| Research Abstract |
In this research project, we studied the effect of a hybrid approach to computationally hard problems. This approach combines three basic approaches (namely, approximation, randomization, and parallelization) to computationally hard problems. Previously, not so many algorithms were based on such a hybrid approach. The main purpose here is to use this hybrid approach to solve computationally hard problems that have not been solved so far. This may lead to the finding of new design techniques of efficient algorithms for hard problems.
We focused on four computationally hard problems. The first is the maximum traveling salesman problem (Max TSP) which is the maximization variant of the famous traveling salesman problem. We obtained a number of results for Max TSP and included the results in three papers. In the first paper, we presented an O(n^3)-time approximation algorithm for Max TSP whose approximation ratio is asymptotically 61/81, and also presented an O(n^3)-time approximation algor
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ithm for the metric vase of Max TSP whose approximation ratio is asymptotically 17/20. In the second paper, we presented an O(n^3)-time randomized approximation algorithm for Max TSP whose expected approximation ratio is asymptotically 251/331. In the third paper, we presented two polynomial-time approximation algorithms for the metric case of Max TSP. One of them is for directed graphs and its approximation ratio is 27/35 while the other is for undirected graphs and its approximation ratio is 7/8 o(1).
The second problem we considered is the following : Given a simple graph G, compute two disjoint matchings M_1 and M_2 of G such that the total number edges in MI and /1/Z is maximized. Motivated by call admittance issues in satellite based telecommunication networks, Feige, et. al. introduced the problem. We presented a polynomial-time approximation algorithm for the problem. It achieves an approximation ratio of 468/575. This improves on the previous best (trivial) ratio of 4/5.
The third problem we considered is the Degree-ΔClosest Phylogenetic k-th Root Problem (ΔCPR) : Given a graph G = (V, E) find a (phylogenetic) tree T such that (1) the degree of each internal node in T is at least 3 and at most Δ, (2) the leaves of T are exactly the elements of V, and (3) the number of disagreements, i.e., the symmetric difference between Eand {{U,V}: u and v are leaves of T and d_T (u, v) , is minimized, where d_T (u, v) denotes the distance between u and v in tree TΔCPR_κ is known to be NP-hard for all fixed constants Δ such that either both Δ≧ 3 and k ≧ 3, or Δ> 3 and k = 2. We presented a polynomial-time 8-approximation algorithm forΔCPR_2 for any fixed Δ > 3, a quadratic-time 12-approximation algorithm for 3CPR_3, and a polynomial-time approximation scheme for the maximization version ofΔCPR_κ for any fixedΔ,K.
The fourth problem we considered is the problem of computing the duplication history of a tandem repeated region. We designed a polynomial-time approximation scheme(PTAS) for the case where the size duplication block is 1. Our PTAS is faster than the previously best PTAS. For example, to achieve a ratio of 1.5, our PTAS takes O(n^5) time while the previous best takes O(n^<11>) time. We also designed a ratio-6 polynomial-time approximation algorithm for the case where the size of each duplication block is at most 2. This is the first polynomial-time approximation algorithm with a guaranteed ratio for this case. Less
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