Practice Final Exam 1

Transcription

1 Algorithm esign Techniques Practice Final xam Instructions. The exam is hours long and contains 6 questions. Write your answers clearly. You may quote any result/theorem seen in the lectures or in the assignments without proving it (unless, of course, it is what the question asks you to prove).. Maximum Flows. onsider the following maximum flow problem. (Arc capacities are shown.) t e f c d a b s (a) Apply the Ford-Fulkerson algorithm (use the shortest augmenting path method to chose the augmenting path in each iteration) to find a maximum s t flow. [ marks] (b) Prove that your solution is a maximum s t flow by giving a certificate that shows it is impossible to find a larger flow. [ marks] (c) Prove that the maximum flow problem has a finite optimal solution if and only if there is no directed path from s to t consisting only of infinite capacity arcs. [ marks]

2 . Parameterised omplexity. (a) When is a problem fixed parameter tractable? [ marks] (b) Use the colour coding method to show the following problem is fixed parameter tractable: [8 marks] inary-tree Subgraph: Given a graph G = (V, ) and an integer k. oes G contain a subgraph T that is a (complete) binary tree on at least k vertices?. The Simplex Algorithm. (a) Apply the simplex method (using land s Rule) to solve the following linear program. [ marks] max x + x s.t. x + x x x x, x 0 (b) Give the optimal dual solution and prove that it is optimal. [ marks]

3 . NP-ompleteness. (a) What is the Vertex over problem? [ marks] (b) Using reductions from Vertex over show that the following two problems are NP-complete. i. Independent Set: Given a graph G = (V, ) and an integer k. oes G contain an independent set S of cardinality at least k? [ marks] [A set S V is independent (stable) if every pair of vertices in S are non-adjacent.] ii. ominating Set: Given a graph G = (V, ) and an integer k. oes G contain a dominating set of cardinality at most k? [ marks] [A set V is dominating if every vertex v V has at least one neighbour in.]. Approximation Algorithms. (a) What is an α-approximation algorithm for a maximisation problem. [ mark] (b) A directed graph H is acyclic if it contains no directed cycles. Prove that an acyclic graph has an acyclic ordering, that is, its vertices can be labelled {v, v,..., v n } such that i < j for any arc (v i, v j ) in H. [ marks] (c) Give a -approximation algorithm for the following problem: [ marks] Maximum Acyclic Subgraph: Given a directed graph G = (V, A), find an acyclic subgraph H of G that contains as many arcs as possible.

4 6. ranch and ound. A In this question, you will use the branch and bound method to solve the Travelling Salesman Problem above using a different combinatorial bound to the one seen in class. To reduce the search space of the branch and bound tree, we may use the following observations: The first vertex of the tour is A. In an undirected graph, we can traverse a tour in either direction. Thus, we may assume that our tour must visit node before node. When we reach the fourth vertex in the tour the fifth (final) vertex of the tour is forced. Thus at the fourth vertex we can find the true value of the tour and not use the lower bound. As the edge lengths are integral, if the lower bound is fractional we may round it up to the nearest integer. Using these observations, the search space is shown in the tree on the next page.

5 NO L: A NO L: L: L: 0 NO NO Val: Here some of the lower bounds have also been filled in. To motivate the new lower bound used, observe that every vertex is incident to exactly two edges in the optimal TSP tour. Therefore, if we sum the cost of the two cheapest edges incident to each vertex and divide by two then this will be a lower bound on the minimum cost tour. Thus, we have a lower bound of (( + ) + ( + 6) + ( + ) + ( + ) + ( + )) = 8 = In general, however, each node of our branch and bound search tree will correspond to a subpath P = {(v, v ), (v, v ),..., (v k, v k )} that we wish to extend into a tour. Then, for a lower bound, each vertex in V P is free to select its two cheapest incident edges, but the internal vertices of P must select both incident edges of P, and each of the two end-vertices of P must select its incident edge in P plus its cheapest other edge. For example, if P contains no edges (this corresponds to the root node of the search tree where the path contains only the vertex A but no edges) then every node selects its two cheapest incident edges and the lower bound on the cost of the minimum cost tour is as above. ut if P = {(A, )} then nodes A and must select the edge (A, ) of cost 9 so a lower bound on the minimum cost of any tour created from this path is (( + 9) + ( + 6) + ( + ) + ( + 9) + ( + )) = 9 = 0 Now use this branch and bound method to find an optimal tour. (Use depth first search and order the children in alphabetical order.) Note that, by pruning, you may not need to find bounds for every node in the search tree shown. [0 marks] We divide by two as otherwise edge costs are double counted.