Improved Approximation Algorithms for Computing k Disjoint Paths Subject to Two Constraints

For a given graph $G$ with positive integral cost and delay on edges, distinct vertices $s$ and $t$, cost bound $C\in Z^{+}$ and delay bound $D\in Z^{+}$, the $k$ bi-constraint path ($k$BCP) problem is to compute $k$ disjoint $st$-paths subject to $C$ and $D$. This problem is known NP-hard, even when $k=1$ \cite{garey1979computers}. This paper first gives a simple approximation algorithm with factor-$(2,2)$, i.e. the algorithm computes a solution with delay and cost bounded by $2*D$ and $2*C$ respectively. Later, a novel improved approximation algorithm with ratio $(1+\beta,\,\max\{2,\,1+\ln\frac{1}{\beta}\})$ is developed by constructing interesting auxiliary graphs and employing the cycle cancellation method. As a consequence, we can obtain a factor-$(1.369,\,2)$ approximation algorithm by setting $1+\ln\frac{1}{\beta}=2$ and a factor-$(1.567,\,1.567)$ algorithm by setting $1+\beta=1+\ln\frac{1}{\beta}$. Besides, by setting $\beta=0$, an approximation algorithm with ratio $(1,\, O(\ln n))$, i.e. an algorithm with only a single factor ratio $O(\ln n)$ on cost, can be immediately obtained. To the best of our knowledge, this is the first non-trivial approximation algorithm for the $k$BCP problem that strictly obeys the delay constraint.Comment: 12 page

Efficient approximation algorithms for computing k-disjoint minimum cost paths with delay constraint

For a given graph G with distinct vertices s, t and a given delay constraint D ∈ R+, the k-disjoint restricted shortest path (kRSP) problem of computing k-disjoint minimum cost stpaths with total delay restrained by D, is known to be NP-hard. Bifactor approximation algorithms have been developed for its special case when k = 2, while no approximation algorithm with constant single factor or bifactor ratio has been developed for general k. This paper firstly presents a (k, (1 + ε)H(k))-approximation algorithm for the kRSP problem by extending Orda's factor(1.5, 1.5) approximation algorithm [9]. Secondly, this paper gives a novel linear programming (LP) formula for the kRSP problem. Based on LP rounding technology, this paper rounds an optimal solution of this formula and obtains an approximation algorithm within a bifactor ratio of (2, 2). To the best of our knowledge, it is the first approximation algorithm with constant bifactor ratio for the kRSP problem. Our results can be applied to serve applications in networks which require quality of service and robustness simultaneously, and also have broad applications in construction of survivable networks and fault tolerance systems.Longkun Guo, Hong She