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