The Hypergraph Assignment Problem · The Hypergraph Assignment Problem Olga Heismann jointworkwith:...

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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

4 Heuristics

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.

. . . 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.

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.

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.

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

Complexity Results

minx∈RE

∑e∈E

cE (e)xe

s. t.∑

e∈δ(v)

xe = 1 ∀v ∈ U ∪ V

x ≥ 0

x ∈ ZE u1

Complexity Results

minx∈RE

∑e∈E

cE (e)xe

s. t.∑

e∈δ(v)

xe = 1 ∀v ∈ U ∪ V

x ≥ 0

x ∈ ZE u1

Complexity Results

minx∈RE

∑e∈E

cE (e)xe

s. t.∑

e∈δ(v)

xe = 1 ∀v ∈ U ∪ V

x ≥ 0

x ∈ ZE u1

Contents

4 Heuristics

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

i=1Ui = U, ·⋃qi=1Vi = V , and E ⊆

⋃pi=1

⋃qj=1 2

v1 v2 v3 v4 v5 v6

u1 u2 u3 u4 u5u5 u6

not partitioned

i=1Ui = U, ·⋃qi=1Vi = V , and E ⊆

⋃pi=1

⋃qj=1 2

v1 v2 v3 v4 v5 v6

u1 u2 u3 u4 u5u5 u6

not partitioned

i=1Ui = U, ·⋃qi=1Vi = V , and E ⊆

⋃pi=1

⋃qj=1 2

v1 v2 v3 v4 v5 v6

u1 u2 u3 u4 u5u5 u6

not partitionedThe Hypergraph Assignment Problem 10 / 23

Results for Partitioned Hypergraphs

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.

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.

Contents

4 Heuristics

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.

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.

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”)

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)}|

⌋xe ≤ k .

Generalization for a hypergraph G = (U,V ,E ), N ′ ⊆ U ∪ V ,|N ′| = 2k + 1 odd:∑

⌊|{v ∈ N ′ : e ∈ δ(v)}|

⌋xe ≤ k

or ∑e∈E

⌊|{v ∈ N ′ : e ∈ δ(v)}|

⌋xe ≤ k .

⌊|{v ∈ N ′ : e ∈ δ(v)}|

⌋xe ≤ k

or ∑e∈E

⌊|{v ∈ N ′ : e ∈ δ(v)}|

⌋xe ≤ k .

⌊|{v ∈ N ′ : e ∈ δ(v)}|

⌋xe ≤ k

Odd Clique Set Inequalities (B., He.)

⌊|{v ∈ N ′ : e ∈ δ(v)}|

⌋xe ≤ k

Replace N ′ by a set of cliques: Q ⊆ 2E , |Q| = 2k + 1 odd number ofcliques in G . Odd clique set cut:∑

⌊|{Q ∈ Q : e ∈ Q}|

⌋xe ≤ k

Odd Clique Set Inequalities (B., He.)

⌊|{v ∈ N ′ : e ∈ δ(v)}|

⌋xe ≤ k

Replace N ′ by a set of cliques: Q ⊆ 2E , |Q| = 2k + 1 odd number ofcliques in G . Odd clique set cut:∑

⌊|{Q ∈ Q : e ∈ Q}|

⌋xe ≤ k

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)

Contents

4 Heuristics

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

Computational ResultsResults of first tests:

instance

bipartite

hypergraph

2-hyperedg

optimal

heuristicresult

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 AssignmentProblem

Olga Heismann

joint work with: Ralf Borndörfer, Achim Hildenbrandt

DFG Research Center MATHEONMathematics for key technologies

January 7–11, 2013

The Hypergraph Assignment Problem · The Hypergraph Assignment Problem Olga Heismann jointworkwith:...

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