A polynomial-time partitioning algorithm for weighted cactus graphs - - PowerPoint PPT Presentation

Leonie Selbach 17.3.2020

A polynomial-time partitioning algorithm for weighted cactus graphs

Maike Buchin, Leonie Selbach

Ruhr University Bochum

EuroCG 2020

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Given: Vertex-weighted graph G = (V, E, w), non-negative integers l and u with l ≤ u, positive integer p

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Given: Vertex-weighted graph G = (V, E, w), non-negative integers l and u with l ≤ u, positive integer p Find a vertex partition into p clusters V1, V2, . . . , Vp such that l ≤ w(Vi) ≤ u for all 1 ≤ i ≤ p.

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Given: Vertex-weighted graph G = (V, E, w), non-negative integers l and u with l ≤ u, positive integer p Find a vertex partition into p clusters V1, V2, . . . , Vp such that l ≤ w(Vi) ≤ u for all 1 ≤ i ≤ p.

connected and pairwise disjoint

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Given: Vertex-weighted graph G = (V, E, w), non-negative integers l and u with l ≤ u, positive integer p Find a vertex partition into p clusters V1, V2, . . . , Vp such that l ≤ w(Vi) ≤ u for all 1 ≤ i ≤ p.

connected and pairwise disjoint =

• v∈Vi

w(v)

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Given: Vertex-weighted graph G = (V, E, w), non-negative integers l and u with l ≤ u, positive integer p Find a vertex partition into p clusters V1, V2, . . . , Vp such that l ≤ w(Vi) ≤ u for all 1 ≤ i ≤ p.

connected and pairwise disjoint =

• v∈Vi

w(v)

Motivation Fragmentation of biomedical structures

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Find a (l, u)-partition with exactly p clusters.

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Find a (l, u)-partition with exactly p clusters.

2 5 4 3 7 8 2 1 4 3 2

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Find a (l, u)-partition with exactly p clusters.

2 5 4 3 7 8 2 1 4 3 2

6-(3, 12)-partition

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Find a (l, u)-partition with exactly p clusters. Minimum/maximum-(l, u)-partition problem Find a (l, u)-partition with the minimal resp. maximal number of clusters.

2 5 4 3 7 8 2 1 4 3 2

6-(3, 12)-partition

Leonie Selbach 17.3.2020

Motivation & Problem

p-(l, u)-partition problem Find a (l, u)-partition with exactly p clusters. Minimum/maximum-(l, u)-partition problem Find a (l, u)-partition with the minimal resp. maximal number of clusters.

2 5 4 3 7 8 2 1 4 3 2 2 5 4 3 7 8 2 1 4 3 2

minimum (3,12)-partition 6-(3, 12)-partition

Leonie Selbach 17.3.2020

(Related) Results

Related results

• series-parallel graphs

NP-hard

Leonie Selbach 17.3.2020

(Related) Results

p-(l, u)-partition problem min/max-(l, u)-partition problem

Related results

• series-parallel graphs

NP-hard O(u4p2n) O(u4n)

• partitial k-trees

O(u2(k+1)p2n) O(u2(k+1)n)

Leonie Selbach 17.3.2020

(Related) Results

Related results

• series-parallel graphs

NP-hard O(u4p2n) O(u4n)

• partitial k-trees

O(u2(k+1)p2n) O(u2(k+1)n)

• trees

polynomial O(p4n) O(n5)

p-(l, u)-partition problem min/max-(l, u)-partition problem (Decision)

Leonie Selbach 17.3.2020

(Related) Results

Related results

• series-parallel graphs

NP-hard O(u4p2n) O(u4n)

• partitial k-trees

O(u2(k+1)p2n) O(u2(k+1)n)

• trees

polynomial O(p4n) O(n5)

p-(l, u)-partition problem min/max-(l, u)-partition problem

Our results

• cactus graphs

polynomial O(p4n2) O(n6)

(Decision)

Leonie Selbach 17.3.2020

(Related) Results

Related results

• series-parallel graphs

NP-hard O(u4p2n) O(u4n)

• partitial k-trees

O(u2(k+1)p2n) O(u2(k+1)n)

• trees

polynomial O(p4n) O(n5)

p-(l, u)-partition problem min/max-(l, u)-partition problem

Our results

• cactus graphs

polynomial O(p4n2) O(n6)

(Decision & computation) (Decision & Computation)

Leonie Selbach 17.3.2020

Preprocessing

DFS on some vertex r ∈ G and store cycles

Leonie Selbach 17.3.2020

Preprocessing

DFS on some vertex r ∈ G and store cycles r

Leonie Selbach 17.3.2020

Preprocessing

DFS on some vertex r ∈ G and store cycles r

Leonie Selbach 17.3.2020

Preprocessing

DFS on some vertex r ∈ G and store cycles r

Leonie Selbach 17.3.2020

Preprocessing

DFS on some vertex r ∈ G and store cycles r

Leonie Selbach 17.3.2020

Preprocessing

DFS on some vertex r ∈ G and store cycles r x y z r x y z

Leonie Selbach 17.3.2020

Preprocessing

DFS on some vertex r ∈ G and store cycles r x y z r x y z

start node end node

C(r, z) = r, x, y, z

Leonie Selbach 17.3.2020

Some Definitions

For a partition P of Tv: |P| = number of clusters in P Cv = cluster containing the node v

Leonie Selbach 17.3.2020

Some Definitions

For a partition P of Tv: |P| = number of clusters in P Cv = cluster containing the node v subtree rooted in node v

Leonie Selbach 17.3.2020

Some Definitions

For a partition P of Tv: |P| = number of clusters in P Cv = cluster containing the node v Extendable (l, u)-partition of Tv:

• w(Cv) ≤ u
• l ≤ w(C′) ≤ u for every cluster C′ = Cv

subtree rooted in node v

Leonie Selbach 17.3.2020

Some Definitions

Partition Sets S(Tv) = {(x, k) | ∃ extendable (l, u)-partition P of Tv such that |P| = k ∧ w(Cv) = x}

Leonie Selbach 17.3.2020

Some Definitions

Partition Sets S(Tv) = {(x, k) | ∃ extendable (l, u)-partition P of Tv such that |P| = k ∧ w(Cv) = x} |S(Tv)| = O(up)

Leonie Selbach 17.3.2020

Some Definitions

Partition Sets S(Tv) = {(x, k) | ∃ extendable (l, u)-partition P of Tv such that |P| = k ∧ w(Cv) = x} Lemma Tv has p-(l, u)-partition ⇔ ∃ (x, p) ∈ S(Tv) such that l ≤ x ≤ u

Leonie Selbach 17.3.2020

Some Definitions

Partition Sets S(Tv) = {(x, k) | ∃ extendable (l, u)-partition P of Tv such that |P| = k ∧ w(Cv) = x} Lemma Tv has p-(l, u)-partition ⇔ ∃ (x, p) ∈ S(Tv) such that l ≤ x ≤ u Idea Compute S(Tr) for r being the root of the DFS-Tree

Leonie Selbach 17.3.2020

Some Definitions

Partition Sets S(Tv) = {(x, k) | ∃ extendable (l, u)-partition P of Tv such that |P| = k ∧ w(Cv) = x} Lemma Tv has p-(l, u)-partition ⇔ ∃ (x, p) ∈ S(Tv) such that l ≤ x ≤ u Idea Compute S(Tr) for r being the root of the DFS-Tree include an efficient procedure for the cycles

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi v vi v1 vi−1 v vi v1 vi−1 T i

v

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi v vi v1 vi−1 v vi v1 vi−1 T i−1

v

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi v vi T i−1

v

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi v vi T i−1

v

Tvi

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi Given partitions P ′ of T i−1

v

and P ′′ of Tvi.

C′

v

C

′′

vi

v vi

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi Given partitions P ′ of T i−1

v

and P ′′ of Tvi. as (x1, k1) ∈ S(T i−1

v

) and (x2, k2) ∈ S(Tvi)

C′

v

C

′′

vi

v vi

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi Option 1: merge w(Cv) = w(C′

v) + w(C′′ vi)

|P| = |P ′| + |P ′′| − 1 (x1 + x2, k1 + k2 − 1) ∈ S(T i

v)

if x1 + x2 ≤ u and k1 + k2 − 1 ≤ p

Cv

v vi

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi Option 2: don’t merge w(Cv) = w(C′

v)

|P| = |P ′| + |P ′′| (x1, k1 + k2) ∈ S(T i

v)

if x2 ≥ l and k1 + k2 ≤ p

Cv

v vi

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi Compute set S(T i

v)

by combining sets S(T i−1

v

) and S(Tvi)

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi Compute set S(T i

v)

by combining sets S(T i−1

v

) and S(Tvi)

S(T i

v) =S(T i−1 v

) ⊕ S(Tvi) ={(x1 + x2, k1 + k2 − 1)|x1 + x2 ≤ u, k1 + k2 − 1 ≤ p} ∪ {(x1, k1 + k2)|l ≤ x2, k1 + k2 ≤ p}

⊕-operation O(u2p2)

Leonie Selbach 17.3.2020

Method

Compute partition P of T i

v by

combining partitions of T i−1

v

and Tvi Compute set S(T i

v)

by combining sets S(T i−1

v

) and S(Tvi) ⊕-operation Dynamic approach S(T 0

v ) = {(w(v), 1)}

S(T i

v) = S(T i−1 v

) ⊕ S(Tvi) for all edges (v, vi) bottom-up

Leonie Selbach 17.3.2020

Method

What about cycles? consider different configurations

Leonie Selbach 17.3.2020

Method

What about cycles? consider different configurations

w0 w3 w1 w2 w0 w3 w1 w2 w0 w3 w1 w2 w0 w3 w1 w2

Leonie Selbach 17.3.2020

Method

What about cycles? consider different configurations

w0 w3 w1 w2 w0 w3 w1 w2 w0 w3 w1 w2 w0 w3 w1 w2

Leonie Selbach 17.3.2020

Method

What about cycles? consider different configurations

w0 w3 w1 w2 w0 w3 w1 w2 w0 w3 w1 w2 w0 w3 w1 w2

Leonie Selbach 17.3.2020

Method

What about cycles? consider different configurations

w0 w3 w1 w2 w0 w3 w1 w2 w0 w3 w1 w2 w0 w3 w1 w2

Leonie Selbach 17.3.2020

Method

What about cycles in the graph? consider different configurations

Leonie Selbach 17.3.2020

Method

w1 w0 w2 w3 T c1−1

w1

T c2−1

w2

Tw3 T k−1

w0

What about cycles in the graph? consider different configurations

Leonie Selbach 17.3.2020

Method

w1 w0 w2 w3 w1 w0 w2 w3 T c1−1

w1

T c2−1

w2

Tw3 T k−1

w0

What about cycles in the graph? consider different configurations

Leonie Selbach 17.3.2020

Method

w1 w0 w2 w3 w1 w0 w2 w3 w1 w0 w2 w3 T c1−1

w1

T c2−1

w2

Tw3 T k−1

w0

What about cycles in the graph? consider different configurations

Leonie Selbach 17.3.2020

Method

Computation of the partition sets

Sj(Twm−1) =

• S(Twm−1)

j = 1, 2 S(Twm−1) ⊕ Sj(Twm−2)

• therwise

Sj(Twi) =      S(T ci−1

wi

) ⊕ Sj(Twi+1) j < m − i S(T ci−1

wi

) ⊕ Sj(Twi−1) j > m − i + 1 S(T ci−1

wi

)

• therwise

Sj(Tw0) =

• S(T k−1

w0 ) ⊕ Sj(Tw1)

j = 1

• S(T k−1

w0 ) ⊕ Sj(Tw1)

• ⊕ Sj(Twm−1)
• therwise

for nodes in some cycle C(w0, wm−1) = w0, w1, . . . , wm−1 in different configurations j

Leonie Selbach 17.3.2020

Method

Computation of the partition sets

Sj(Twm−1) =

• S(Twm−1)

j = 1, 2 S(Twm−1) ⊕ Sj(Twm−2)

• therwise

Sj(Twi) =      S(T ci−1

wi

) ⊕ Sj(Twi+1) j < m − i S(T ci−1

wi

) ⊕ Sj(Twi−1) j > m − i + 1 S(T ci−1

wi

)

• therwise

Sj(Tw0) =

• S(T k−1

w0 ) ⊕ Sj(Tw1)

j = 1

• S(T k−1

w0 ) ⊕ Sj(Tw1)

• ⊕ Sj(Twm−1)
• therwise

S(T k

w0) = m−1 j=1 Sj(T k w0)

for nodes in some cycle C(w0, wm−1) = w0, w1, . . . , wm−1 in different configurations j

Leonie Selbach 17.3.2020

Results

Theorem Given a weighted cactus graph G, a positive integer p and two non-negative integers l and u (with l ≤ u). The p-(l, u)-partition problem can be decided in time O(u2p2n2).

Leonie Selbach 17.3.2020

Results

Theorem Given a weighted cactus graph G, a positive integer p and two non-negative integers l and u (with l ≤ u). The p-(l, u)-partition problem can be decided in time O(u2p2n2). reduce size of partition sets from O(up) to O(p2) O(p4n2)

Leonie Selbach 17.3.2020

Results

Theorem Given a weighted cactus graph G, a positive integer p and two non-negative integers l and u (with l ≤ u). The p-(l, u)-partition problem can be decided in time O(u2p2n2). reduce size of partition sets from O(up) to O(p2) O(p4n2)

before: x1 x2 x3 xm m ≤ u for each k (≤ p)

• verall O(up)

Leonie Selbach 17.3.2020

Results

Theorem Given a weighted cactus graph G, a positive integer p and two non-negative integers l and u (with l ≤ u). The p-(l, u)-partition problem can be decided in time O(u2p2n2). reduce size of partition sets from O(up) to O(p2) O(p4n2)

before: x1 x2 x3 xm m ≤ u for each k (≤ p)

• verall O(up)

before: x1 x2 x3 xm d = u − l > d ≤ d

Leonie Selbach 17.3.2020

Results

Theorem Given a weighted cactus graph G, a positive integer p and two non-negative integers l and u (with l ≤ u). The p-(l, u)-partition problem can be decided in time O(u2p2n2). reduce size of partition sets from O(up) to O(p2) O(p4n2)

before: x1 x2 x3 xm m ≤ u for each k (≤ p)

• verall O(up)

before: x1 x2 x3 xm d = u − l > d ≤ d now: [a1, a2] [a2, a3] [al, al+1] l ≤ k

• verall O(p2)

for each k (≤ p)

Leonie Selbach 17.3.2020

Results

by storing additional information and using backtracking Theorem Given a weighted cactus graph G, a positive integer p and two non-negative integers l and u (with l ≤ u). The p-(l, u)-partition problem can be decided in time O(p4n2)and space O(p5n2). solved

Leonie Selbach 17.3.2020

Results

Theorem Given a weighted cactus graph G and two non-negative integers l and u (with l ≤ u). The minimum and maximum (l, u)-partition problem can solved in time O(n6) and space O(n7).

Leonie Selbach 17.3.2020

Results

Open problems

• NP-hard
• Polynomial-time algorithms

for other graph classes?