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Homework #1 in Design and Analysis of Algorithms Question 1 A clique in a graph G = (V, E) is a subset C ⊆ V such that there is an edge between every pair of vertices in C. A maximum clique is a clique with a maximum number of vertices. (a) Show that the minimum vertex-cover problem and the maximum clique problem have the same complexity up to polynomial factors. That is: If you had an algorithm for solving the minimum vertex-cover problem in time T (n) (where n = |V |) then you would have an algorithm for solving the maximum clique problem in time T (n) + poly(n) (where poly(n) denotes some polynomial in n), and that the reverse statement also holds. Hint: think of the complementary graph (that is, in which non-edges are replaced by edges and edges by non-edges). (b) Given that we have an approximation algorithm for the minimum vertex-cover with approxima- tion factor 2, does this imply, given the above item, that we have approximation algorithm for the maximum clique problem? Question 2 Let G = (V, E) be a graph such that every vertex in the graph has degree at most d. Show that the m minimum vertex cover of G is of size at least 2d−1 (where m = |E|). (Think of the approximation algorithm described in class and the relation between the number of edges that are included in the maximal matching that the algorithm determines compared to the number of edges that not included in the matching.) Question 3 Let G = (V, E) be a connected graph (that is, there is a path between every pair of vertices), where n = |V | ≥ 2. Recall that a Depth First Search (DFS ) in a graph is a procedure for visiting all vertices in the graph (assuming it is connected), in the following manner. Starting from an arbitrary initial vertex v0 , at each step, if the current vertex has some neighbors that have not yet been visited, then we continue to one of those neighbors. Otherwise, we return back to the previous vertex. The search ends when we have returned to v0 for the last time (and can’t continue since all its neighbors have already been visited). The DFS procedure deﬁnes a tree over all graph vertices: The root of the tree is v0 , and the parent of every other vertex is the vertex from which it was ﬁrst reached. Given a connected graph G = (V, E), suppose we run DFS, obtain the resulting DFS tree T , and let C ⊂ V be all internal vertices in T (that is, vertices that are not leaves of T ). (a) Prove that C is a vertex cover of G. (All you need to know about DFS is what is written above.) (b) Prove that the size of C is at most twice the size of a minimum vertex cover of G. Hint: show that there exists a matching1 T (and hence in G) that consists of at least |C|/2 edges. Prove that such a matching exists by induction on the number of internal vertices |C|. Question 4 Let F = {S1 , . . . , Sm } be a family of sets such that Si ⊆ {1, . . . , n} for every 1 ≤ i ≤ m. We shall say that H ⊂ {1, . . . , n} is a hitting set with respect to F if for every set Si ∈ F there exists an element 1 Recall that a matching in a graph G = (V, E) is a subset of edges M ⊆ E such that no two edges in M have a common end-point. 1 j ∈ H such that j ∈ Si . In other words, for every 1 ≤ i ≤ m we have that Si ∩ H = ∅. The Minimum Hitting Set problem is deﬁned as follows: Given a family of sets F = {S1 , . . . , Sm } the goal is to ﬁnd a hitting set with respect to F with minimum size. This problem is NP-hard. Suppose the input is restricted in the following manner: For some ﬁxed integer k, we have that |Si | ≤ k for every 1 ≤ i ≤ m. Describe and analyze an algorithm for ﬁnding a hitting set with respect to F such that H is at most k times larger than the minimum hitting set. 2

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posted: | 2/13/2010 |

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