Exam problems for the course Combinatorial Optimization I (DM208)

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1 Exam problems for the course Combinatorial Optimization I (DM208) Jørgen Bang-Jensen Department of Mathematics and Computer Science University of Southern Denmark The problems are available form the course page as of Monday October at 5 p.m. Two copies of the solutions must be returned in Jørgen Bang-Jensen s mailbox in the secretary s office by Monday November 1, 2010 at 1 p.m. Note the following important points: Exactly one of problems 5A and 5B can be handed in. Thus there are 100 points to earn. It is important that you explain how you obtain your answers and argue why they are correct. If you are asked to describe an algorithm, then you must supply enough details so that a reader who does not already know the algorithm (but knows basic algorithms such a searching for a path from p to q, finding a maximum (s,t)-flow or a minumim (s,t)-cut, an alternating path with respect to a given matching in a bipartite graph, etc) can understand it (but you do not have to give pseudo code). You should also give the complexity of the algorithm when relevant. In particular, when you are asked to describe a polynomial algorithm, you should argue why the algorithm you describe is in fact polynomial. Note also that illustrating an algorithm means that one has to follow the steps of the algorithm meticulously (DK: slavisk). It is strictly forbidden to work in groups and any exchange of results/ideas before 1 p.m. November 1, 2010 will be considered as exam fraud. 1

2 PROBLEM 1 20 point) This problem is about edge-connectivity in graphs. Notice that we allow multiple edges! a 4 b 2 c d 3 e 2 f Figure 1: A graph with 6 vertices. The numbers on the edges indicate the number of parallel edges. Thus there are 4 edges between vertex a and b. Let G = (V,E) be the graph in Figure 1. Show how to find the edge-connectivity λ(g) of G via max-back orderings. You should start from vertex a in each iteration and give a max-back numbering of the current graph in each step. Question b: Give a certificate G = (V,E ) which shows that G is λ(g)-edge-connected. You must explain how you obtained the set of edges. The set E may not have more than λ(g)( V 1) edges. Explain shortly (e.g. by referring to the appropriate result in the course material) why it is always possible to find such a certificate for every input graph G = (V,E). Question c: Is G above minimally λ(g)-connected? Question d: Find a Gomory Hu tree for the graph G. You should start by explaining the algorithm briefly and then show all the steps that lead to the final tree. You may find the cuts by inspection in each step (and say which cut you consider next). You should argue in the end that your solution is correct. 2

3 PROBLEM 2 (20 point) This problem is about the greedy algorithm for subset systems. Recall that (S,F) is a subset system if Y F and X Y implies that X F (Definition 12.1 in Papadimitriou and Steiglitz). Start by describing the greedy algorithm on a subset system (S,F) and a weight function ω : S R. Next give (without proof) a (correct) characterization of subset systems for which the greedy algorithm always works, no matter the choice of ω as long as ω is non-negative. Question b: For each of the problems below you should first show that one can associate a subset system to each of them and then either prove that the greedy algorithm always works, no matter the choice of (non-negative) weight function or show by an example or other correct method that the greedy algorithm does not always work for the problem. (a) Given a set S, a natural number k S and non-negative weight function ω : S R + {0}. Find a maximum weight subset S S such that S k. (b) Given an undirected graph G = (V,E) and a non-negative weight function ω : E R + {0}. Find a maximum weight subset E of E such that the graph H = (V,E E ) (the graph obtained by deleting the edges in E ) is still connected. (c) Given an undirected graph G = (V, E) containing a pair of edge-disjoint spanning trees and a non-negative weight function ω : E R + {0}. Find a pair of edge-disjoint spanningtreest,t suchthat e E(T) E(T ) ω(e)isassmallaspossible(thatis, apairof edge-disjoint spanning trees with minimum total weight among all pairs of edge-disjoint spanning trees). (d) Given an undirected graph G = (V,E) and a non-negative weight function ω : V R + {0}. Findamaximumweight independentset ofg(asetx V(G) isindependent if no edge of G has both endvertices in X). (e) Given a bipartite graph G = (X,Y,E), where X and Y are the two bipartition classes (all edges are between X and Y) and a non-negative weight function ω : X R + {0}. Find a maximum weight subset X X such that G has a matching M covering all vertices of X (every vertex of X is matched to a different vertex in Y by M). (f) Given a digraph D = (V,A), a special vertex s V and a non-negative weight function ω : A R + {0}. Find a minimum cost out-branching rooted at s. 3

4 PROBLEM 3 (20 point) Figure 2: A graph G. The dotted edges indicate the matching M. Explain briefly how to find a maximum matching in a given undirected graph and illustrate the method on the graph G in Figure 2 where a matching M is already given and your algorithm should start from this matching and extend it to a maximum matching. Question b: Give a certificate that shows that the matching you found above is indeed a maximum matching in G. Question c: Describe in words how one can obtain a polynomial algorithm for deciding whether a given graph G = (V,E) contains a collection C 1,C 2,...,C k, k 1 of vertex disjoint cycles such that V = V(C 1 )... V(C k ). The algorithm should return a set of disjoint cycles covering V if one exists. You should explain the important steps in the algorithm and give the complexity. Question d: Suppose we are given a bipartite graph B = (X,Y,E) and we want to know whether B has two edge-disjoint perfect matchings. Suggest an algorithm for solving this problem. Question e: Does the algorithm for question d always work when B is not bipartite? 4

5 Problem 4 (20 point) Question a : Give a brief description of an algorithm to find a minimum cost out-branching from a given root s in a digraph D = (V,A) with weights on the arcs. e 1 f 3 g a 2 d 4 c b 5 5 s Figure 3: A weighted digraph with special vertex s. Question b : Demonstrate how the algorithm works by applying it to the digraph in Figure 3 with the root as the vertex s. Below we let D = (V,A) be a digraph, let s V be a special vertex and let k be a natural number. Question c : Explain how one can decide in polynomial time, using flows, whether D has k arc-disjoint outbranchings. What is the complexity of your algorithm expressed in terms of the complexity of finding a maximum flow in a network on n vertices and m arcs and V, A? Remember that you are not asked to find the branchings. 5

6 Question d : Describe a polynomial algorithm which given D, s and k, either shows that D does not have k arc-disjoint out branchings rooted at s, or returns a subset A A such that D = (V,A ) is the union of k arc-disjoint out-branchings with root s. Figure 4: A digraph D formed by the union of two out-branchings rooted at s. s Question e : Show by an example, based on Figure 4 with k = 2, that the algorithm which just greedily finds one out-branching from s in D, deletes the edges of F and then repeats this may not succeed in finding 2 edge-disjoint out-branchings from s. Question f : Describe an algorithm which given the arc set A from question d can produce a list of k- edge-disjoint out-branchings rooted at s in D. Show how your algorithm works by applying it to the digraph D shown in Figure 4 (here k = 2). 6

7 PROBLEM 5A (20 point) This problem deals with the maximum weight bipartite matching problem. First explain why the problem is closely related to the minimum weight bipartite matching problem in the sense that if we have any algorithm for the minimization version then we can solve the maximization version and conversely. Question b: Select one of the algorithms for finding a maximum (minimum) weight perfect matching in a bipartite graph that you have seen in the course and explain in words how the algorithm works and how one can verify optimality once the algorithm terminates. Question c: Illustrate the algorithm by applying it to the assignment problem shown in Figure 5 where the goal is to find a perfect matching of maximum weight Figure 5: An instance of the assignment problem. The weights listed are c ij, where c ij is the weight of assigning the ith row to the jth column (e.g. c 24 = 13). Below B = (X,Y,E) is a bipartite graph, ω : E R is a weight function on its edges and M is a maximum weight matching of B. Note that (depending on the weight function ω, M may or may not also be a maximum cardinality matching of B). Question d: Suppose that the weight of one of the edges in M is changed. Explain how one can check whether M is still a maximum weight matching of B with respect to the new weight function without running an algorithm for finding a maximum weight matching from scratch. Question e: Suppose instead that a new edge e is added to B and ω is updated (giving a weight to e also). Devise a fast method for checking whether M is still a maximum weight matching in the new graph. 7

8 PROBLEM 5B (20 point) Show how the problem of findingamaximum matching in a bipartite graph can beformulated as a matroid intersection problem. Figure 6: The 3 dotted edges indicate the current matching. Question b : Give a detailed explanation of the matroid intersection algorithm specialized to the problem above. Indicate how the algorithm works by showing how the algorithm finds a maximum matching in the graph in Figure 6 when it is started from the matching shown. Does it matter if we take minimal (s, t)-paths or not for the specialization of the algorithm? Question c : Give a proof of König s theorem (the size of a maximum matching in a bipartite graph equals the size of a minimum vertex cover) based on Edmonds min-max formula for the maximum size of a common independent set of two matroids on the same ground set (Theorem 10.9 in Schrijver s notes). Hint: use your reduction above. Question d : Give a proof of Hall s Theorem (A bipartite graph B = (U,V,E) has a perfect matching if and only if U = V and for all X U the number of neighbours of X is at least the size of X) again based on Theorem 10.9 in Schrijver s notes. Note that you must do this directly. It is not sufficient to show how Hall s theorem follows from König s theorem. Question e : What is wrongwiththe following algorithm to findamaximum matching in ageneral graph G = (V,E): LetF = {E E : no vertex in V is incident to more than one edge in E }and find a maximum cardinality set in F using the greedy algorithm? 8

9 Question f : Describe briefly in words how to use matroid intersection to decide whether a given bipartite graph has a spanning (i.e. every vertex is covered) collection of vertex-disjoint cycles. 9