# Computing a most probable delay constrained path: NP-hardness and approximation schemes

Ying Xiao, Krishnaiya Thulasiraman, Xi Fang, Dejun Yang, Guoliang Xue

Research output: Contribution to journalArticle

22 Citations (Scopus)

### Abstract

Delay constrained path selection is concerned with finding a source-to-destination path so that the delay of the path is within a given delay bound. When the network is modeled by a directed graph where the delay of a link is a random variable with a known mean and a known variance, the problem becomes that of computing a most probable delay constrained path. In this paper, we present a comprehensive theoretical study of this problem. First, we prove that the problem is NP-hard. Next, for the case where there exists a source-to-destination path with a delay mean no more than the given delay bound, we present a fully polynomial time approximation scheme. In other words, for any given constant ε such that 0 < ε < 1, our algorithm computes a path whose probability of satisfying the delay constraint is at least (1-\epsilon ) times the probability that the optimal path satisfies the delay constraint, with a time complexity bounded by a polynomial in the number of network nodes and 1/ε. Finally, for the case where any source-to- destination path has a delay mean larger than the given delay bound, we present a simple approximation algorithm with an approximation ratio bounded by the square root of the hop count of the optimal path.

Original language English (US) 5740849 738-744 7 IEEE Transactions on Computers 61 5 https://doi.org/10.1109/TC.2011.61 Published - 2012

### Fingerprint

NP-hardness
Approximation Scheme
Probable
Hardness
Polynomials
Path
Computing
Directed graphs
Approximation algorithms
Random variables
Computational complexity
Optimal Path
Fully Polynomial Time Approximation Scheme
Square root
Directed Graph
Time Complexity
Approximation Algorithms
Count
NP-complete problem
Random variable

### Keywords

• approximation schemes
• computational complexity
• Delay constrained path selection

### ASJC Scopus subject areas

• Hardware and Architecture
• Software
• Computational Theory and Mathematics
• Theoretical Computer Science

### Cite this

Computing a most probable delay constrained path : NP-hardness and approximation schemes. / Xiao, Ying; Thulasiraman, Krishnaiya; Fang, Xi; Yang, Dejun; Xue, Guoliang.

In: IEEE Transactions on Computers, Vol. 61, No. 5, 5740849, 2012, p. 738-744.

Research output: Contribution to journalArticle

Xiao, Ying ; Thulasiraman, Krishnaiya ; Fang, Xi ; Yang, Dejun ; Xue, Guoliang. / Computing a most probable delay constrained path : NP-hardness and approximation schemes. In: IEEE Transactions on Computers. 2012 ; Vol. 61, No. 5. pp. 738-744.
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