Topics in Combinatorial Optimization Orlando Lee Unicamp 15 de - - PowerPoint PPT Presentation

Topics in Combinatorial Optimization

Orlando Lee – Unicamp 15 de abril de 2014

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Agradecimentos

Este conjunto de slides foram preparados originalmente para o curso T´

• picos de Otimiza¸

c˜ ao Combinat´

• ria no primeiro

semestre de 2014 no Instituto de Computa¸ c˜ ao da Unicamp. Preparei os slides em inglˆ es simplesmente porque me deu vontade, mas as aulas ser˜ ao em portuguˆ es (do Brasil)! Agradecimentos especiais ao Prof. M´ ario Leston Rey. Sem sua ajuda, certamente estes slides nunca ficariam prontos a tempo. Qualquer erro encontrado nestes slide ´ e de minha inteira responsabilidade (Orlando Lee, 2014).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matching theory

Matching in graphs is one of subjects of graph theory and combinatorial optimization that accumulated a large body of theory. We will start with the study of matchings in bipartite graphs. We postpone the study of matchings for general graphs for a later part of the course.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matchings

A matching in a graph G = (V , E) is a subset of edges which contains no edges with a common end.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matchings

u v If uv ∈ M then we say that u is matched with v (through M).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matchings

We say that a vertex incident to an edge of M is covered by M. An uncovered vertex is free of M.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matchings

A matching M is perfect if covers every vertex of G.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matchings

P2n C2n K2n Kn,n Graphs that have perfect matchings.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Emparelhamentos

C2n+1 Kn,m Not every graph has a perfect matching.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matchings

A matching is maximal if it cannot be extended to a larger matching. A matching M is maximum if there is no matching larger than M.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Problems of interest

Maximum Matching Problem: given a graph G, find a maximum matching of G. Perfect Matching Problem: given a graph G, find, if it exists, a perfect matching of G. Maximum Weight Matching: given an edge-weighted graph (Ga, w), find a maximum weight matching of Ga. Maximum Weight Perfect Matching: given an edge-weighted graph (G, w), find a maximum weight perfect matching of G.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Alternating paths

Let M be a matching in a graph G. An M-alternating path or just alternating path is a path whose edges are alternatingly in M and out of M.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Augmenting paths

An M-alternating path whose ends are free of M is called M-augmenting path or just augmenting path.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Augmenting paths

M M′ P M′ := M △ EP is a matching with |M′| = |M| + 1 Notation: A △ B := (A ∪ B) − (A ∩ B) (symmetric difference).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Berge’s theorem

• Theorem. (Berge, 1957) A matching M in a graph G is maximum

if and only if G contains no M-augmenting path. Proof. M is maximum = ⇒ G contains no M-augmenting path Suppose that G contains an augmenting path P. Let M′ := M △ EP. Then |M′| = |M| + 1 and M is not maximum.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Berge’s theorem

G contains no M-augmenting path = ⇒ M is maximum Suppose thatM is not maximum and let M∗ be a maximum matching of G. So |M∗| > |M|. let us show that G contains an M-augmenting path. Let H := G[M △ M∗].

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Berge’s theorem

Example: M M∗ H := G[M △ M∗]

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Berge’s theorem

Every vertex in H = G[M △ M∗] has degree one or two. Each component of H is

either a cycle with edges alternating in M and M∗,

• r a path with edges alternating in M and M∗.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Berge’s theorem

Since |M∗| > |M|, H contains more edges from M∗ than from M. Thus, some path component P in H must start and end with edges of M∗. This is an M-aumenting path.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Maximum matching in bipartite graphs

Consider the problem of finding a maximum matching in an (X, Y )-bipartite graph G. We have seen that a matching M is maximum if and only if G contains no M-augmenting paths. However, it would be much better if we had a simple certificate of a maximality of a given matching.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Vertex cover

A vertex cover in a graph G = (V , E) is a subset U of V such that every edge of G has one end in U.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matching versus vertex cover

• Lemma. Let M be a matching and U a vertex cover in a graph G.

Then |M| ≤ |U|. Furthermore, if equality holds then M is a maximum matching and U is a minimum vertex cover.

• Proof. Each edge of M has one end in U. Hence,

|M| ≤ |U|.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matching versus vertex cover

M∗ = U∗ M∗ < U∗ It is not always true that the size of a maximum matching is equal to the size of a minimum vertex cover. We will show that equality holds for bipartite graphs.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Maximum matching in bipartite graphs

Let G be a (X, Y )-bipartite graph. Let M be a matching. Let DM be the digraph obtained from G by orienting each edge e = xy of G (with x ∈ X and y ∈ Y ) as follows: if e ∈ M then orient e as (y, x), if e ∈ M then orient e as (x, y). Let XM and YM be the subsets of vertices of X and Y (resp.) that are free of M. Clearly, G contains an M-augmenting path if and

• nly if DM contains an XMYM-path.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Maximum matching in bipartite graphs

X Y G contains an M-augmenting path if and only if DM contains an XMYM-path.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Maximum matching in bipartite graphs

The following algorithm obviously produces a maximum matching

• f a (X, Y )-bipartite graph G.

MaxBipartiteMatching(G)

• 1. M ← ∅

⊲ or any matching

• 2. while there exists an XMYM-path Q in DM do

3. let P be the augmenting path corresponding to Q 4. M ← M △ E(P)

• 5. return M
• Theorem. (Kuhn, 1955) A maximum matching in a bipartite graph

can be found in time O(nm).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

K˝

• nig’s theorem
• Theorem. (K˝
• nig, 1931) Let G be a bipartite graph. Then the size
• f a maximum matching is equal to the size of a minimum vertex
• cover. Moreover, given a maximum matching, we can find a

minimum vertex cover of G in time O(m).

• Proof. We have seen that the maximum is at most the minimum.

Let M be a maximum matching and let DM, XM and YM as previously defined. Let RM be the set of vertices reachable from XM in DM. Thus, RM ∩ YM = ∅.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

K˝

• nig’s theorem

X Y RM So each edge of M has both ends in RM or none at all. Moreover, no edge of G connects X ∩ RM and Y \ RM. Hence, (X \ RM) ∪ (Y ∩ RM) is a vertex cover with the same size

• f M.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hall’s condition

X Y Does there exist a matching that covers X?

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hall’s condition

X Y S Does there exist a matching that covers X?

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hall’s condition

X Y S Γ(S) Γ(S) := set of neighbors of S |Γ(S)| < |S| = ⇒ no matching covers X

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hall’s condition

X Y S Γ(S) If there exists a matching that covers X, then |Γ(S)| ≥ |S| for all S ⊆ X.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hall’s theorem

• Theorem. (Hall, 1935)

A (X, Y )-bipartite graph has a matching that covers X if and only if |Γ(S)| ≥ |S| for all S ⊆ X.

• Proof. We have seen that condition is necessary. Let us show

sufficiency. Let M be a maximum matching. If it covers X then the result follows. Let DM, XM and YM as defined in the MaxBipartiteMatching

• algorithm. Let RM be the set of vertices reachable from XM in DM.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

K˝

• nig’s theorem

X Y S So each edge of M has both ends in RM or none at all. Moreover, no edge of G connects X ∩ RM and Y \ RM. Let S := RM ∩ X. Then |Γ(S)| < |S|.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Some consequences

• Corollary. There exists an O(nm) algorithm that given an

(X, Y )-bipartite graph, returns either a matching that covers X or a subset of X that violates Hall’s condition.

• Corollary. (Frobenius, 1917) A (X, Y )-bipartite graph with

|X| = |Y | has a perfect matching if and only if |Γ(S)| ≥ |S| for all S ⊆ X. A graph G is k-regular if every vertex has degree k. A graph G is regular if is k-regular for some k 0.

• Corollary. A regular bipartite graph has a perfect matching.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Equivalence between Menger’s and K˝

• nig’s theorems

We have seen that Menger’s theorem implies K˝

• nig’s theorem.

More precisely, we have shown how to reduce the maximum bipartite matching problem to the problem of finding a maximum packing of st-paths in a digraph D. Moreover, the reduction is polynomial. The reverse reduction is also possible, but much more

• difficult. See Schrijver’s book (chapter 16).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Maximum weight matching

Maximum weight matching problem: given a bipartite graph G = (V , E) and an edge-weight function w : E → Q+, find a maximum weight matching. Maximum weight perfect matching problem (MWPM): given a bipartite graph G = (V , E) containing a perfect matching and an edge-weight function w : E → Q, find a maximum weight perfect matching. The first problem can be easily reduced to the second one. So we will describe a solution for the second problem.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Incidence matrix of a graph

Let G = (V , E) be a graph. The incidence matrix of G is the {0, 1}-matrix M with rows indexed by V and columns indexed by E such that: M[v, e] = 1 if e is incident to v,

• therwise.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

ILP formulation

The following ILP is equivalent to the MWPM. max

• e∈E w(e)x(e)

s.t. x(δ(v)) = 1 for every v ∈ V , x(e) ∈ {0, 1} for every e ∈ E. This is equivalent to: max wx s.a Mx = 1 x ∈ {0, 1}E

Orlando Lee – Unicamp Topics in Combinatorial Optimization

LP formulation

Consider the relaxed linear programming (LP). max

• e∈E w(e)x(e)

s.t. x(δ(v)) = 1 for every v ∈ V , x(e) 0 for every e ∈ E. This is equivalent to: max wx s.a Mx = 1 x 0

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Dual problem

The dual problem (DP) is the following. min

• v∈V y(v)

s.t. y(u) + y(v) w(e) for every e = uv ∈ E, Or equivalently, min y1 s.a yM w We say that a feasible dual solution of (DP) is a w-cover.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Weak duality theorem

• Lemma. Let M be a perfect matching and let y be a w-cover.

Then w(M) y(V ).

• Proof. Just observe that

w(M) =

• e∈M

w(e)

• e=uv∈M

(y(u) + y(v)) =

• v∈V

y(v) = y(V ). Or, in matricial notation, for x = χM, we have wx yMx = y1.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hungarian method

Egerv´ ary (1931) proved that the strong duality holds even if we require x to be integral. Kuhn (1955) described the first (strongy) polynomial algorithm for the MWPM (inspired on Egerv´ ary’s proof). We present this algorithm which is often called the Hungarian method. As a byproduct, we have that the polytope of the LP formulation is integral, namely, every extreme point (vertex)

• f the polytope is integral.

Moreover, if the weight function w is integral, then there exists an optimal dual solution (w-cover) which is integral.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Complementary slackness

The primal and dual problems are max

• e∈E w(e)x(e)

s.t. x(δ(v)) = 1 for every v ∈ V , x(e) 0 for every e ∈ E. and min

• v∈V y(v)

s.t. y(u) + y(v) w(e) for every e = uv ∈ E, Complementary Slackness x(e) > 0 ⇒ y(u) + y(v) = w(e) for every e = uv ∈ E.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Complementary slackness

So two solutions x, y of the primal and dual formulation are both

• ptimal if and only if they satisfy complementary slackness

conditions x(e) > 0 ⇒ y(u) + y(v) = w(e) for every e = uv ∈ E. Let y be a w-cover. Let ¯ w(e) := y(u) + y(v) − w(e) for each e = uv ∈ E. So ¯ w(e) 0 for each edge e ∈ E. Let E = := {e ∈ E : ¯ w(e) = 0}. We say that an edge in E = is tight.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Subgraph of tight edges

Complementary slackness x(e) > 0 ⇒ y(u) + y(v) = w(e) for every e = uv ∈ E. If x = χM for some perfect matching then this is equivalent to: M ⊆ E =. So we could try to find a perfect matching M in the equality subgraph G = = (V , E =) using MaxBipartiteMatching. If it finds a perfect matching M then it has maximum weight and we are done.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Updating the dual variables

X Y S If no perfect matching exists in G =, then there exists S ⊆ X such that |ΓG =(S)| < |S|.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Updating the dual variables

X Y S Since G has a perfect matching, there must exist some edge e in G from S to Y \ ΓG =(S). Clearly, ¯ w(e) = y(u) + y(v) − w(e) > 0 where e = uv.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Updating the dual variables

X Y S Hence ǫ := min{¯ w(e) : e = uv, u ∈ S, v ∈ Y \ ΓG =(S)}> 0.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Updating the dual variables

X Y −ǫ −ǫ −ǫ −ǫ −ǫ +ǫ +ǫ +ǫ UpdateDual(y, w, S)

• 1. ǫ := min{¯

w(e) : e = uv, u ∈ S, v ∈ Y \ ΓG =(S)}

• 2. for each v ∈ S do y(v) ← y(v) − ǫ
• 3. for each v ∈ ΓG =(S) do y(v) ← y(v) + ǫ
• 4. return y

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Updating the dual variables

• Lemma. If y is a feasible dual solution and S is a set violating

Hall’s condition in G =, then the vector y ′ returned by UpdateDual is also a feasible dual solution. Moreover, (a) every tight edge for y remains tight for y ′ (b) at least one new edge with an end in Y \ ΓG =(S) becomes tight, (c) if w is integral and y is integral, then y′ is also integral. .

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hungarian algorithm – first version

HungarianAlgorithm(G, X, Y , w)

• 1. let y be a w-cover
• 2. compute G =
• 3. while TRUE do

4. if G = has a perfect matching M 5. then 6. return M and y 7. else 8. let S a subset of X violating Hall’s condition 9. UpdateDual(y, w, S) 10. update G =

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Perfect matching in G =

We have seen how to find either a perfect matching or a set violating Hall’s condition. In order to analyze the efficiency of the algorithm, it is more convenient if we explicit this process in the code. Recall the algorithm MaxBipartiteMatching. For a matching M of G = let DM, XM and YM as previously defined but now with respect to G = and not G. So G = has an augmenting path if and only if there exists an XMYM-path in DM.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hungarian algorithm – second version

HungarianAlgorithm(G, X, Y , w)

• 1. let y be a w-cover
• 2. M ← ∅
• 3. compute G =, DM, XM, YM
• 4. while M is not perfect do

5. if there exists an XMYM-path P in G = 6. then M ← M △ E(P) 7. else 8. let S the set of vertices in X reachable from XM 9. UpdateDual(y, w, S) 10. update G =

• 11. return M and y

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Analysis

An initial feasible w-cover is y(v) = max{w(e) : e ∈ E} for v ∈ X and y(v) = 0 for v ∈ Y . So if w is integral, then y is integral throughout the execution of the algorithm. When M is a perfect matching in G = then M and y satisfy complementary slackness and both are optimal solutions. By (a) and (b) from the lemma, after an update of y and G =, at least one new vertex becomes reachable from XM. Hence the number of consecutive updates in lines 9-10 until an augmentation

• ccurs in line 5-6 occurs is at most |X|.

The number of augmentations is at most |X|. An XY -path can be found in linear time by breadth-first search.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Consequences

• Theorem. Given a weighted bipartite graph (G, w) containing one

perfect matching, HungarianAlgorithm returns a maximum perfect matching M and a minimum w-cover y in time O(nm2). Moreover, if w is integral, then y is also integral.

• Corollary. Let G be a bipartite graph containing a perfect
• matching. Then every extreme point of the polytope

x(δ(v)) = 1 for every v ∈ V , x(e) 0 for every e ∈ E is integral.

• Exercise. Find a very small non-bipartite graph for which the above

polytope is not integral.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

References

The proofs of Hall’s and K˝

• nig’s theorem can be found in several
• books. The proofs given here are essentially the proofs in Bondy

and Murty’s book. I used the definition of DM in Schrijver’s book (page 264) to simplify some algorithmic aspects (but probably this was used before). For the Hungarian Algorithm, I followed Franks’s book (page 111–113). J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, Norht-Holland, 1976.

• A. Schrijver, Combinatorial Optimization, Vol. A, Springer.
• A. Frank, Connections in Combinatorial Optimization, Oxford.

Orlando Lee – Unicamp Topics in Combinatorial Optimization