Abstract
Let \(G=(V,\, E)\) be a given directed graph in which every edge e is associated with two nonnegative costs: a weight w(e) and a length l(e). For a pair of specified distinct vertices \(s,\, t\in V\), the k-(edge) disjoint constrained shortest path (kCSP) problem is to compute k (edge) disjoint paths between s and t, such that the total length of the paths is minimized and the weight is bounded by a given weight budget \(W\in \mathbb {R}_{0}^{+}\). The problem is known to be \({\mathcal {NP}}\)-hard, even when \(k=1\) (Garey and Johnson in Computers and intractability, 1979). Approximation algorithms with bifactor ratio \(\left( 1\,+\,\frac{1}{r},\, r\left( 1\,+\,\frac{2(\log r\,+\,1)}{r}\right) (1\,+\,\epsilon )\right) \) and \((1\,+\,\frac{1}{r},\,1\,+\,r)\) have been developed for \(k=2\) in Orda and Sprintson (IEEE INFOCOM, pp. 727–738, 2004) and Chao and Hong (IEICE Trans Inf Syst 90(2):465–472, 2007), respectively. For general k, an approximation algorithm with ratio \((1,\, O(\ln n))\) has been developed for a weaker version of kCSP, the k bi-constraint path problem which is to compute k disjoint st-paths satisfying a given length constraint and a weight constraint simultaneously (Guo et al. in COCOON, pp. 325–336, 2013). This paper first gives an approximation algorithm with bifactor ratio \((2,\,2)\) for kCSP using the LP-rounding technique. The algorithm is then improved by adopting a more sophisticated method to round edges. It is shown that for any solution output by the improved algorithm, there exists a real number \(0\le \alpha \le 2\) such that the weight and the length of the solution are bounded by \(\alpha \) times and \(2-\alpha \) times of that of an optimum solution, respectively. The key observation of the ratio proof is to show that the fractional edges, in a basic solution against the proposed linear relaxation of kCSP, exactly compose a graph in which the degree of every vertex is exactly two. At last, by a novel enhancement of the technique in Guo et al. (COCOON, pp. 325–336, 2013), the approximation ratio is further improved to \((1,\,\ln n)\).
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Notes
Note that \(\widetilde{G}\) may contain pairs of parallel edges in the same direction, which are probably with different lengths and weights.
References
Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice Hall, Englewood Cliffs
Bhatia R, Kodialam M, Lakshman TV (2006) Finding disjoint paths with related path costs. J Comb Optim 12(1):83–96
Chao P, Hong S (2007) A new approximation algorithm for computing 2-restricted disjoint paths. IEICE Trans Inf Syst 90(2):465–472
Garey MR, Johnson DS (1979) Computers and intractability. Freeman, San Francisco
Guo L, Shen H (2012) On the complexity of the edge-disjoint min-min problem in planar digraphs. Theor Comput Sci 432:58–63
Guo Longkun, Shen Hong (2013) On finding min-min disjoint paths. Algorithmica 66(3):641–653
Guo L, Hong S, Kewen L (2013) Improved approximation algorithms for computing k disjoint paths subject to two constraints. In: Du DZ, Zhang G (eds) COCOON. Springer, Heidelberg
Korte B, Vygen J (2012) Combinatorial optimization, vol 21. Springer, Heidelberg
Li CL, McCormick TS, Simich-Levi D (1989) The complexity of finding two disjoint paths with min-max objective function. Discrete Appl Math 26(1):105–115
Lorenz DH, Raz D (2001) A simple efficient approximation scheme for the restricted shortest path problem. Oper Res Lett 28(5):213–219
Misra S, Xue G, Yang D (2009) Polynomial time approximations for multi-path routing with bandwidth and delay constraints. In: IEEE INFOCOM 2009, p 558–566
Orda A, Sprintson A (2004) Efficient algorithms for computing disjoint QoS paths. In: IEEE INFOCOM, vol 1, p 727–738. Citeseer
Schrijver A (1998) Theory of linear and integer programming. Wiley, New York
Suurballe JW (1974) Disjoint paths in a network. Networks 4(2):125
Suurballe JW, Tarjan RE (1984) A quick method for finding shortest pairs of disjoint paths. Networks 14(2):325
Xu D, Chen Y, Xiong Y, Qiao C, He X (2006) On the complexity of and algorithms for finding the shortest path with a disjoint counterpart. IEEE/ACM Trans Netw 14(1):147–158
Xue G, Zhang W, Tang J, Thulasiraman K (2008) Polynomial time approximation algorithms for multi-constrained qos routing. IEEE/ACM Trans Netw 16(3):656–669
Acknowledgments
This research was partially supported by Natural Science Foundation of China #61300025, Natural Science Foundation of Fujian Province #2012J05115, and Doctoral Funds of Ministry of Education of China for Young Scholars #20123514120013.
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Guo, L. Efficient approximation algorithms for computing k disjoint constrained shortest paths. J Comb Optim 32, 144–158 (2016). https://doi.org/10.1007/s10878-015-9934-2
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DOI: https://doi.org/10.1007/s10878-015-9934-2