The Hypergraph AssignmentProblem
Olga Heismann
joint work with: Ralf Borndörfer, Achim Hildenbrandt
DFG Research Center MATHEONMathematics 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 oneincident edge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident edge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident edge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident edge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident edge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident hyperedge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident hyperedge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident hyperedge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident hyperedge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
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 oneincident hyperedge in H for each vertex.
u1
v1
u2
v2
u3
v3
u4
v4
u5
v5
u6
v6
The Hypergraph Assignment Problem 5 / 23
Definition of a Bipartite Hypergraph
DefinitionA bipartite hypergraph G = (U,V ,E ) is a triple of two disjoint vertexsets U, V and a set of hyperedges E ⊆ 2U ·∪V . We assume that thevertex sets have the same size |U| = |V |, and that every hyperedgee ∈ E has the same number |e ∩ U| = |e ∩ V | > 0 of vertices in Uand V . We denote by |e| the size of the hyperedge e ∈ E , and call ahyperedge of size 2 an edge.
DefinitionFor a vertex subset W ⊆ U ∪ V we define the incident hyperedgesδ(W ) := {e ∈ E : e ∩W 6= ∅, e \W 6= ∅} to be the set of allhyperedges 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)
DefinitionLet G = (U,V ,E ) be a bipartite hypergraph. A hyperassignment inG is a subset H ⊆ E of hyperedges such that every v ∈ U ∪ V iscontained in exactly one hyperedge e ∈ H.
Hypergraph Assignment ProblemInput: A pair (G , cE ) consisting of a bipartite hypergraphG = (U,V ,E ) and a cost function cE : E → R.Output: A minimum cost hyperassignment in G w. r. t. cE , i. e., ahyperassignment 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. 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.
minx∈RE
∑e∈E
cE (e)xe
s. t.∑
e∈δ(v)
xe = 1 ∀v ∈ U ∪ V
x ≥ 0
x ∈ ZE u1
v1
u2
v2
u3
v3
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.
minx∈RE
∑e∈E
cE (e)xe
s. t.∑
e∈δ(v)
xe = 1 ∀v ∈ U ∪ V
x ≥ 0
x ∈ ZE u1
v1
u2
v2
u3
v3
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.
minx∈RE
∑e∈E
cE (e)xe
s. t.∑
e∈δ(v)
xe = 1 ∀v ∈ U ∪ V
x ≥ 0
x ∈ ZE u1
v1
u2
v2
u3
v3
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.
minx∈RE
∑e∈E
cE (e)xe
s. t.∑
e∈δ(v)
xe = 1 ∀v ∈ U ∪ V
x ≥ 0
x ∈ ZE u1
v1
u2
v2
u3
v3
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 HypergraphsDefinitionG = (U,V ,E ) is called a partitioned bipartite hypergraph withmaximum part size d ∈ N if additionally there exist pairwise disjoint≤ d -element sets U1, . . . ,Up and V1, . . . ,Vq called the parts of Hsuch that ·⋃p
i=1Ui = U, ·⋃qi=1Vi = V , and E ⊆
⋃pi=1
⋃qj=1 2
Ui∪Vj , i. e.,every hyperedge intersects only one part in U and one part in V .
v1 v2 v3 v4 v5 v6
u1 u2 u3 u4 u5u5 u6
not partitioned
The Hypergraph Assignment Problem 10 / 23
Partitioned HypergraphsDefinitionG = (U,V ,E ) is called a partitioned bipartite hypergraph withmaximum part size d ∈ N if additionally there exist pairwise disjoint≤ d -element sets U1, . . . ,Up and V1, . . . ,Vq called the parts of Hsuch that ·⋃p
i=1Ui = U, ·⋃qi=1Vi = V , and E ⊆
⋃pi=1
⋃qj=1 2
Ui∪Vj , i. e.,every hyperedge intersects only one part in U and one part in V .
v1 v2 v3 v4 v5 v6
u1 u2 u3 u4 u5u5 u6
not partitioned
The Hypergraph Assignment Problem 10 / 23
Partitioned HypergraphsDefinitionG = (U,V ,E ) is called a partitioned bipartite hypergraph withmaximum part size d ∈ N if additionally there exist pairwise disjoint≤ d -element sets U1, . . . ,Up and V1, . . . ,Vq called the parts of Hsuch that ·⋃p
i=1Ui = U, ·⋃qi=1Vi = V , and E ⊆
⋃pi=1
⋃qj=1 2
Ui∪Vj , i. e.,every hyperedge intersects only one part in U and one part in V .
v1 v2 v3 v4 v5 v6
u1 u2 u3 u4 u5u5 u6
not partitioned
The Hypergraph Assignment Problem 10 / 23
Partitioned HypergraphsDefinitionG = (U,V ,E ) is called a partitioned bipartite hypergraph withmaximum part size d ∈ N if additionally there exist pairwise disjoint≤ d -element sets U1, . . . ,Up and V1, . . . ,Vq called the parts of Hsuch that ·⋃p
i=1Ui = U, ·⋃qi=1Vi = V , and E ⊆
⋃pi=1
⋃qj=1 2
Ui∪Vj , i. e.,every hyperedge intersects only one part in U and one part in V .
v1 v2 v3 v4 v5 v6
u1 u2 u3 u4 u5u5 u6
not partitionedThe Hypergraph Assignment Problem 10 / 23
Results for Partitioned Hypergraphs
Theorem (B., He. [2012])
Every HAP can be polynomially transformed into a HAP on apartitioned hypergraph.
A clique Q ⊆ E is a set of hyperedges such that every pair ofhyperedges in Q has a nonempty intersection.
Theorem (B., He. [2011])
Every clique in a partitioned hypergraph is a subset of the incidenthyperedges δ(P) of some part P.There exists an extended formulation with O(|U|d+1) variables thatimplies all clique inequalities.
The Hypergraph Assignment Problem 11 / 23
Contents
1 Definition and Complexity of the HAP
2 Results for Partitioned Hypergraphs
3 Polyhedral Investigation
4 Heuristics
The Hypergraph Assignment Problem 12 / 23
Polyhedral InvestigationLet G2,3 = (U,V ,E ) be the complete partitioned bipartite hypergraphwith. parts {u11, u12}, {u21, u22}, {u31, u32} in U and. parts {v11, v12}, {v21, v22}, {v31, v32} in V .
u11 u12
v11 v12
u21 u22
v21 v22
u31 u32
v31 v32
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 13 / 23
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∑
e∈E1
xe −∑e∈E2
xe ≤ 1.
. So far we have no normal form.
The Hypergraph Assignment Problem 14 / 23
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 ofthe 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 15 / 23
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 16 / 23
Generalization of Odd Set InequalitiesOdd set cuts for the matching polytope of a graph G = (N,E ),N ′ ⊆ N, |N ′| = 2k + 1 odd (Edmonds [1965]):∑
e∈E :e⊆N′xe ≤ k
or ∑e∈E
⌊|{v ∈ N ′ : e ∈ δ(v)}|
2
⌋xe ≤ k .
Generalization for a hypergraph G = (U,V ,E ), N ′ ⊆ U ∪ V ,|N ′| = 2k + 1 odd:∑
e∈E
⌊|{v ∈ N ′ : e ∈ δ(v)}|
2
⌋xe ≤ k
The Hypergraph Assignment Problem 17 / 23
Generalization of Odd Set InequalitiesOdd set cuts for the matching polytope of a graph G = (N,E ),N ′ ⊆ N, |N ′| = 2k + 1 odd (Edmonds [1965]):∑
e∈E :e⊆N′xe ≤ k
or ∑e∈E
⌊|{v ∈ N ′ : e ∈ δ(v)}|
2
⌋xe ≤ k .
Generalization for a hypergraph G = (U,V ,E ), N ′ ⊆ U ∪ V ,|N ′| = 2k + 1 odd:∑
e∈E
⌊|{v ∈ N ′ : e ∈ δ(v)}|
2
⌋xe ≤ k
The Hypergraph Assignment Problem 17 / 23
Generalization of Odd Set InequalitiesOdd set cuts for the matching polytope of a graph G = (N,E ),N ′ ⊆ N, |N ′| = 2k + 1 odd (Edmonds [1965]):∑
e∈E :e⊆N′xe ≤ k
or ∑e∈E
⌊|{v ∈ N ′ : e ∈ δ(v)}|
2
⌋xe ≤ k .
Generalization for a hypergraph G = (U,V ,E ), N ′ ⊆ U ∪ V ,|N ′| = 2k + 1 odd:∑
e∈E
⌊|{v ∈ N ′ : e ∈ δ(v)}|
2
⌋xe ≤ k
The Hypergraph Assignment Problem 17 / 23
Odd Clique Set Inequalities (B., He.)
Generalization for a hypergraph G = (U,V ,E ), N ′ ⊆ U ∪ V ,|N ′| = 2k + 1 odd:∑
e∈E
⌊|{v ∈ N ′ : e ∈ δ(v)}|
2
⌋xe ≤ k
Replace N ′ by a set of cliques: Q ⊆ 2E , |Q| = 2k + 1 odd number ofcliques in G . Odd clique set cut:∑
e∈E
⌊|{Q ∈ Q : e ∈ Q}|
2
⌋xe ≤ k
The Hypergraph Assignment Problem 18 / 23
Odd Clique Set Inequalities (B., He.)
Generalization for a hypergraph G = (U,V ,E ), N ′ ⊆ U ∪ V ,|N ′| = 2k + 1 odd:∑
e∈E
⌊|{v ∈ N ′ : e ∈ δ(v)}|
2
⌋xe ≤ k
Replace N ′ by a set of cliques: Q ⊆ 2E , |Q| = 2k + 1 odd number ofcliques in G . Odd clique set cut:∑
e∈E
⌊|{Q ∈ Q : e ∈ Q}|
2
⌋xe ≤ k
The Hypergraph Assignment Problem 18 / 23
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 p0 ≤ J ≤ p − r
Ep := {e ∈ E : |{Q ∈ Q : e ∈ Q}| ≥ p}Ep−j := {e ∈ E : |{Q ∈ Q : e ∈ Q}| = p − j}∑
0≤j≤J
(p − r − j)∑
e∈Ep−j
xe ≤ b
do not lead to facets of P(G2,3)
The Hypergraph Assignment Problem 19 / 23
Contents
1 Definition and Complexity of the HAP
2 Results for Partitioned Hypergraphs
3 Polyhedral Investigation
4 Heuristics
The Hypergraph Assignment Problem 20 / 23
Overview of Heuristic Approaches
. Constructive heuristics:I greedy with coefficients per vertexI Hungarian method with vertex groups
. Local search:I Hungarian method with vertex groupsI 2-optI dynamic k-opt
. Perturbation heuristics:I greedy insert with randomization
The Hypergraph Assignment Problem 21 / 23
Computational ResultsResults of first tests:
instance
name
bipartite
hypergraph
arcs
2-hyperedg
es
optimal
value
heuristicresult
gap
runtime(sec.)
Random10 G2,10 400 100 88 88 0% 52.9Random20 G2,20 1600 400 84 85 1.2% 53.7Random35 G2,35 4900 1225 92 129 40.2% 57.8Random50 G2,50 10000 2500 112 144 28.6% 54.4Random75 G2,75 22500 5625 95 140 47.4% 105.8
Random100 G2,100 40000 10000 93 155 66.7% 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 22 / 23
The Hypergraph AssignmentProblem
Olga Heismann
joint work with: Ralf Borndörfer, Achim Hildenbrandt
DFG Research Center MATHEONMathematics for key technologies
January 7–11, 2013