be routed either clockwise or counterclockwise around the cycle from its source

# Be routed either clockwise or counterclockwise around

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be routed either clockwise or counterclockwise around the cycle from its source node to its destination node. Our goal is to route the calls so as to minimize the overall load on the network. The load L i on any edge ( i , ( i + 1 ) mod n ) is the number of calls routed through that edge, and the overall load is max i L i . Describe and analyze an efficient -approximation algorithm for this problem. . The linear arrangement problem asks, given an n -vertex directed graph as input, for an ordering v 1 , v 2 ,..., v n of the vertices that maximizes the number of forward edges: directed edges v i v j such that i < j . Describe and analyze an efficient - approximation algorithm for this problem. (Solving this problem exactly is NP-hard.)
J. . An FPTAS for Subset Sum . A three-dimensional matching in an undirected graph G is a collection of vertex- disjoint triangles (cycles of length 3 ) in G . A three-dimensional matching is maximal if it is not a proper subgraph of a larger three-dimensional matching in the same graph. (a) Let M and M 0 be two arbitrary maximal three-dimensional matchings in the same underlying graph G . Prove that | M | ≤ 3 · | M 0 | . (b) Finding the largest three-dimensional matching in a given graph is NP-hard. Describe and analyze a fast -approximation algorithm for this problem. (c) Finding the smallest maximal three-dimensional matching in a given graph is NP-hard. Describe and analyze a fast -approximation algorithm for this problem. . Consider the following optimization version of the P problem. Given a set X of positive integers, our task is to partition X into disjoint subsets A and B such that max A , B is as small as possible. Solving this problem exactly is clearly NP-hard. (a) Prove that the following algorithm yields a ( 3 / 2 ) -approximation. G P ( X [ 1.. n ]) : a 0 b 0 for i 1 to n if a < b a a + X [ i ] else b b + X [ i ] return max { a , b } (b) Prove that the approximation ratio 3 / 2 cannot be improved, even if the input array X is sorted in non-decreasing order. In other words, give an example of a sorted array X where the output of G P is 50% larger than the cost of the optimal partition. (c) Prove that if the array X is sorted in non-increasing order, then G P achieves an approximation ratio of at most 4 / 3 . (d) Prove that if the array X is sorted in non-increasing order, then G P achieves an approximation ratio of exactly 7 / 6 . . The chromatic number χ ( G ) of a graph G is the minimum number of colors required to color the vertices of the graph, so that every edge has endpoints with different colors. Computing the chromatic number exactly is NP-hard. Prove that the following problem is also NP-hard: Given an n -vertex graph G , return any integer between χ ( G ) and χ ( G )+ 31337 . [Note: This does not contradict the possibility of a constant factor approximation algorithm.]
J. A A . Let G = ( V , E ) be an undirected graph, each of whose vertices is colored either red, green, or blue. An edge in G is boring if its endpoints have the same color, and interesting if its endpoints have different colors. The most interesting -coloring is the