Combinatorial Optimization and Graph Theory ORCO Applications of submodular functions

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Combinatorial Optimization and Graph Theory ORCO

Applications of submodular functions

Zolt´an Szigeti

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Applications of submodular functions

Planning

1 Definitions, Examples,

2 Uncrossing technique,

3 Splitting off technique,

4 Constructive characterization,

5 Orientation,

6 Augmentation,

7 Submodular function minimization.

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Submodular functions

Definitions

1 A function m: 2^S →Ris modular if for allX,Y ⊆S, m(X) +m(Y) =m(X ∩Y) +m(X ∪Y).

2 A function b: 2^S →R∪ {+∞} issubmodularif for allX,Y ⊆S, b(X) +b(Y)≥b(X ∩Y) +b(X ∪Y).

3 A function p: 2^S →R∪ {−∞} issupermodular if for allX,Y ⊆S, p(X) +p(Y)≤p(X∩Y) +p(X ∪Y).

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Modular functions

Examples for Modular functions

1 m(X) =k: constant function, where X ⊆S,k ∈R,

2 m(X) =|X|: cardinalityfunction on a set S,

3 m(X) =m(∅) +P

v∈Xm(v): whereX ⊆S,m(∅),m(v) ∈R ∀v ∈S.

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Submodular functions

Examples for Submodular functions

1 d_G(X): degreefunction of an undirected graphG,

(byd_G(X) +d_G(Y) =d_G(X∩Y) +d_G(X∪Y) + 2dG(X\Y,Y \X))

2 d_D⁺(X) :out-degree function of a directed graphD,

3 d_g⁺(X) :capacityfunction of a network (D,g),

4 |Γ(X)|:number ofneighborsofX in a bipartite graph, (by modularity of | · |,Γ(X)∪Γ(Y) = Γ(X ∪Y) andΓ(X)∩Γ(Y)⊇Γ(X ∩Y)),

5 r(X) : rankfunction of a matroid,

6 r₁(X) +r₂(S\X) :for rank functionsr₁ andr₂ of two matroids on S,

7 g(|X|) : for aconcavefunctiong :R+ →R+.

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Supermodular functions

Examples for Supermodular functions

1 |E(X)|:where E(X) is the set ofedges ofG insideX ⊆V, (by |E(X)|= ¹₂(P

v∈X dG(v)−dG(X)),

2 c_G(F) :the number of connected componentsof the subgraph of G = (V,E)induced by F ⊆E.

(by c_G(F) =|V| −r_G(F), where r_G is the rank function of the forest matroid of G).

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Uncrossing technique: Flows

Theorem

In a network (D,g), the intersection and the union of twominimum capacity(s,t)-cuts are minimum capacity(s,t)-cuts.

Proof:

1 Let X andY be two(s,t)-cuts of capacitymin.

2 Then d_g⁺(X) =min andd_g⁺(Y) =min.

3 Since X∩Y andX ∪Y are (s,t)-cuts,

4 d_g⁺(X ∩Y)≥min andd_g⁺(X ∪Y)≥min.

5 min+min=d_g⁺(X) +d_g⁺(Y)≥d_g⁺(X∩Y) +d_g⁺(X ∪Y)

≥min+min by (2), submodularity and (4).

6 Hence equality holds everywhere: d_g⁺(X ∩Y) =minand d_g⁺(X ∪Y) =min.

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Uncrossing technique: Matchings

Theorem (Frobenius)

A bipartite graph B = (U,V;E) has a matching coveringU if and only if (∗) |Γ(X)| ≥ |X|for allX ⊆U.

Proof:

1 We show only the difficult direction.

2 We call a setX ⊆U tight if |Γ(X)|=|X|.

3 If X andY are tight, then X∩Y andX ∪Y are also tight:

1 By the tightness ofX andY, the submodularity of|Γ(·)|,(∗)and the modularity of| · |, we have

|X|+|Y|=|Γ(X)|+|Γ(Y)|

≥ |Γ(X∩Y)|+|Γ(X∪Y)|

≥ |X ∩Y|+|X∪Y|

=|X|+|Y|,

2 hence equality holds everywhere andX∩Y andX ∪Y are tight.

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Uncrossing technique: Matchings

Proof:

4 We may suppose that after deleting any edge of B,(∗)doesn’t hold anymore.

5 Then every edge uv of B enters a tight setXuv such thatu is the only neighbor of v in Xuv:

1 Since after deletinguv fromB,(∗)doesn’t hold,

2 ∃Xuv ⊆U:|Xuv| −1≥ |ΓB−uv(Xuv)|.

3 Moreover, |Γ_B−uv(Xuv)| ≥ |Γ_B(Xuv)| −1, and

4 by(∗), |Γ_B(Xuv)| −1≥ |X_uv| −1,

5 hence equality holds everywhere, that is

6 Xuv is tight anduis the only neighbor ofv inXuv.

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Uncrossing technique: Matchings

Proof:

6 We show that every vertex ofU is of degree 1 in B.

1 Suppose thatu∈U is incident to two edgesuv anduw inB.

2 By (5),X:=Xuv∩Xuw is tight,uis unique neighbor ofv (ofw) inX.

3 Then, by(∗)and the tightness ofX, we have a contradiction:

|X| −1 =|X\u| ≤ |ΓB(X \u)| ≤ |ΓB(X)| −2 =|X| −2.

7 Two vertices u andu^′ inU can not have a common neighbor since

|ΓB({u,u^′})| ≥2.

8 By (6) and (7), E is a matching ofB coveringU.

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Uncrossing technique: General lemma

Definitions:

1 A graphG coversa function p onV if d_G(X)≥p(X) for allX ⊆V.

2 X ⊆V is tightif d_G(X) =p(X).

3 Two sets X and Y ofV are crossing if none ofX \Y,Y \X,X∩Y andV \(X∪Y) is empty.

4 A function iscrossing supermodularif the supermodular inequality holds for any crossing sets X andY.

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Uncrossing technique: General lemma

Uncrossing Lemma

If G covers a crossing supermodular functionp then the intersection and the union of crossing tight sets are tight.

Proof:

1 Let X andY be two crossing tight sets ofV.

2 Since they are tight,d_G(·) is submodular,G coversp andp is crossing supermodular, we have

p(X) +p(Y) =d_G(X) +d_G(Y)

≥d_G(X ∩Y) +d_G(X ∪Y)

≥p(X∩Y) +p(X∪Y)

≥p(X) +p(Y),

3 hence equality holds everywhere and the lemma follows.

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Uncrossing technique for minimum tight sets

Definitions:

1 A graph G is calledk-edge-connectedifd_G(X)≥k ∀∅ 6=X ⊂V(G).

2 G isminimallyk-edge-connected if

1 G isk-edge-connected and

2 for each edgee ofG, G−e is notk-edge-connected anymore.

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Uncrossing technique for minimum tight sets

Theorem (Mader)

A minimally k-edge-connected graphG has a vertex of degree k.

Proof:

1 Let p(X):=k if∅ 6=X ⊂V and0 otherwise.

2 Then p is crossing supermodular.

3 Since G is k-edge-connected, d_G(X)≥k =p(X), so G coversp.

4 By minimality of G, each edge ofG enters a tight set.

5 Let X be a minimal non-empty tight set.

6 Suppose that X is not a vertex.

7 By minimality of X,there exists an edge uv in X.

8 Let Y be a tight set thatuv enters.

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Uncrossing technique for minimum tight sets

Proof:

9 By minimality of X,X andY are crossing.

1 Sinceuv entersY, we may suppose thatu∈X∩Y andv ∈X\Y.

2 By the minimality ofX,X∩Y is not tight, soY\X 6=∅.

3 By the minimality ofX,X\Y is not tight, soV \(X ∪Y)6=∅.

10 Then, by the Uncrossing Lemma, X∩Y is a tight set that contradicts the minimality of X.

11 Then X =v andd_G(v) =p(v) =k.

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Uncrossing technique for minimum tight sets

Definitions:

1 A directed graph D isk-arc-connected ifd_D⁺(X)≥k ∀∅ 6=X ⊂V.

2 D isminimallyk-arc-connected if

1 D isk-arc-connected and

2 for each arce ofD,D−eis notk-arc-connected anymore.

Theorem (Mader)

A minimally k-arc-connected directed graph has a vertex of in- and out-degree k.

Remark

1 One can easily show that there exists a vertex of in-degree k and a vertex of out-degree k but

2 it is not so easy to see that there exists a vertex with both in- and out-degree k.

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Splitting off technique

Definitions: for G := (V ∪s,E)

1 Operation splitting off at s: for su,sv ∈E,we replacesu,sv by an edge uv, that isG_uv := (V ∪s,(E\ {su,sv})∪ {uv}).

2 Operation complete splitting off ats:

1 dG(s)is even,

2 dG(s)

2 splitting off atsand

3 deleting the vertexs.

3 The graphG isk-edge-connected inV ifd_G(X)≥k ∀∅ 6=X ⊂V.

Definitions

s s

u u

v v

G G_uv

✲ Splitting off

V V

s s

u u

v v

G G^′

✲ Splitting offComplete

V V

w z

Theorem (Mader)

If D = (V ∪s,A) is k-arc-connected (k ≥1) andd_D⁺(s) =d_D⁻(s), then there is a complete directed splitting off ats preserving k-arc-connectivity.

Proof

Similar to previous one.

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Constructive characterization

Theorem (Lov´asz)

A graph is2k-edge-connected if and only if it can be obtained from K₂^2k by a sequence of the following two operations:

(a) adding a new edge,

Proof:

1 We show that G can be reduced toK₂^2k via 2k-edge-connected graphs by the inverse operations:

1 deleting an edge and

2 complete splitting off at a vertex of degree2k.

2 While G 6=K₂^2k repeat the following.

1 By deleting edges we get a minimally2k-edge-connected graph.

2 By Theorem of Mader, it contains a vertex of degree2k.

3 By Theorem of Lov´asz, there exists a complete splitting off at that vertex that preserves2k-edge-connectivity.

4 LetG be the graph obtained after this complete splitting off.

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Constructive characterization

Theorem (Mader)

For k ≥1,a graph is k-arc-connected if and only if it can be obtained from K₂^k,k, the directed graph on2 vertices withk arcs between them in both directions, by a sequence of the following two operations:

1 adding a new arc,

2 pinching k arcs.

Proof

Similar to previous one, by applying Mader’s results on

1 minimallyk-arc-connected graphs and,

2 complete directed splitting off.

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Orientation

Theorem (Nash-Williams)

G has a k-arc-connected orientation if and only if G is2k-edge-connected.

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Orientation

Necessity :

X V−X k

k

G~

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Edge-connectivity augmentation problem:

Given a graph G = (V,E) andk ∈Z+, what is the minimum numberγ of new edges whose addition results in ak-edge-connected graph?

Theorem (Watanabe-Nakamura)

Let G = (V,E) be a graph and k≥2an integer.

min{|F|: (V,E ∪F) isk-edge-conn.}=₁

2maxP

X∈X(k−d_G(X)) , where X is a subpartition of V.

GraphG andk = 4

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Augmentation

2maxP

1

Deficient sets, deficiency = 4−d_G(X)

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2maxP

X∈X(k−dG(X)) , where X is a subpartition of V.

1

2

Opt ≥ ⌈⁵₂⌉= 3

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Augmentation

2maxP

1

2

Graph G +F is 4-edge-connected and|F|= 3

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Augmentation

2maxP

X∈X(k−dG(X)) , where X is a subpartition of V.

1

2

Opt =⌈¹₂maximum deficiency of a subpartition of V⌉

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Augmentation

Proof:

1 First we provide the lower bound onγ.

2 Suppose that G is not k-edge-connected.

3 This is because there is a set X of degreed_G(X) less thank.

4 Then the deficiency ofX isk −dG(X),that is, we must add at least k−d_G(X) edges between X andV \X.

5 Let {X1, . . . ,X_ℓ} be a subpartition of V.

6 The deficiency of {X₁, . . . ,X_ℓ} is the sum of the deficiencies of X_i’s.

7 By adding a new edge we may decrease the deficiency of at most two Xi’s so we may decrease the deficiency of{X1, . . . ,X_ℓ}by at most2,

8 hence we obtain the following lower bound:

γ ≥α:=⌈half of the maximum deficiency of a subpartition of V⌉.

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Augmentation

Frank’s algorithm

1 Minimal extension,

1 Add a new vertex s,

2 Add a minimum number of new edges incident tos to satisfy the edge-connectivity requirements,

3 If the degree ofsis odd, then add an arbitrary edge incident to s.

2 Complete splitting off preserving the edge-connectivity requirements.

G = (V,E)

✲ w Extension

s

Minimal v

u z

G^′andG^′′arek-e-c inV

✲

Complete Splitting off v

u w z

G^∗isk-e-c

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Augmentation

Minimal extension:

1 Add a new vertex s toG and connect it to each vertex ofG by k edges. The resulting graph is k-edge-connected inV.

2 Delete as many new edges as possible preserving k-edge-connectivity in V to getG^′ = (V ∪s,E∪F^′).

3 If d_G′(s) is odd, then add an arbitrary new edge incident tos to get G^′′= (V ∪s,E ∪F^′′) that is k-edge-connected inV andd_G′′(s) is even.

G = (V,E)

✲ w Extension

s

Minimal v

u z

✲

u w z

G^∗isk-e-c

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Augmentation

Splitting off:

1 By Theorem of Lov´asz, there exists in G^′′a complete splitting off ats that preserves k-edge-connectivity.

2 This way we obtain a k-edge-connected graphG^∗= (V,E∪F) with

|F|= ^|F₂^′′^| =⌈^|F₂^′^|⌉.

G = (V ,E)

w Extension✲

s

Minimal v

u z

✲

u w z

G^∗isk-e-c

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Augmentation

Optimality:

1 In G^′,no edge incident tos can be deleted without violating k-edge-connectivity inV, so each edgee ∈F^′ enters a maximal proper subsetXe in V of degree k,that is, dG(Xe) +d_F′(Xe) =k.

2 ByUncrossing Lemma,these sets form a subpartition{X1, . . . ,Xℓ}ofV.

1 Suppose thatXi∩Xj 6=∅.

2 Then, by Uncrossing Lemma and the maximality ofXi,Xi∪Xj =V.

3 Byk+k =dG^′(Xi) +dG^′(Xj)

=dG^′(Xi\Xj) +dG^′(Xj\Xi) + 2dG^′(Xi∩Xj,Xi∪Xj)

≥k+k+ 0,

4 dG^′(Xi\Xj) =k=dG^′(Xj\Xi)and every edge incident tos enters eitherXi\Xj orXj\Xi, that is{Xi\Xj,Xj\Xi}is the required subpartition.

3 γ ≤ |F|=⌈^|F₂^′^|⌉=⌈¹₂Pℓ

1d_F′(Xi)⌉=⌈¹₂Pℓ

1(k−d_G(Xi))⌉ ≤α ≤γ.

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Augmentation

Theorem (Frank)

Let D= (V,A) be a directedgraph andk ≥1an integer.

min{|F|: (V,A∪F) is k-arc-connected}= max{P

X∈X(k−d_D⁺(X)),P

X∈X(k−d_D⁻(X))}

where X is a subpartition of V.

Proof

Similar to previous one, by applying Mader’s directed splitting off theorem.

Generalizations

1 localedge-connectivity; polynomially solvable,

2 hypergraphs; polynomially solvable,

3 partition constrained; polynomially solvable,

4 weighted; NP-complete even fork = 2.

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Submodular function minimization

Theorem (Gr¨otschel-Lov´asz-Schrijver, Fujishige-Fleicher-Iwata, Schrijver) The minimum value of a submodular function can be found in poly. time.

Corollary: One can decide in polynomial time whether

1 a graphG isk-edge-connected

(by minimizing d_G(X ∪u)X ⊆V −v ∀u,v ∈V),

2 a network (D,g) has a feasible flow of valuek (by minimizing d_g⁺(Z∪s) Z ⊆V \ {s,t}),

3 a bipartite graph G has a perfect matching (by minimizing |Γ(X)| − |X|),

4 two matroids have a common independent set of size k (by minimizing r₁(X) +r₂(S−X)),

5 a digraph D has a packing of k spannings-arborescences (by minimizing d_D⁻(X∪u) X ⊆V −s ∀u∈V −s).