Topics in Combinatorial Optimization Orlando Lee – Unicamp 20 de maio 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
Dicuts and dijoins Let D = ( V , A ) be a digraph. A dicut in D is a subset F of arcs such that there exists ∅ � = X ⊂ V with F = δ in ( X ) and δ out ( X ) = ∅ . We say that X is the in-shore of the dicut. A dijoin in D is a subset F of arcs such that F ∩ C � = ∅ for every dicut C , that is, F intersects every dicut. Lemma. Let C be a dicut and let F be a collection of disjoint dijoins. Then | C | ≥ |F| . Orlando Lee – Unicamp Topics in Combinatorial Optimization
Packing dijoins Conjecture. (Woodall, 1978) Let D = ( V , A ) be a digraph. Then the size of a minimum dicut is equal to the size of a maximum packing of dijoins. Equivalently, we have the following. Conjecture. (Woodall, 1978) Let D = ( V , A ) be a digraph such that every dicut contains at least k arcs. Then D contains k disjoint dijoins. For k = 1 this is trivial. For k = 2 it follows from the fact that if F is a minimal dijoin then A − F is also a dijoin. For k � 3 the problem is still open. This is one of the most challenging open problems of the area. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Packing dijoins Let κ ( D ) denote the size of a minimum dicut of D . The following somewhat weaker versions of Woodall’s conjecture are open as well. Conjecture. There exists some integer k such that every digraph with κ ( D ) � k contains 3 disjoint dijoins. Conjecture. For every integer k � 1, there exists some integer f ( k ) such that every digraph with κ ( D ) � f ( k ) contains k disjoint dijoins. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Planar case Woodall’s conjecture restricted to planar digraphs is still open as well. By planar duality, we have the following. Conjecture. Let D be a planar digraph. Then the size of a shortest cycle is equal to the size of a maximum collection of disjoint transversal of cycles. If we remove the hypothesis of planarity, then the statement above is false. Orlando Lee – Unicamp Topics in Combinatorial Optimization
The weighted version Let D = ( V , A ) be a digraph and let c : A �→ Z + be an arc weight function. We say that a collection F of dijoins is a c -packing of dijoins if each arc a belongs to at most c ( a ) dijoins in F . The following is a natural extension of Woodall’s conjecture. Conjecture. (Edmonds-Giles, 1977, and Younger) Let D = ( V , A ) be a digraph and let c : A �→ Z + be an arc weight function. Then the minimum weight of a dicut is equal to the maximum size of a c -packing of dijoins. However, Schrijver (1980) found a counterexample to this conjecture. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Counterexample for the weighted version y x z The blue arcs have weight 1 and others have weight 0. The weight of a minimum dicut is 2. The set F of blue arcs cannot be split into two dijoins F 1 and F 2 . Orlando Lee – Unicamp Topics in Combinatorial Optimization
Counterexample for the weighted version (Schrijver) Suppose for a contradiction that F can be split into two dijoins F 1 and F 2 . Note that there exist three sources and three sinks, each one with exactly two blue arcs incident to them. So each F i , if they exist, must contain exactly one blue arc incident to each one of those sources and sinks. Moreover, each F i contains at least one of the arcs x , y , z , because they are in the dicut leaving the internal hexagon. Hence, by symmetry we may assume that F 1 contains x , y but not z . But then, F 1 does not intersect the dicut that goes from the right half to the left half of the figure. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Counterexample for the weighted version Obviously if the weighted version were true then Woodall’s conjecture would be true. On the other hand, unlike some other results we have seen, we do not know if the converse is true. If all arcs have positive weight then we can replace each arc a by c ( a ) parallel arcs on the same ends. The problem are the arcs with weight zero. What should we do with them? We cannot delete them because this could introduce new dicuts. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Counterexample for the weighted version Cornu´ ejols and Guenin (2002) found two other counterexamples for Edmonds-Giles conjecture (by computer). Aaron Williams (2003) in his master’s thesis studied several operations that a minimal counterexample must satisfy. These three digraphs (Schrijver’s and Cornu´ ejols and Guenin’s) are the only known minimal counterexamples (under a set of specific operations). Conjecture. Let ( D , c ) be a minimal counterexample for the Edmonds-Giles conjecture. Then the weight of a minimum dicut in ( D , c ) is two. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Source-sink connected digraphs A digraph D is source-sink connected if for every source s and every sink t , there exists an st -path in D . Schrijver (1982) and independently Feofiloff fand Younger (1987) proved that Edmonds-Giles conjecture holds for source-sink connected digraphs. Theorem. Let D = ( V , A ) be a source-sink connected digraph and let c : A �→ Z + be an arc weight function. Then the minimum weight of a dicut is equal to the maximum size of a c -packing of dijoins. This is the most important result related to this problem, but we will not prove it here. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Guenin’s conjecture A super-source in a digraph is a source that can reach every sink. A super-sink in a digraph is a sink that can be reached by every source. Conjecture. (Guenin, 2002) Let D = ( V , A ) be a digraph that contains a super-source and a super-sink, and let c : A �→ Z + be an arc weight function. Then the minimum weight of a dicut is equal to the maximum size of a c -packing of dijoins. Orlando Lee – Unicamp Topics in Combinatorial Optimization
A special case One special case of source-sink connected digraph is one that contains a spanning arborescence. Proposition. Let D = ( V , A ) be a digraph containing a spanning r -arborescence. Then the size of a minimum dicut is equal to the size of a maximum packing of dijoins. Proof. Let k be the size of a minimum dicut. Add to D , for each arc ( u , v ) k parallel reverse arcs ( v , u ). Let D ′ be the resulting digraph. Let X be an ¯ r -set. If X is an in-shore of D , then d in D ′ ( X ) = d in D ( X ) � k . Otherwise, there exists k parallel arcs entering X and hence d in D ′ ( X ) � k . Thus, by Edmonds’ arborescence theorem, there exists k spanning r -arborescences in D ′ , say B 1 , . . . , B k . Orlando Lee – Unicamp Topics in Combinatorial Optimization
A special case We have that B i ∩ A is a dijoin for i = 1 , . . . , k . In fact, let δ in D ( X ) be a dicut in D . Then δ in D ′ ( X ) = δ in D ( X ) and hence δ in B i ∩ A ( X ) = δ in B ( X ) � = ∅ . So each B i is a dijoin and the result holds. Theorem. Let D = ( V , A ) be a digraph containing a spanning r -arborescence and let c : A �→ Z + be an arc weight function. Then the minimum weight of a dicut is equal to the maximum size of a c -packing of dijoins. The same idea works here by applying the weighted version of Edmonds’ arborescence theorem. Orlando Lee – Unicamp Topics in Combinatorial Optimization
Other special cases Edmonds-Giles conjecture holds for the following classes of digraphs. (a) Series-parallel digraphs (underlying graph has no minor of K 4 ) (O. Lee). (b) Digraphs whose underlying graph has no minor of K 5 − e (O. Lee and A. Williams). Orlando Lee – Unicamp Topics in Combinatorial Optimization
References A. Schrijver, Combinatorial Optimization, Vol. B, Springer. (chapter 56) Orlando Lee – Unicamp Topics in Combinatorial Optimization
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