Topics in Combinatorial Optimization Orlando Lee Unicamp 27 de - PowerPoint PPT Presentation

Topics in Combinatorial Optimization Orlando Lee – Unicamp 27 de fevereiro 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

Goals Minimax theorems, structure results and efficient algorithms. Pervasive role of submodularity in graph theory and combinatorial optimization. Linear description of integral polyhedra associated to several combinatorial problems (if time allows). Discussion of open problems (hopefully). So, without further delay, shall we start? Orlando Lee – Unicamp Topics in Combinatorial Optimization

Menger’s theorem A classical result proved by Karl Menger in 1927 (whose interest was in topology). It relates maximum packing of st -paths and minimum st -cuts. It precedes the classical MaxFlow MinCut theorem of Ford and Fulkerson (1950). There are several versions of the theorem depending on whether (i) the graph is directed or not, (ii) the paths must be arc/edge or vertex disjoint, and (iii) the source/target is a single vertex or a set of vertices. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Some notation Let D = ( V , A ) be a digraph and s , t ∈ V , s � = t . Let X ⊆ V . We write ¯ X instead of V \ X when V is clear from context. Let D ( X ) := { a = ( u , v ) ∈ A : u ∈ ¯ δ in X , v ∈ X } and D ( X ) := { a = ( u , v ) ∈ A : u ∈ X , v ∈ ¯ δ out X } . We denote d in D ( X ) = | δ in D ( X ) | and d out D ( X ) = | δ out D ( X ) | . When D is understood from the context, we may omit the subscript D . For example, d in ( X ) , δ out ( X ). When X = { v } we may write for example, d out ( v ) , δ in ( v ) instead of d out ( { v } ) , δ in ( { v } ). Orlando Lee – Unicamp Topics in Combinatorial Optimization

Paths and cuts Let D = ( V , A ) be a digraph and s , t ∈ V , s � = t . An st -path is a (di)path that starts at s and ends at t . We say that X is an s ¯ t -set if s ∈ X and t �∈ X . In this case we also say that δ out ( X ) is an st -cut. Lemma 1. Let D = ( V , A ) be a digraph and s , t ∈ V , s � = t . Then there exists an st -path in D if and only if d out ( X ) � 1 for every s ¯ t -set X . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Packings and cuts We say that two paths are arc-disjoint if they have no arc in common. A packing of st -paths is a collection P of pairwise arc-disjoint st -paths. We are interested in the problem of finding a packing of st -paths with maximum size , or simply, a maximum packing of st -paths. Convention: the size of a set B is its cardinality | B | . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Packings and cuts Lemma. Let P be a packing of st -paths and δ out ( X ) be an st -cut. Then |P| � | δ out ( X ) | . Furthermore, if |P| = | δ out ( X ) | then P is a maximum packing and δ out ( X ) is a minimum cut. The first part follows from the following observation. Each path in P in P must intersect δ out ( X ). The second part follows from the first one. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Menger’s theorem – directed arc-version We shall prove the following. Theorem. (Menger, 1927) Let D = ( V , A ) be a digraph and s , t ∈ V , s � = t . Then the size of a maximum packing of st -paths is equal to size of a minimum st -cut. We will present three different proofs, each one using an important proof technique. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Menger’s theorem – directed arc-version For convenience, let us restate the theorem as follows. Theorem. (Menger, 1927) Let D = ( V , A ) be a digraph and s , t ∈ V , s � = t . Then there exists k arc-disjoint st -paths if and only if d out ( X ) � k for every s ¯ t -set X . As we have seen, necessity is obvious. We will prove sufficency. Orlando Lee – Unicamp Topics in Combinatorial Optimization

First proof Let D = ( V , A ) be a digraph and s , t ∈ V , s � = t . Suppose that d out ( X ) � k for every s ¯ t -set X . We will use induction on | V | + | A | . The base case is trivial. By Lemma 1 the assertion is trivial for k = 0 , 1 . So suppose that k > 1 and there exists at least one st -path in D . Orlando Lee – Unicamp Topics in Combinatorial Optimization

First proof Suppose first that there exists an st -path P with length 1 or 2. (The length of a path is its number of arcs.) Let D ′ := D − A ( P ). It is easy to check that D ′ ( X ) � k − 1 for every s ¯ d out t -set X . By induction, D ′ has a packing P ′ of st -paths with size k − 1. Then P := P ′ ∪ { P } is a packing of st -paths with size k in D . Orlando Lee – Unicamp Topics in Combinatorial Optimization

First proof So we may assume the every st -path has length at least 3. Let a be an arc incident to neither s nor t . We analyze two cases. Suppose first that d out ( X ) � k + 1 for every s ¯ t -set X with a ∈ δ out ( X ). Let D ′ := D − a . Then d out D ′ ( X ) � k for every s ¯ t -set X . So by induction, D ′ has a packing P of st -paths with size k . Clearly, P is also a packing of st -paths in D . Orlando Lee – Unicamp Topics in Combinatorial Optimization

First proof Now suppose that there exists an s ¯ t -set X such that d out ( X ) = k a ∈ δ out ( X ). and ¯ X X s t Orlando Lee – Unicamp Topics in Combinatorial Optimization

First proof Let D 1 the digraph obtained from D by shrinking X to a single vertex s ′ . Note that every cut in D 1 corresponds to cut in D with the ”same arcs”. Moreover, if Y is an s ′ t -set in D 1 then Y \ { s ′ } ∪ X is an s ¯ t -set in D and d out D 1 ( Y ) � k . By induction there exists a packing P 1 of s ′ t -paths with size k . ¯ X s ′ t Orlando Lee – Unicamp Topics in Combinatorial Optimization

First proof Let D 2 be the digraph obtained by shrinking ¯ X to a single vertex t ′ . Similarly, if Y is an s ¯ t ′ -set in D 2 then d out D 2 ( Y ) � k . By induction there exists a packing P 2 of st ′ -paths with size k . X t ′ s Orlando Lee – Unicamp Topics in Combinatorial Optimization

First proof Note that each arc in δ out ( X ) is used by exactly one path in P 1 and by exactly one path in P 2 . Moreover, any arc which is common to a path in P 1 and a path in P 2 must be in δ out ( X ). So by taking the union of each path in P 1 with the one in P 2 which intersects it, we obtain a packing of st -paths with size k . This concludes the proof. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Second proof The next proof is due to Andr´ as Frank. We need the following observation. For any subsets of vertices X and Y of D we have d out ( X ) + d out ( Y ) � d out ( X ∩ Y ) + d out ( X ∪ Y ) . We say that d out is submodular (for every pair X , Y ). X Y Remark: an analogous inequality holds for d in . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Second proof The proof is by induction on | A | . If every arc has tail in s and head in t , then each one of these arcs is an st -path and the result follows. So we may assume that there exists an arc a = ( s , u ) such that u � = t . Let D ′ := D − a . If d out ( X ) � k + 1 for every s ¯ t -set such that a ∈ δ out ( X ) then by induction D ′ (and hence, D ) has a packing of st -paths with size k . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Second proof So assume that there exists an s ¯ t -set X such that a ∈ δ out ( X ) d out ( X ) = k . and These kind of cuts play an important role in the proof(s) of Menger’s theorem. This motivates the following concept. Tight sets. We say that an s ¯ t -set X is tight if d out ( X ) = k . Orlando Lee – Unicamp Topics in Combinatorial Optimization

Second proof Next we have a simple but very important result. Claim 1. If X and Y are tight sets then X ∩ Y and X ∪ Y are also tight sets. Proof. We have k + k = d out ( X )+ d out ( Y ) ≥ d out ( X ∩ Y )+ d out ( X ∪ Y ) ≥ k + k . Thus equality holds throughout and X ∩ Y and X ∪ Y are tight sets. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Second proof Claim 2. If X is a tight s ¯ t -set and a = ( s , u ) ∈ δ out ( X ) then there exists an arc b = ( u , v ) such that v ∈ V \ X . Proof. Suppose that there exists no such arc. Then d out ( X ∪ { u } ) < d out ( X ) because every arc that leaves X ∪ { u } also leaves X and a does not leave X ∪ { u } . But then d out ( X ∪ { u } ) < k which is a contradiction. Orlando Lee – Unicamp Topics in Combinatorial Optimization

Second proof Let X a maximal tight s ¯ t -set with a = ( s , u ) ∈ δ out ( X ). Let b = ( u , v ) be an arc with v ∈ V \ X (Claim 2). Let e = ( s , v ) be a new arc and let D ′ := D − { a , b } + e . X u X u s v s v Remark: this operation is called splitting off a and b . Orlando Lee – Unicamp Topics in Combinatorial Optimization