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

Tropical Connected Sets problem Eugene Vagin 07.04.2020 Eugene Vagin Tropical Connected Sets problem 07.04.2020 1 / 42

Outline Exact exponential algorithms to find tropical connected sets of minimum 1 size Enumerating Minimal Tropical Connected Sets 2 Algorithm which enumerates all MTCS on chordal graphs 3 Eugene Vagin Tropical Connected Sets problem 07.04.2020 2 / 42

Paper: Exact exponential algorithms to find tropical connected sets of minimum size Eugene Vagin Tropical Connected Sets problem 07.04.2020 3 / 42

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 Eugene Vagin Tropical Connected Sets problem 07.04.2020 4 / 42

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 Eugene Vagin Tropical Connected Sets problem 07.04.2020 5 / 42

Extra preliminaries N ( v ) set of all neighbors of v N [ v ] N ( v ) ∪ { v } N [ X ] ∪ x ∈ X N [ x ] N ( X ) N [ X ] \ X L 1 ( G ) vertices whose colors appears only once in ( G , c ) l 1 ( G ) | L 1 ( G ) | l 2 ( G ) number of colors appearing at least twice in ( G , c ) Eugene Vagin Tropical Connected Sets problem 07.04.2020 6 / 42

An exact exponential algorithm for general graphs Naive brute force: O ∗ ( 2 n ) time In paper [1] described algorithm which works in O ∗ ( 1 . 5359 n ) via reductions to Connected red-blue dominating set Steiner tree balancing technique depending on the value of l 1 ( G ) Eugene Vagin Tropical Connected Sets problem 07.04.2020 7 / 42

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 | ) Eugene Vagin Tropical Connected Sets problem 07.04.2020 8 / 42

Connected red-blue dominating set Input Graph G = ( R , B , E ) where vertices are colored either red (vertices in R ) or 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 . 36443 n ) Eugene Vagin Tropical Connected Sets problem 07.04.2020 9 / 42

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. Eugene Vagin Tropical Connected Sets problem 07.04.2020 10 / 42

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 ′ { r i | i ∈ C } E ′ { u ′ v ′ | uv ∈ E } ∪ { v ′ r i | v is of color i in G } Eugene Vagin Tropical Connected Sets problem 07.04.2020 11 / 42

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 r i from B ′′ , where c ( v ) = iinG B 1 , . . . , B p - components of the subgraph induced in G ′′ [ B ′′ ] by those vertices that had been moved to B ′′ ∀ i = 1 , 2 , . . . , p contract the component B i in G ′′ [ B ′′ ] so that it remains only one vertex and call this vertex b i ∀ b i , 1 ≤ i ≤ p : turn N G ′′ ( b i ) ⊆ R ′′ into a clique The resulting graph is G ′′ = ( R ′′ ∪ B ′′ , E ′′ ) Eugene Vagin Tropical Connected Sets problem 07.04.2020 12 / 42

Three algorithms to solve MTCS Brute force Using Steiner tree Using Connected red-blue dominating set Eugene Vagin Tropical Connected Sets problem 07.04.2020 13 / 42

Brute force U = L 1 ( G ) ∀ A ⊆ V \ U verify in polynomial time whether U ∪ A is a MTCS runs in time O ∗ ( 2 n − l 1 ( G ) ) Eugene Vagin Tropical Connected Sets problem 07.04.2020 14 / 42

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 | ) � ∀ e = u ′ v ′ : u ′ , v ′ ∈ R ′ 1 , w ( e ) = n = | V | else Eugene Vagin Tropical Connected Sets problem 07.04.2020 15 / 42

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 | ) � ∀ e = u ′ v ′ : u ′ , v ′ ∈ R ′ 1 , w ( e ) = n = | V | else Best known time : O ∗ ( W · 2 | K | ) reduction yields an algorithm solving MTCS on (G, c) in O ∗ ( 2 | C | ) Eugene Vagin Tropical Connected Sets problem 07.04.2020 15 / 42

Correctness of reduction Lemma 1 ⇒ ( G ′ , w , B ′ ) admits a Steiner tree of ( G , c ) admits a MTCS of size k ⇐ weight k ′ = k − 1 + | C | n Eugene Vagin Tropical Connected Sets problem 07.04.2020 16 / 42

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 r i ∈ B ′ where i is a color of C , choose an edge v ′ r i ∈ E ′ where v ∈ S is a vertex of color i in G : such an edge exists ∀ r i ∈ 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 ′ Eugene Vagin Tropical Connected Sets problem 07.04.2020 17 / 42

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 r i ∈ B ′ an edge incident to r i . 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 Eugene Vagin Tropical Connected Sets problem 07.04.2020 18 / 42

Using Connected red-blue dominating set ( G , c ) - instance of MTCS G ′′ = ( R ′′ ∪ B ′′ , E ′′ ) - instance of CRBDS Eugene Vagin Tropical Connected Sets problem 07.04.2020 19 / 42

Using Connected red-blue dominating set ( G , c ) - instance of MTCS G ′′ = ( R ′′ ∪ B ′′ , E ′′ ) - instance of CRBDS Best known time : O ∗ ( 1 . 36443 n ) reduction yields an algorithm solving MTCS on (G, c) in O ∗ ( 1 . 36443 n + l 2 ( G ) ) time Eugene Vagin Tropical Connected Sets problem 07.04.2020 19 / 42

Correctness of reduction Lemma 2 ⇒ G ′′ = ( R ′′ ∪ B ′′ , E ′′ ) admits a ( G , c ) admits a MTCS of size k ⇐ CRBDS set of size k ′ = k − l 1 ( G ) Eugene Vagin Tropical Connected Sets problem 07.04.2020 20 / 42

Balancing three algorithms if l 1 ( G ) < 0 . 23814, then reduce to Steiner Tree n solving MTCS in time O ∗ ( 2 | C | ) | C | = l 1 ( G ) + l 2 ( G ) < 0 . 23814 · n + 1 − 0 . 23814 · n = 0 . 61907 · n 2 running time is bounded by 2 0 . 61907 · n < 1 . 53589 n if 0 . 23814 ≤ l 1 ( G ) ≤ 0 . 42218, then reduce to CRBDS n solving MTCS in time O ∗ ( 1 . 36443 n + l 2 ( G ) ) n + l 2 ( G ) ≤ n + 1 2 · n − 1 2 · l 1 ( G ) ≤ 1 . 38093 · n running time is bounded by 1 . 36443 1 . 38093 n < 1 . 53589 n if 0 . 42218 < l 1 ( G ) n , then use brute force algorithm solving MTCS in O ∗ ( 2 n − l 1 ( G ) ) time n − l 1 ( G ) ≤ ( 1 − 0 . 42218 ) · n = 0 . 57782 · n running time is bounded by 2 0 . 57782 n < 1 . 49460 n Eugene Vagin Tropical Connected Sets problem 07.04.2020 21 / 42