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´ opicos de Otimiza¸ c˜ ao Combinat´ oria 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 P 2 n C 2 n K 2 n K n , n Graphs that have perfect matchings. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Emparelhamentos C 2 n +1 K n , 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 P M ′ M 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. ⇒ G contains no M -augmenting path M is maximum = 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 ⇒ M is maximum G contains no M -augmenting path = Suppose that M 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 ∗ , or 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 D M 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 X M and Y M be the subsets of vertices of X and Y (resp.) that are free of M . Clearly, G contains an M -augmenting path if and only if D M contains an X M Y M -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 D M contains an X M Y M -path. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Maximum matching in bipartite graphs The following algorithm obviously produces a maximum matching of a ( X , Y )-bipartite graph G . MaxBipartiteMatching ( G ) 1. M ← ∅ ⊲ or any matching 2. while there exists an X M Y M -path Q in D M do 3. let P be the augmenting path corresponding to Q M ← M △ E ( P ) 4. 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˝ onig’s theorem Theorem. (K˝ onig, 1931) Let G be a bipartite graph. Then the size of 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 D M , X M and Y M as previously defined. Let R M be the set of vertices reachable from X M in D M . Thus, R M ∩ Y M = ∅ . Orlando Lee – Unicamp Topics in Combinatorial Optimization
K˝ onig’s theorem R M X Y So each edge of M has both ends in R M or none at all. Moreover, no edge of G connects X ∩ R M and Y \ R M . Hence, ( X \ R M ) ∪ ( Y ∩ R M ) is a vertex cover with the same size of 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 S X Y Does there exist a matching that covers X ? Orlando Lee – Unicamp Topics in Combinatorial Optimization
Hall’s condition S X Y 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 S X Y 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 D M , X M and Y M as defined in the MaxBipartiteMatching algorithm. Let R M be the set of vertices reachable from X M in D M . Orlando Lee – Unicamp Topics in Combinatorial Optimization
K˝ onig’s theorem S X Y So each edge of M has both ends in R M or none at all. Moreover, no edge of G connects X ∩ R M and Y \ R M . Let S := R M ∩ X . Then | Γ( S ) | < | S | . Orlando Lee – Unicamp Topics in Combinatorial Optimization
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