶𝐶

𝑘− ¿𝐶𝐶in: digraph G, out:

matchings path-cycle covers

implies

inoutsplitG S(G)(𝑢 ,𝑣 ) {𝑢𝑜𝑢𝑡 ,𝑣 𝑖𝑛}

standard reduction

reduction chain

in: weighted bipartite G, out: in: weighted digraph G, out:

𝑘− ¿𝑤 h𝑀𝑎𝑡𝑐

𝑘− ¿𝑤𝑃𝐶𝐶

𝑘− ¿𝐶𝐶in: digraph G, out:

analysis via path-cycle polynomial transform by gadgets

path-cycle covers cycle covers

interpolation along allows to track out paths

b-b b-b

b -bb

-b

b-b b

-b

-bbb

recipe for path-cycle covers in 1. path-cycle cover in as core2. at path ends: add nothing or 3. at isolated vts: add nothing or or

reduction chain

𝑘− ¿𝐶𝐶

𝑘− ¿𝑡𝑦𝑝𝑈𝐶𝑊

𝑘− ¿𝐶𝑙𝑖𝑞𝑢𝑒𝑠

in: digraph G, out:

in: graph G, out:

in: digraph G, type , out:

cycle-like structuresfor tuple , let cyclic shifts of

cycle no repeating vts. closed walk CW

UCW

types of UCWssimilar to types of cyclic walks defined by Flum, Grohe

type #visits of at size (omitting 0s is fine)

11

1

23

0

size

size

reduction from cliquesproof adapted from Flum, Grohe In G, replace all edges by , and add . Want to count induced subgraphs isomorphic to . Consider set of UCWs of type in G.

k Partition according to visited vertices

is some graph on vertices. If , then „same“ UCWs in and .

Partition according to (isomorphism type of)

within partition class of H

GH

𝛽(𝑙)= ∑𝐻∈ h𝐺𝑟𝑎𝑝 𝑠𝑘

𝑥𝐻 ⋅ 𝛽𝐻(𝑙)

UCWs of type in

UCWs of type in induced in

𝑥𝐻 ⋅ 𝛽𝐻(𝑙)

k vertices

𝛽(𝑙)= ∑𝐻∈ h𝐺𝑟𝑎𝑝 𝑠𝑘

𝑥𝐻 ⋅ 𝛽𝐻(𝑙)

typed UCWs cliques

UCWs of type in oracle callUCWs of type i oracle callisomorphic copies of in wanted

(𝛽𝐻1

(1) ⋯ 𝛽𝐾(1)

⋮ ⋱ ⋮𝛽𝐻1

(𝑙) ⋯ 𝛽𝐾(𝑙) )(𝑥𝐻1

⋮𝒙𝑲

)=(𝛽❑(1)

⋮𝛽❑

( 𝑙 ))prove that this column is lin. indep. for some

proof adapted from Flum, Grohe

reduction chain

in: weighted bipartite G, out: in: weighted digraph G, out:

in: digraph G, out:

in: graph G, out: in: digraph G, type , out:

future work

in: weighted bipartite G, out: in: weighted digraph G, out:

in: digraph G, out:

in: graph G, out: in: digraph G, type , out: