By Witold Bednorz
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Extra info for Advances in Greedy Algorithms
As a result, the returned assignment is, A Greedy Scheme for Designing Delay Monitoring Systems of IP Networks 27 Theorem 5 The approximation ratio of the simple probe assignment algorithm is 2. Proof: For monitoring the delay of any edge e ∈ E, at least one station s ∈ S must send two probe messages, one to each endpoint of e. As a result, in any feasible probe assignment at least one probe message can be associated with each edge e. Let it be the message that is sent to the farthest endpoint of e from the monitoring station.
Since the set of monitoring stations cover all the element edges, the collection S covers all the elements of Z, and is a solution to the SC problem of size at most k. □ The above reduction R(I) can be extended to derive a lower bound for the best approximation ratio achievable by any algorithm. This reduction and the proof of Theorem 2 are given in . Theorem 2 The lower bound of any approximation algorithm for the LM problem is · ln(│V│). 2 A greedy algorithm for the LM and WLM problems We turn to present an efficient algorithm for solving the LM and the WLM problems.
If neither of these two situations occur, an unassigned variable is chosen and the algorithm is called recursively after adding a unit clause containing this variable and its negation. If all branches are explored and no satisfying assignment has been reached, the formula is found to be unsatisfiable. For efficiency reasons, the search tree is explored in depth-first search manner. Since we are only interested in whether the SAT problem is satisfiable or not, we stop as soon as the first solution is found.
Advances in Greedy Algorithms by Witold Bednorz