Tropical Connected Sets problem Eugene Vagin 07.04.2020 Eugene - - PowerPoint PPT Presentation

Tropical Connected Sets problem

Eugene Vagin 07.04.2020

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Outline

1

Exact exponential algorithms to find tropical connected sets of minimum size

2

Enumerating Minimal Tropical Connected Sets

3

Algorithm which enumerates all MTCS on chordal graphs

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Paper: Exact exponential algorithms to find tropical connected sets of minimum size

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Preliminaries

Let assume that G = (V , E) - an undirected graph G[X] - the subgraph of G induced by X, where X ⊆ V S is connected if the subgraph G[S] is connected (G, c) - vertex-colored graph

c : V → N: coloring of G (not necessarily proper)

C = {c(v) : v ∈ V } c(S) = {c(v) : v ∈ S} - set of colors of S, where S ⊆ V

S is tropical if c(S) = C

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Minimal Tropical Connected Set problem (MTCS)

Input

Graph G = (V , E) with a coloring c : V → N and set of colors C

Question

Find a minimum size subset S ⊆ V such that G[S] is connected, and S contains at least one vertex of each color in C

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Extra preliminaries

N(v) set of all neighbors of v N[v] N(v) ∪ {v} N[X] ∪x∈XN[x] N(X) N[X] \ X L1(G) vertices whose colors appears only once in (G, c) l1(G) |L1(G)| l2(G) number of colors appearing at least twice in (G, c)

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An exact exponential algorithm for general graphs

Naive brute force: O∗(2n) time In paper [1] described algorithm which works in O∗(1.5359n) via

reductions to

Connected red-blue dominating set Steiner tree

balancing technique

depending on the value of l1(G)

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Steiner tree problem

Input

Graph G = (V , E), weight function w : E → N set of terminals K ⊆ V

Question

Find a connected subtree T = (V ′, E ′) of G with V ′ ⊆ V and E ′ ⊆ E,

such that K ⊆ V ′ and

e∈E ′ w(e) is minimum

Best known time: O∗(W · 2|K|)

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Connected red-blue dominating set

Input

Graph G = (R, B, E) where vertices are colored either red (vertices in R)

• r blue (vertices in B).

Question

Find the smallest subset S ⊆ R of red vertices such that G[S] is connected, and every vertex in B has at least one neighbor in S, that is B ⊆ N(S) Best known time: O∗(1.36443n)

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Reduction example

Figure: An example of reduction: from a vertex-colored graph (G, c) with color set C = { , •, ◦, ×} , to the intermediate graph G’ (middle) with vertex set R′ ∪ B′ , and to the final graph G” (right) with vertex set R′′ ∪ B′′ . Highlighted edges correspond to edges newly added at the corresponding step of the construction.

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Reduction to Steiner Tree

G = (V , E) - graph, c - coloring of G, C = c(G) G ′ (R′ ∪ B′, E ′) - new graph R′ {v′|v ∈ V } B′ {ri|i ∈ C} E ′ {u′v′|uv ∈ E} ∪ {v′ri| v is of color i in G }

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Reduction to Connected Red-Blue Dominating Set

R′′ = R′, B′′ = B′, E ′′ = E ′ ∀v ∈ V : c(v) appears exactly once in G

move the corresponding v ′ from R′′ to B′′ remove the vertex ri from B′′, where c(v) = iinG

B1, . . . , Bp - components of the subgraph induced in G ′′[B′′] by those vertices that had been moved to B′′ ∀i = 1, 2, . . . , p contract the component Bi in G ′′[B′′] so that it remains only one vertex and call this vertex bi ∀bi, 1 ≤ i ≤ p: turn NG ′′(bi) ⊆ R′′ into a clique The resulting graph is G ′′ = (R′′ ∪ B′′, E ′′)

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Three algorithms to solve MTCS

Brute force Using Steiner tree Using Connected red-blue dominating set

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Brute force

U = L1(G) ∀A ⊆ V \ U

verify in polynomial time whether U ∪ A is a MTCS

runs in time O∗(2n−l1(G))

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Using Steiner tree

(G, c) - an instance of MTCS (G ′, w, K) instance for Steiner Tree, where G ′ = (R′ ∪ B′, E ′); terminal set K = B′ (|K| = |B′| = |C|) w(e) =

• 1,

∀e = u′v′ : u′, v′ ∈ R′ n = |V | else

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Using Steiner tree

(G, c) - an instance of MTCS (G ′, w, K) instance for Steiner Tree, where G ′ = (R′ ∪ B′, E ′); terminal set K = B′ (|K| = |B′| = |C|) w(e) =

• 1,

∀e = u′v′ : u′, v′ ∈ R′ n = |V | else Best known time: O∗(W · 2|K|) reduction yields an algorithm solving MTCS on (G, c) in O∗(2|C|)

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Correctness of reduction

Lemma 1

(G, c) admits a MTCS of size k ⇐ ⇒ (G ′, w, B′) admits a Steiner tree of weight k′ = k − 1 + |C|n

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Correctness of reduction

Proof: ⇒

S - MTCS of (G, c), |S| = k edge set of a Steiner tree T for (G ′, w, B′) can be obtained by first taking all k − 1 edges of a spanning tree of G[S] = G ′[S]; those edges have weight 1 ∀ terminal ri ∈ B′ where i is a color of C, choose an edge v′ri ∈ E ′ where v ∈ S is a vertex of color i in G: such an edge exists ∀ri ∈ B′ since S is tropical in G, and each such edge has weight n. Hence the Steiner tree T has weight k − 1 + |C|n = k′

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Correctness of reduction

Proof: ⇐

• E - edge set of a Steiner tree

T of (G ′, w, B′) having weight k′ B′ - independent set in G ′, hence E contains for each vertex of ri ∈ B′ an edge incident to ri. There are at least |B′| = |C| edges of weight n due to the value of k′, there are indeed precisely |C| edges of weight n in T S - set of vertices of R′ incident to the edges of weight n in

• E. By the

construction of G ′ , S is tropical in G. Steiner tree T contains k − 1 edges in G ′[R] = G connecting the vertices of S Consequently the set of vertices in the Steiner tree within R′ is connected, contains S, and is therefore tropical Since the Steiner tree in G ′[R] has k − 1 edges, the MTCS has k vertices

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Using Connected red-blue dominating set

(G, c) - instance of MTCS G ′′ = (R′′ ∪ B′′, E ′′) - instance of CRBDS

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Using Connected red-blue dominating set

(G, c) - instance of MTCS G ′′ = (R′′ ∪ B′′, E ′′) - instance of CRBDS Best known time: O∗(1.36443n) reduction yields an algorithm solving MTCS on (G, c) in O∗(1.36443n+l2(G)) time

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Correctness of reduction

Lemma 2

(G, c) admits a MTCS of size k ⇐ ⇒ G ′′ = (R′′ ∪ B′′, E ′′) admits a CRBDS set of size k′ = k − l1(G)

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Balancing three algorithms

if l1(G)

n

< 0.23814, then reduce to Steiner Tree

solving MTCS in time O∗(2|C|) |C| = l1(G) + l2(G) < 0.23814 · n + 1−0.23814

2

· n = 0.61907 · n running time is bounded by 20.61907·n < 1.53589n

if 0.23814 ≤ l1(G)

n

≤ 0.42218, then reduce to CRBDS

solving MTCS in time O∗(1.36443n+l2(G)) n + l2(G) ≤ n + 1

2 · n − 1 2 · l1(G) ≤ 1.38093 · n

running time is bounded by 1.364431.38093n < 1.53589n

if 0.42218 < l1(G)

n , then use brute force algorithm

solving MTCS in O∗(2n−l1(G)) time n − l1(G) ≤ (1 − 0.42218) · n = 0.57782 · n running time is bounded by 20.57782n < 1.49460n

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Balancing three algorithms

if l1(G)

n

< 0.23814, then reduce to Steiner Tree

solving MTCS in time O∗(2|C|) |C| = l1(G) + l2(G) < 0.23814 · n + 1−0.23814

2

· n = 0.61907 · n running time is bounded by 20.61907·n < 1.53589n

if 0.23814 ≤ l1(G)

n

≤ 0.42218, then reduce to CRBDS

solving MTCS in time O∗(1.36443n+l2(G)) n + l2(G) ≤ n + 1

2 · n − 1 2 · l1(G) ≤ 1.38093 · n

running time is bounded by 1.364431.38093n < 1.53589n

if 0.42218 < l1(G)

n , then use brute force algorithm

solving MTCS in O∗(2n−l1(G)) time n − l1(G) ≤ (1 − 0.42218) · n = 0.57782 · n running time is bounded by 20.57782n < 1.49460n

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Running time plot

Figure: The dark curve corresponds to the running time of algorithm. The x-axis corresponds to the ratio α in l1(G) = αn , while the y-axis corresponds to the constant c in the running time O∗(cn). a → α = 0.23814, b → α = 1

3, c → α = 0.42218

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No sub-exponential time algorithm for trees for MTCS

Theorem 3

The MTCS problem on trees of height at most 3 admits no sub-exponential time algorithm, unless SNP ⊆ SUBEXP , which would imply that the Exponential Time Hypothesis fails.

Proof: part 1

G = (V , E) σ : V → N: arbitrary ordering on the vertices of G T: vertex-colored tree (T, c). Initially it contains a unique vertex r colored R ∈ {0, 1, . . . , |V |} ∀v ∈ V (G), add to T a star whose center is colored σ(v), leafs: N(u) with colors defined with σ instance (T, c) admits a solution of size at most n + k + 1 to MTCS ⇐ ⇒ graph G = (V , e) admits a solution of size ≤ k to Dominating Set.

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No sub-exponential time algorithm for trees for MTCS

Theorem 3

The MTCS problem on trees of height at most 3 admits no sub-exponential time algorithm, unless SNP ⊆ SUBEXP , which would imply that the Exponential Time Hypothesis fails.

Proof: part 2

Claim: Dominating Set admits no sub-exp time algo on graphs with max degree 6, unless SNP ⊆ SUBEXP Graph G = (V , E) of max degree 6: instance for Dominating Set. → tree T of height 3, instance for MTCS, containing ≤ 8n + 1 = O∗(n) vertices ⇒ if ∃ a sub-exp time algo for MTCS on trees of max height 3 → ∃ a sub-exp time algo for Dominating Set on graphs with max degree 6 ⇒ SNP ⊆ SUBEXP ⇒ Exponential Time Hypothesis fails

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Example

Figure: The reduction from Dominating Set to Tropical Connected Set

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Paper: Enumerating Minimal Tropical Connected Sets

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Enumerating Minimal Tropical Connected Sets

Algorithms for the enumeration and combinatorial lower and upper bounds

• n graph classes for various objects in graphs have been established in the

last years, among them minimal feedback vertex sets, minimal dominating sets, minimal separators, minimal transversals, minimal connected vertex covers and minimal steiner trees

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Summarized results

Graph class Lower Bound Upper bound Enumeration Chordal 1.4961n 2n O∗(2n) Split 1.4766n 1.6042n O(1.6042n) Cobipartite 3n/3 n2 · 3n/3 + n2 O∗(3n/3) Inteval 3n/3 1.8613n O(1.8613n) Block 3n/3 3n/3 O∗(3n/3) 31/3 ∼ 1.44225

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Summarized results

Graph class Lower Bound Upper bound Enumeration Chordal 1.4961n 2n O∗(2n) Split 1.4766n 1.6042n O(1.6042n) Cobipartite 3n/3 n2 · 3n/3 + n2 O∗(3n/3) Inteval 3n/3 1.8613n O(1.8613n) Block 3n/3 3n/3 O∗(3n/3) 31/3 ∼ 1.44225

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Chordal graphs

Figure: A Chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle.

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Research: Algorithm which enumerates all MTCS on chordal graphs

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Common exact exponential algorithms idea

Introduce some parameter of graph

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Common exact exponential algorithms idea

Introduce some parameter of graph Design an algorithms that work well for some specific parameter values

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Common exact exponential algorithms idea

Introduce some parameter of graph Design an algorithms that work well for some specific parameter values Balance algorithms

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Chordal graphs properties

Each minimal separator is a clique They may be recognized in polynomial time treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it.

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Tree decomposition intuition

Figure: A subgraph of a 4 × 8 grid, with the third and the fourth columns

• highlighted. When computing c[4, Y ] for the forbidden set Y = {(2, 4)} (crossed
• ut in the fourth column), one of the sets Swe consider is S = {(1, 4), (4, 4)},

depicted with blue circles. Then, when looking at the previous column we need to forbid picking neighbors (1, 3) and (4, 3) (crossed out in the third column), since this would violate the independence constraint.

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Tree decomposition example

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Another exact exponential algorithms idea

Try build tree decomposition of graph treewidth is small: algorithms based on dynamic programming perform well treewidth is large: there must be vertices of high degree in the graph: good for branching algorithms

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Parameters

γ ∈ (0, 1] γn - count of colors in graph ε ∈ (0, 1] εn - treewidth

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Algorithms

γ is big - iterate over all subsets of vertices of all colors

algorithm with time O∗((21/γ − 1)γ)

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Algorithms

γ is big - iterate over all subsets of vertices of all colors

algorithm with time O∗((21/γ − 1)γ)

ε is small - dynamic algorithm on tree decomposition

algorithm with time O∗(2k · 4c · n)

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Algorithms

γ is big - iterate over all subsets of vertices of all colors

algorithm with time O∗((21/γ − 1)γ)

ε is small - dynamic algorithm on tree decomposition

algorithm with time O∗(2k · 4c · n)

ε is large - iterate over all subsets of vertices beyound the clique, get vertices from clique to provide connectivity and tropicality

algorithm with time O∗(( 1

α − 1 2)α · eα · 21−σ)

where α = 1+γ−σ

σ

> 1

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Working Time Diagram

Figure: For given ε and γ enumerating algorithm will works O∗(cn) time.

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Materials

[1] Exact exponential algorithms to find tropical connected sets of minimum size (Chapelle, Cochefert, Kratsch, Letourneur, Liedloff) [2] Enumerating Minimal Tropical Connected Sets (Kratsch, Liedloff, Sayadi)

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Thanks for your attention!

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