The Hypergraph Assignment Problem Olga Heismann joint work with: - PowerPoint PPT Presentation

The Hypergraph Assignment Problem Olga Heismann joint work with: Ralf Borndörfer, Achim Hildenbrandt DFG Research Center M ATHEON Mathematics for key technologies January 7–11, 2013

Contents Definition and Complexity of the HAP 1 Results for Partitioned Hypergraphs 2 Polyhedral Investigation 3 Heuristics 4 The Hypergraph Assignment Problem 2 / 23

Contents Definition and Complexity of the HAP 1 Results for Partitioned Hypergraphs 2 Polyhedral Investigation 3 Heuristics 4 The Hypergraph Assignment Problem 3 / 23

From Assignments . . . Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of edges connecting U and V , an assignment is a subset H of E such that there is exactly one incident edge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 4 / 23

From Assignments . . . Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of edges connecting U and V , an assignment is a subset H of E such that there is exactly one incident edge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 4 / 23

From Assignments . . . Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of edges connecting U and V , an assignment is a subset H of E such that there is exactly one incident edge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 4 / 23

From Assignments . . . Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of edges connecting U and V , an assignment is a subset H of E such that there is exactly one incident edge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 4 / 23

From Assignments . . . Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of edges connecting U and V , an assignment is a subset H of E such that there is exactly one incident edge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 4 / 23

. . . to Hyperassignments Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of hyperedges connecting U and V , a hyperassignment is a subset H of E such that there is exactly one incident hyperedge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 5 / 23

. . . to Hyperassignments Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of hyperedges connecting U and V , a hyperassignment is a subset H of E such that there is exactly one incident hyperedge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 5 / 23

. . . to Hyperassignments Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of hyperedges connecting U and V , a hyperassignment is a subset H of E such that there is exactly one incident hyperedge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 5 / 23

. . . to Hyperassignments Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of hyperedges connecting U and V , a hyperassignment is a subset H of E such that there is exactly one incident hyperedge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 5 / 23

. . . to Hyperassignments Given ⊲ two equally sized sets U , V of vertices of ⊲ a set E of hyperedges connecting U and V , a hyperassignment is a subset H of E such that there is exactly one incident hyperedge in H for each vertex. v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 6 The Hypergraph Assignment Problem 5 / 23

Definition of a Bipartite Hypergraph Definition A bipartite hypergraph G = ( U , V , E ) is a triple of two disjoint vertex sets U , V and a set of hyperedges E ⊆ 2 U · ∪ V . We assume that the vertex sets have the same size | U | = | V | , and that every hyperedge e ∈ E has the same number | e ∩ U | = | e ∩ V | > 0 of vertices in U and V . We denote by | e | the size of the hyperedge e ∈ E , and call a hyperedge of size 2 an edge . Definition For a vertex subset W ⊆ U ∪ V we define the incident hyperedges δ ( W ) := { e ∈ E : e ∩ W � = ∅ , e \ W � = ∅} to be the set of all hyperedges having at least one vertex in both U and ( U ∪ V ) \ W . We also write δ ( v ) = δ ( { v } ) if v is a vertex. The Hypergraph Assignment Problem 6 / 23

Hypergraph Assignment Problem (HAP) Definition Let G = ( U , V , E ) be a bipartite hypergraph. A hyperassignment in G is a subset H ⊆ E of hyperedges such that every v ∈ U ∪ V is contained in exactly one hyperedge e ∈ H . Hypergraph Assignment Problem Input: A pair ( G , c E ) consisting of a bipartite hypergraph G = ( U , V , E ) and a cost function c E : E → R . Output: A minimum cost hyperassignment in G w. r. t. c E , i. e., a hyperassignment H ∗ in G such that c E ( H ∗ ) = min { c E ( H ) : H is a hyperassignment in G } , or the information that no hyperassignment exists. The Hypergraph Assignment Problem 7 / 23

Complexity Results Theorem (B., He. [2011]) 1. The hypergraph assignment problem (HAP) is NP-hard. 2. The HAP is APX-hard. 3. The LP/IP gap of HAP can be arbitrarily large. 4. The determinants of basis matrices of HAP can be arbitrarily large. v 1 v 2 v 3 � min c E ( e ) x e x ∈ R E e ∈ E � ∀ v ∈ U ∪ V s. t. x e = 1 e ∈ δ ( v ) x ≥ 0 u 1 u 2 u 3 x ∈ Z E The Hypergraph Assignment Problem 8 / 23

Complexity Results Theorem (B., He. [2011]) 1. The hypergraph assignment problem (HAP) is NP-hard. 2. The HAP is APX-hard. 3. The LP/IP gap of HAP can be arbitrarily large. 4. The determinants of basis matrices of HAP can be arbitrarily large. v 1 v 2 v 3 � min c E ( e ) x e x ∈ R E e ∈ E � ∀ v ∈ U ∪ V s. t. x e = 1 e ∈ δ ( v ) x ≥ 0 u 1 u 2 u 3 x ∈ Z E The Hypergraph Assignment Problem 8 / 23

Complexity Results Theorem (B., He. [2011]) 1. The hypergraph assignment problem (HAP) is NP-hard. 2. The HAP is APX-hard. 3. The LP/IP gap of HAP can be arbitrarily large. 4. The determinants of basis matrices of HAP can be arbitrarily large. v 1 v 2 v 3 � min c E ( e ) x e x ∈ R E e ∈ E � ∀ v ∈ U ∪ V s. t. x e = 1 e ∈ δ ( v ) x ≥ 0 u 1 u 2 u 3 x ∈ Z E The Hypergraph Assignment Problem 8 / 23

Complexity Results Theorem (B., He. [2011]) 1. The hypergraph assignment problem (HAP) is NP-hard. 2. The HAP is APX-hard. 3. The LP/IP gap of HAP can be arbitrarily large. 4. The determinants of basis matrices of HAP can be arbitrarily large. v 1 v 2 v 3 � min c E ( e ) x e x ∈ R E e ∈ E � ∀ v ∈ U ∪ V s. t. x e = 1 e ∈ δ ( v ) x ≥ 0 u 1 u 2 u 3 x ∈ Z E The Hypergraph Assignment Problem 8 / 23

Contents Definition and Complexity of the HAP 1 Results for Partitioned Hypergraphs 2 Polyhedral Investigation 3 Heuristics 4 The Hypergraph Assignment Problem 9 / 23

Partitioned Hypergraphs Definition G = ( U , V , E ) is called a partitioned bipartite hypergraph with maximum part size d ∈ N if additionally there exist pairwise disjoint ≤ d -element sets U 1 , . . . , U p and V 1 , . . . , V q called the parts of H � p � q i = 1 V i = V , and E ⊆ � p � q j = 1 2 U i ∪ V j , i. e., such that · i = 1 U i = U , · i = 1 every hyperedge intersects only one part in U and one part in V . v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 5 u 6 The Hypergraph Assignment Problem 10 / 23

Partitioned Hypergraphs Definition G = ( U , V , E ) is called a partitioned bipartite hypergraph with maximum part size d ∈ N if additionally there exist pairwise disjoint ≤ d -element sets U 1 , . . . , U p and V 1 , . . . , V q called the parts of H � p � q i = 1 V i = V , and E ⊆ � p � q j = 1 2 U i ∪ V j , i. e., such that · i = 1 U i = U , · i = 1 every hyperedge intersects only one part in U and one part in V . v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 5 u 6 The Hypergraph Assignment Problem 10 / 23

Partitioned Hypergraphs Definition G = ( U , V , E ) is called a partitioned bipartite hypergraph with maximum part size d ∈ N if additionally there exist pairwise disjoint ≤ d -element sets U 1 , . . . , U p and V 1 , . . . , V q called the parts of H � p � q i = 1 V i = V , and E ⊆ � p � q j = 1 2 U i ∪ V j , i. e., such that · i = 1 U i = U , · i = 1 every hyperedge intersects only one part in U and one part in V . v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 5 u 6 The Hypergraph Assignment Problem 10 / 23

Partitioned Hypergraphs Definition G = ( U , V , E ) is called a partitioned bipartite hypergraph with maximum part size d ∈ N if additionally there exist pairwise disjoint ≤ d -element sets U 1 , . . . , U p and V 1 , . . . , V q called the parts of H � p � q i = 1 V i = V , and E ⊆ � p � q j = 1 2 U i ∪ V j , i. e., such that · i = 1 U i = U , · i = 1 every hyperedge intersects only one part in U and one part in V . v 1 v 2 v 3 v 4 v 5 v 6 u 1 u 2 u 3 u 4 u 5 u 5 u 6 The Hypergraph Assignment Problem 10 / 23 not partitioned