The Hypergraph Assignment Problem
The Hypergraph Assignment Problem
Olga Heismann joint work with: Ralf Borndörfer, Achim Hildenbrandt
DFG Research Center MATHEON Mathematics for key technologies January 7–11, 2013
Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
The Hypergraph Assignment Problem
2 / 23
Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
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The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
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The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
v4
v5
v6
u1
u2
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u4
u5
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The Hypergraph Assignment Problem
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. . . 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. v1
v2
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u1
u2
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u4
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The Hypergraph Assignment Problem
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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 ⊆ 2U ∪· 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 6= ∅, e \ W 6= ∅} 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
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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 , cE ) consisting of a bipartite hypergraph G = (U, V , E ) and a cost function cE : E → R. Output: A minimum cost hyperassignment in G w. r. t. cE , i. e., a hyperassignment H ∗ in G such that cE (H ∗ ) = min{cE (H) : H is a hyperassignment in G }, or the information that no hyperassignment exists. The Hypergraph Assignment Problem
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Complexity Results Theorem (B., He. [2011]) 1. 2. 3. 4.
The The The The
min
x∈RE
s. t.
hypergraph assignment problem (HAP) is NP-hard. HAP is APX-hard. LP/IP gap of HAP can be arbitrarily large. determinants of basis matrices of HAP can be arbitrarily large. X
v1
v2
v3
u1
u2
u3
cE (e)xe
e∈E
X
xe = 1
∀v ∈ U ∪ V
e∈δ(v )
x ≥0 x ∈ ZE The Hypergraph Assignment Problem
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Complexity Results Theorem (B., He. [2011]) 1. 2. 3. 4.
The The The The
min
x∈RE
s. t.
hypergraph assignment problem (HAP) is NP-hard. HAP is APX-hard. LP/IP gap of HAP can be arbitrarily large. determinants of basis matrices of HAP can be arbitrarily large. X
v1
v2
v3
u1
u2
u3
cE (e)xe
e∈E
X
xe = 1
∀v ∈ U ∪ V
e∈δ(v )
x ≥0 x ∈ ZE The Hypergraph Assignment Problem
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Complexity Results Theorem (B., He. [2011]) 1. 2. 3. 4.
The The The The
min
x∈RE
s. t.
hypergraph assignment problem (HAP) is NP-hard. HAP is APX-hard. LP/IP gap of HAP can be arbitrarily large. determinants of basis matrices of HAP can be arbitrarily large. X
v1
v2
v3
u1
u2
u3
cE (e)xe
e∈E
X
xe = 1
∀v ∈ U ∪ V
e∈δ(v )
x ≥0 x ∈ ZE The Hypergraph Assignment Problem
8 / 23
Complexity Results Theorem (B., He. [2011]) 1. 2. 3. 4.
The The The The
min
x∈RE
s. t.
hypergraph assignment problem (HAP) is NP-hard. HAP is APX-hard. LP/IP gap of HAP can be arbitrarily large. determinants of basis matrices of HAP can be arbitrarily large. X
v1
v2
v3
u1
u2
u3
cE (e)xe
e∈E
X
xe = 1
∀v ∈ U ∪ V
e∈δ(v )
x ≥0 x ∈ ZE The Hypergraph Assignment Problem
8 / 23
Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
The Hypergraph Assignment Problem
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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 U1 , . . . , Up and V1 , . . . , Vq called the parts of H S S S S such that · pi=1 Ui = U, · qi=1 Vi = V , and E ⊆ pi=1 qj=1 2Ui ∪Vj , i. e., every hyperedge intersects only one part in U and one part in V . v1
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The Hypergraph Assignment Problem
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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 U1 , . . . , Up and V1 , . . . , Vq called the parts of H S S S S such that · pi=1 Ui = U, · qi=1 Vi = V , and E ⊆ pi=1 qj=1 2Ui ∪Vj , i. e., every hyperedge intersects only one part in U and one part in V . v1
v2
u1
u2
The Hypergraph Assignment Problem
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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 U1 , . . . , Up and V1 , . . . , Vq called the parts of H S S S S such that · pi=1 Ui = U, · qi=1 Vi = V , and E ⊆ pi=1 qj=1 2Ui ∪Vj , i. e., every hyperedge intersects only one part in U and one part in V . v1
v2
u1
u2
The Hypergraph Assignment Problem
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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 U1 , . . . , Up and V1 , . . . , Vq called the parts of H S S S S such that · pi=1 Ui = U, · qi=1 Vi = V , and E ⊆ pi=1 qj=1 2Ui ∪Vj , i. e., every hyperedge intersects only one part in U and one part in V . v1
v2
u1
u2
The Hypergraph Assignment Problem
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Results for Partitioned Hypergraphs Theorem (B., He. [2012]) Every HAP can be polynomially transformed into a HAP on a partitioned hypergraph. A clique Q ⊆ E is a set of hyperedges such that every pair of hyperedges in Q has a nonempty intersection.
Theorem (B., He. [2011]) Every clique in a partitioned hypergraph is a subset of the incident hyperedges δ(P) of some part P. There exists an extended formulation with O(|U|d+1 ) variables that implies all clique inequalities. The Hypergraph Assignment Problem
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Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
The Hypergraph Assignment Problem
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Polyhedral Investigation Let G2,3 = (U, V , E ) be the complete partitioned bipartite hypergraph with . parts {u11 , u12 }, {u21 , u22 }, {u31 , u32 } in U and . parts {v11 , v12 }, {v21 , v22 }, {v31 , v32 } in V . v11
v12
v21
v22
v31
v32
u11
u12
u21
u22
u31
u32
Let P(G2,3 ) be the HAP polytope associated with G2,3 . P(G2,3 ) is completely described by 14049 facets. The Hypergraph Assignment Problem
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Polyhedral Results cont. (B., He.) . Every facet of P(G2,3 ) can be described by many different inequalities (polytope description includes 11 equations). . All facets can be described in the form X X xe − xe ≤ 1. e∈E1
e∈E2
. So far we have no normal form.
The Hypergraph Assignment Problem
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Polyhedral Results cont. (B., He.) The polytope is highly symmetric. The symmetries are generated by: . uij 7→ vij , vij 7→ uij for all i, j . uij 7→ uij , vij 7→ vσ(i)j for some σ ∈ S3 . u11 7→ u11 , u12 7→ u11 , uij 7→ uij for i 6= 1, vij 7→ vij This results in 4608 vertex permutations, which imply permutations of the hyperedge variables. The 14049 facets of P(G2,3 ) fall into 30 symmetry classes. We have understood 16 facet classes: . trivial facets: hyperedge ≥ 0 . cliques . odd clique set inequalities (see next slides) 14 facet classes are still to be understood. The Hypergraph Assignment Problem
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Classification of Facets Without Normal Form Given: . permutation of variables . vertices of the polytope . facet inequalities of the polytope How to classify the facets into symmetry classes? . identify every facet with the incident vertices of the polytope . permutation of variables implies permutation of vertices . permutation of vertices implies permutation of facets . implemented in general (“HUHFA”)
The Hypergraph Assignment Problem
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Generalization of Odd Set Inequalities Odd set cuts for the matching polytope of a graph G = (N, E ), N 0 ⊆ N, |N 0 | = 2k + 1 odd (Edmonds [1965]): X xe ≤ k e∈E :e⊆N 0
or
X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k. 2 e∈E
Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E The Hypergraph Assignment Problem
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Generalization of Odd Set Inequalities Odd set cuts for the matching polytope of a graph G = (N, E ), N 0 ⊆ N, |N 0 | = 2k + 1 odd (Edmonds [1965]): X xe ≤ k e∈E :e⊆N 0
or
X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k. 2 e∈E
Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E The Hypergraph Assignment Problem
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Generalization of Odd Set Inequalities Odd set cuts for the matching polytope of a graph G = (N, E ), N 0 ⊆ N, |N 0 | = 2k + 1 odd (Edmonds [1965]): X xe ≤ k e∈E :e⊆N 0
or
X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k. 2 e∈E
Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E The Hypergraph Assignment Problem
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Odd Clique Set Inequalities (B., He.) Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E Replace N 0 by a set of cliques: Q ⊆ 2E , |Q| = 2k + 1 odd number of cliques in G . Odd clique set cut: X |{Q ∈ Q : e ∈ Q}| xe ≤ k 2 e∈E
The Hypergraph Assignment Problem
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Odd Clique Set Inequalities (B., He.) Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E Replace N 0 by a set of cliques: Q ⊆ 2E , |Q| = 2k + 1 odd number of cliques in G . Odd clique set cut: X |{Q ∈ Q : e ∈ Q}| xe ≤ k 2 e∈E
The Hypergraph Assignment Problem
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Odd Clique Set Inequalities (cont.) . Not all odd clique set inequalities are facets for the HAP polytope. . Separation? . Different generalization of odd set cuts: “Generalized clique family inequalities for claw-free graphs” (Pêcher, Wagler [2006]) p ≤ |Q| 0 ≤ r ≤ R = |Q| mod p 0≤J ≤p−r Ep := {e ∈ E : |{Q ∈ Q : e ∈ Q}| ≥ p} Ep−j := {e ∈ E : |{Q ∈ Q : e ∈ Q}| = p − j} X X (p − r − j) xe ≤ b 0≤j≤J
e∈Ep−j
do not lead to facets of P(G2,3 ) The Hypergraph Assignment Problem
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Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
The Hypergraph Assignment Problem
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Overview of Heuristic Approaches . Constructive heuristics: I I
greedy with coefficients per vertex Hungarian method with vertex groups
. Local search: I I I
Hungarian method with vertex groups 2-opt dynamic k-opt
. Perturbation heuristics: I
greedy insert with randomization
The Hypergraph Assignment Problem
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Computational Results instance name
bipartite hypergraph
arcs
2-hyperedges
optimal value
heuristic result
gap
run time (sec.)
Results of first tests:
Random10 Random20 Random35 Random50 Random75 Random100
G2,10 G2,20 G2,35 G2,50 G2,75 G2,100
400 1600 4900 10000 22500 40000
100 400 1225 2500 5625 10000
88 84 92 112 95 93
88 85 129 144 140 155
0% 1.2 % 40.2 % 28.6 % 47.4 % 66.7 %
52.9 53.7 57.8 54.4 105.8 223.5
. costs of hyperedges i. i. d. from {0, . . . , 100} . some variability in results and run times due to randomization . many parameter changes possible The Hypergraph Assignment Problem
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The Hypergraph Assignment Problem
Olga Heismann joint work with: Ralf Borndörfer, Achim Hildenbrandt
DFG Research Center MATHEON Mathematics for key technologies January 7–11, 2013
Olga Heismann joint work with: Ralf Borndörfer, Achim Hildenbrandt
DFG Research Center MATHEON Mathematics for key technologies January 7–11, 2013
Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
The Hypergraph Assignment Problem
2 / 23
Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
u6
The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
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u1
u2
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u4
u5
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The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
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The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
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The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
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The Hypergraph Assignment Problem
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. . . 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. v1
v2
v3
v4
v5
v6
u1
u2
u3
u4
u5
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The Hypergraph Assignment Problem
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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 ⊆ 2U ∪· 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 6= ∅, e \ W 6= ∅} 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
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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 , cE ) consisting of a bipartite hypergraph G = (U, V , E ) and a cost function cE : E → R. Output: A minimum cost hyperassignment in G w. r. t. cE , i. e., a hyperassignment H ∗ in G such that cE (H ∗ ) = min{cE (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. 2. 3. 4.
The The The The
min
x∈RE
s. t.
hypergraph assignment problem (HAP) is NP-hard. HAP is APX-hard. LP/IP gap of HAP can be arbitrarily large. determinants of basis matrices of HAP can be arbitrarily large. X
v1
v2
v3
u1
u2
u3
cE (e)xe
e∈E
X
xe = 1
∀v ∈ U ∪ V
e∈δ(v )
x ≥0 x ∈ ZE The Hypergraph Assignment Problem
8 / 23
Complexity Results Theorem (B., He. [2011]) 1. 2. 3. 4.
The The The The
min
x∈RE
s. t.
hypergraph assignment problem (HAP) is NP-hard. HAP is APX-hard. LP/IP gap of HAP can be arbitrarily large. determinants of basis matrices of HAP can be arbitrarily large. X
v1
v2
v3
u1
u2
u3
cE (e)xe
e∈E
X
xe = 1
∀v ∈ U ∪ V
e∈δ(v )
x ≥0 x ∈ ZE The Hypergraph Assignment Problem
8 / 23
Complexity Results Theorem (B., He. [2011]) 1. 2. 3. 4.
The The The The
min
x∈RE
s. t.
hypergraph assignment problem (HAP) is NP-hard. HAP is APX-hard. LP/IP gap of HAP can be arbitrarily large. determinants of basis matrices of HAP can be arbitrarily large. X
v1
v2
v3
u1
u2
u3
cE (e)xe
e∈E
X
xe = 1
∀v ∈ U ∪ V
e∈δ(v )
x ≥0 x ∈ ZE The Hypergraph Assignment Problem
8 / 23
Complexity Results Theorem (B., He. [2011]) 1. 2. 3. 4.
The The The The
min
x∈RE
s. t.
hypergraph assignment problem (HAP) is NP-hard. HAP is APX-hard. LP/IP gap of HAP can be arbitrarily large. determinants of basis matrices of HAP can be arbitrarily large. X
v1
v2
v3
u1
u2
u3
cE (e)xe
e∈E
X
xe = 1
∀v ∈ U ∪ V
e∈δ(v )
x ≥0 x ∈ ZE The Hypergraph Assignment Problem
8 / 23
Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
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 U1 , . . . , Up and V1 , . . . , Vq called the parts of H S S S S such that · pi=1 Ui = U, · qi=1 Vi = V , and E ⊆ pi=1 qj=1 2Ui ∪Vj , i. e., every hyperedge intersects only one part in U and one part in V . v1
v2
u1
u2
The Hypergraph Assignment Problem
v3
u3
u4
v4
v5
u5
u6
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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 U1 , . . . , Up and V1 , . . . , Vq called the parts of H S S S S such that · pi=1 Ui = U, · qi=1 Vi = V , and E ⊆ pi=1 qj=1 2Ui ∪Vj , i. e., every hyperedge intersects only one part in U and one part in V . v1
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The Hypergraph Assignment Problem
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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 U1 , . . . , Up and V1 , . . . , Vq called the parts of H S S S S such that · pi=1 Ui = U, · qi=1 Vi = V , and E ⊆ pi=1 qj=1 2Ui ∪Vj , i. e., every hyperedge intersects only one part in U and one part in V . v1
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u1
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The Hypergraph Assignment Problem
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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 U1 , . . . , Up and V1 , . . . , Vq called the parts of H S S S S such that · pi=1 Ui = U, · qi=1 Vi = V , and E ⊆ pi=1 qj=1 2Ui ∪Vj , i. e., every hyperedge intersects only one part in U and one part in V . v1
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u1
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The Hypergraph Assignment Problem
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Results for Partitioned Hypergraphs Theorem (B., He. [2012]) Every HAP can be polynomially transformed into a HAP on a partitioned hypergraph. A clique Q ⊆ E is a set of hyperedges such that every pair of hyperedges in Q has a nonempty intersection.
Theorem (B., He. [2011]) Every clique in a partitioned hypergraph is a subset of the incident hyperedges δ(P) of some part P. There exists an extended formulation with O(|U|d+1 ) variables that implies all clique inequalities. The Hypergraph Assignment Problem
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Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
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Polyhedral Investigation Let G2,3 = (U, V , E ) be the complete partitioned bipartite hypergraph with . parts {u11 , u12 }, {u21 , u22 }, {u31 , u32 } in U and . parts {v11 , v12 }, {v21 , v22 }, {v31 , v32 } in V . v11
v12
v21
v22
v31
v32
u11
u12
u21
u22
u31
u32
Let P(G2,3 ) be the HAP polytope associated with G2,3 . P(G2,3 ) is completely described by 14049 facets. The Hypergraph Assignment Problem
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Polyhedral Results cont. (B., He.) . Every facet of P(G2,3 ) can be described by many different inequalities (polytope description includes 11 equations). . All facets can be described in the form X X xe − xe ≤ 1. e∈E1
e∈E2
. So far we have no normal form.
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Polyhedral Results cont. (B., He.) The polytope is highly symmetric. The symmetries are generated by: . uij 7→ vij , vij 7→ uij for all i, j . uij 7→ uij , vij 7→ vσ(i)j for some σ ∈ S3 . u11 7→ u11 , u12 7→ u11 , uij 7→ uij for i 6= 1, vij 7→ vij This results in 4608 vertex permutations, which imply permutations of the hyperedge variables. The 14049 facets of P(G2,3 ) fall into 30 symmetry classes. We have understood 16 facet classes: . trivial facets: hyperedge ≥ 0 . cliques . odd clique set inequalities (see next slides) 14 facet classes are still to be understood. The Hypergraph Assignment Problem
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Classification of Facets Without Normal Form Given: . permutation of variables . vertices of the polytope . facet inequalities of the polytope How to classify the facets into symmetry classes? . identify every facet with the incident vertices of the polytope . permutation of variables implies permutation of vertices . permutation of vertices implies permutation of facets . implemented in general (“HUHFA”)
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Generalization of Odd Set Inequalities Odd set cuts for the matching polytope of a graph G = (N, E ), N 0 ⊆ N, |N 0 | = 2k + 1 odd (Edmonds [1965]): X xe ≤ k e∈E :e⊆N 0
or
X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k. 2 e∈E
Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E The Hypergraph Assignment Problem
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Generalization of Odd Set Inequalities Odd set cuts for the matching polytope of a graph G = (N, E ), N 0 ⊆ N, |N 0 | = 2k + 1 odd (Edmonds [1965]): X xe ≤ k e∈E :e⊆N 0
or
X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k. 2 e∈E
Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E The Hypergraph Assignment Problem
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Generalization of Odd Set Inequalities Odd set cuts for the matching polytope of a graph G = (N, E ), N 0 ⊆ N, |N 0 | = 2k + 1 odd (Edmonds [1965]): X xe ≤ k e∈E :e⊆N 0
or
X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k. 2 e∈E
Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E The Hypergraph Assignment Problem
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Odd Clique Set Inequalities (B., He.) Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E Replace N 0 by a set of cliques: Q ⊆ 2E , |Q| = 2k + 1 odd number of cliques in G . Odd clique set cut: X |{Q ∈ Q : e ∈ Q}| xe ≤ k 2 e∈E
The Hypergraph Assignment Problem
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Odd Clique Set Inequalities (B., He.) Generalization for a hypergraph G = (U, V , E ), N 0 ⊆ U ∪ V , |N 0 | = 2k + 1 odd: X |{v ∈ N 0 : e ∈ δ(v )}| xe ≤ k 2 e∈E Replace N 0 by a set of cliques: Q ⊆ 2E , |Q| = 2k + 1 odd number of cliques in G . Odd clique set cut: X |{Q ∈ Q : e ∈ Q}| xe ≤ k 2 e∈E
The Hypergraph Assignment Problem
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Odd Clique Set Inequalities (cont.) . Not all odd clique set inequalities are facets for the HAP polytope. . Separation? . Different generalization of odd set cuts: “Generalized clique family inequalities for claw-free graphs” (Pêcher, Wagler [2006]) p ≤ |Q| 0 ≤ r ≤ R = |Q| mod p 0≤J ≤p−r Ep := {e ∈ E : |{Q ∈ Q : e ∈ Q}| ≥ p} Ep−j := {e ∈ E : |{Q ∈ Q : e ∈ Q}| = p − j} X X (p − r − j) xe ≤ b 0≤j≤J
e∈Ep−j
do not lead to facets of P(G2,3 ) The Hypergraph Assignment Problem
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Contents
1
Definition and Complexity of the HAP
2
Results for Partitioned Hypergraphs
3
Polyhedral Investigation
4
Heuristics
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Overview of Heuristic Approaches . Constructive heuristics: I I
greedy with coefficients per vertex Hungarian method with vertex groups
. Local search: I I I
Hungarian method with vertex groups 2-opt dynamic k-opt
. Perturbation heuristics: I
greedy insert with randomization
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Computational Results instance name
bipartite hypergraph
arcs
2-hyperedges
optimal value
heuristic result
gap
run time (sec.)
Results of first tests:
Random10 Random20 Random35 Random50 Random75 Random100
G2,10 G2,20 G2,35 G2,50 G2,75 G2,100
400 1600 4900 10000 22500 40000
100 400 1225 2500 5625 10000
88 84 92 112 95 93
88 85 129 144 140 155
0% 1.2 % 40.2 % 28.6 % 47.4 % 66.7 %
52.9 53.7 57.8 54.4 105.8 223.5
. costs of hyperedges i. i. d. from {0, . . . , 100} . some variability in results and run times due to randomization . many parameter changes possible The Hypergraph Assignment Problem
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The Hypergraph Assignment Problem
Olga Heismann joint work with: Ralf Borndörfer, Achim Hildenbrandt
DFG Research Center MATHEON Mathematics for key technologies January 7–11, 2013
