Topics in Combinatorial Optimization Orlando Lee Unicamp 4 de - - PowerPoint PPT Presentation

Topics in Combinatorial Optimization

Orlando Lee – Unicamp 4 de junho 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

Independent sets

A matroid is a pair M = (E, I) in which I ⊆ 2E that satisfies the following properties: (I1) ∅ ∈ I. (I2) If I ∈ I and I ′ ⊆ I, then I ′ ∈ I. (I3) If I 1, I 2 ∈ I and |I 1| < |I 2|, then there exists e ∈ I 2 − I 1 such that I 1 ∪ {e} ∈ I. (Independence augmenting axiom) We say that the members of I are the independent sets of M.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matroid intersection

We have seen how to determine efficiently (via an independence

• racle) a maximum basis (independent set) of a matroid.

Let M1 = (E, I1) and M2 = (E, I2) be two matroids. We are interested in finding a maximum element I ∈ I1 ∩ I2, that is, a maximum common independent set I of both matroids. We consider the unweighted version, in which we want to maximize |I| for I ∈ I1 ∩ I2, and the weighted version in which given c : E → R we want to maximum c(I) for I ∈ I1 ∩ I2. Let us describe some applications of matroid intersection.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Partition matroids

Recall the following definiton. Let E1, . . . , Et be a partition of a finite set E and let k1, . . . , kt be non-negative integers (ki |E i|). Let I := {I ⊆ E : |I ∩ E i| ki, i = 1, . . . , t}. Then M = (E, I) is a matroid, called partition matroid.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matchings in bipartite graphs

Let G = (V , E) be a bipartite graph with bipartition (U1, U2). For i = 1, 2 let Mi := (E, Ii) be the matroid in which I ⊆ E is independent if each vertex of Ui is covered by at most one edge. In other words, Mi is a partition matroid in which we take the partition {δ(ui

1), . . . , δ(ui n)} of E where Ui = {ui 1, . . . , ui n} and an

independent set can pick at most one element of each star. So a set of edges that is independent in both partition matroids must be a matching. Hence, a maximum common independent set

• f both matroids is a maximum matching in G.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Colored trees

Let G = (V , E) be a graph and let E1, . . . , Ek be a partition of E. Refer to each E i as a color and so G is k-edge-colored (arbitrarily). We want to find a (non-necessarily spanning) tree with all edges having distinct colors. Such a tree is a common independent set of the graphic matroid M(G) and the partition matroid induced by E1, . . . , Ek (in which each independent set can pick at most one edge of each color).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Arborescences

Let D = (V , A) be a digraph with V = {v 1, . . . , v n} and δin(v 1) = ∅. Let N the partition matroid induced by the partition {δin(v 2), . . . , δin(v n)} of A in which each independent set can pick at most one arc of each in-star. So a spanning v 1-arborescence in D (assuming that it exists) is a maximum common independent of N and the graphich matroid of the underlying graph of G.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Edmonds’ matroid intersection theorem

• Theorem. (Edmonds, 1970) Let M1 = (E, r 1) and M2 = (E, r 2)

be two matroids. Then the maximum size of a common independent set of M1 and M2 is equal to min

X⊆E(r 1(X) + r 2(E \ X)).

To see that max min let I be a common independent set and let X be an arbitrary subset of E. Then |I| = |I ∩ X| + |I \ X| r 1(X) + r 2(E \ X). We show that max min by describing an algorithm.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Oracle

Let Ii denote the collection of independent sets of Mi, for i = 1, 2. We assume that each matroid is given by an oracle that given as input an independent set I and a element x ∈ E − I, answer whether I + x is independent or not, and in the latter case, the

• racle returns the fundamental circuit C(I, x).

It is easy to construct such an oracle by using an independence

• racle, right?

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Auxiliary lemma

• Lemma. (Simultaneous exchange) Let x1, . . . , xk be elements of

an independent set I of a matroid M and let y 1, . . . , y k ∈ E − I such that I + y i is dependent for i = 1, . . . , k. Suppose that each xi ∈ C(I, y i) but xi ∈ C(I, y j) whenever j < i. Then I − {x1, . . . , xk} ∪ {y 1, . . . , y k} is independent in M.

• Proof. We prove by induction on k. For k = 1, the result is
• bvious. So assume that k 2 and the result holds for k − 1. By

hypothesis, the set I ′ := I − xk + y k is independent, and I ′ + y i (i k − 1) contains C(I, y i), and hence, C(I ′, y i) = C(I, y i). Therefore, I ′ along with {x1, . . . , xk−1} and {y 1, . . . , y k−1} satisfy the hypotheses and by induction we have that I ′ − {x1, . . . , xk−1} ∪ {y 1, . . . , y k−1} is independent.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Matroid intersection algorithm

We will describe an algorithm that finds a common independent set I and a subset X ⊆ E for which r 1(X) = |I ∩ X| and r 2(E \ X) = |I \ X| whence the result follows. The algorithm consists of at most |E| phases. It starts with any common independent set I (such as the empty set) . In each phase, the algorithm either finds a common independent I ′ for which |I ′| = |I| + 1 (and go to the next phase) or finds X as above and halts.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Phase

Let I be a common independent set. Let E 1 := {e ∈ E − I : I + e ∈ I1} and E 2 := {e ∈ E − I : I + e ∈ I2}. Note that if e ∈ E 1 ∩ E 2 then I + e is a common independent set and the phase ends. So suppose that E 1 ∩ E 2 = ∅.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Phase – auxiliary digraph

For i = 1, 2 let C i(I, e) denote the fundamental circuit (if any) of I + e in Mi. Define the auxiliary digraph DM1,M2(I) whose vertex-set is E and whose arc-set consists of the union of the following sets: {(u, v) : u ∈ E − (I ∪ E 1), v ∈ I, v ∈ C 1(I, u)} and {(x, y) : x ∈ I, y ∈ E − (I ∪ E 2), x ∈ C 2(I, y)}. The algorithm uses breadth-first search to compute the set X of vertices that are reachable from E 2. We have two cases.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Phase

Case 1: E 1 ∩ X = ∅. Since no arc leaves X, the fundamental circuit C 1(I, u) in M1 of each u ∈ X − I is contained in X. We claim that I ∩ X is a maximal independent subset of X in M1. Suppose for a contradiction that there exists u ∈ X − I such that (I ∩ X) + u is independent in M1. Because of the hypothesis of the case, we have that I + u is dependent and so some element of C 1(I, u) lies in E − X, which is a contradiction. So r 1(X) = r 1(X). Similarly, C 2(I, y) is contained in E − X for each y ∈ E − (I ∪ X). We claim that I \ X is a maximal independent subset of E \ X in

• M2. Suppose for a contradiction that there exists y ∈ E \ X such

that (I \ X) + y is independent in M2. Since E 2 ⊆ X, we have that (I \ X) + y is dependent in M2 and so some element of C 2(I, y) lies in X, which is a contradiction. So r 2(E \ X) = r 2(I \ X).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Phase

Case 2: E 1 ∩ X = ∅. So there exists a E 2E 1-path P and it is a shortest one. Let I ′ := I △ V (P). Note that I ′ is larger than I because D(I) is bipartite with I being one of the color classes. So it remains to show that I ′ is a common independent set. We only prove that I ′ is independent in M1. The proof that I ′ is independent in M2 is analogous. Let y 1, x1, y 2, x − 2, . . . , y k, xk, z denote the vertices of P traversed in this order with y 1 ∈ E 2 and z ∈ E 1. Let I ′′ := I + z. Now I ′′, x1, . . . , xk, y 1, . . . , y k satisfy the hypotheses of the Simultaneous Exchange Lemma for M1 because P is a shortest

• path. Hence, the Lemma implies that

I ′ = I − {x1, . . . , xk} ∪ {y 1, . . . , y k} is independent in M1. From the above comments, it follows that the described algorithm returns I and X as desired. So the theorem is proved.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Intersection of three matroids

Let D = (V , A) be a digraph and let V := {v 1, . . . , v n}. The problem of finding hamiltonian path problem in a digraph D can be reduced to the problem of finding a maximum common independent set of three matroids: (a) the graphic matroid on the underlying graph of D, (b) the partition matroid induced by {δin(v 1), . . . , δin(v n)} in which each independent set can pick at most one arc of each in-star, and (c) the partition matroid induced by {δout(v 1), . . . , δout(v n)} in which each independent set can pick at most one arc of each

• ut-star.

There exists an hamiltonian path in D if and only if there exists a common independent set of size |V | − 1. So the matroid intersection problem for three matroids is NP-hard.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

References

• A. Frank, Connections in Combinatorial Optimization, Oxford

(page 416).

Orlando Lee – Unicamp Topics in Combinatorial Optimization