Consider a graph $G=(V,E)$ and a source-destination pair $(s,t)$. A set of edges $E'\subseteq E$ are vulnerable in the sense that at most $k$ of them may fail. My problem is to find a set of $(s,t)$ paths of minimum total cost such that no matter which subset of $k$ edges in $E'$ fail, we can ensure that there exist at least a functional $(s,t)$ path. My questions are: (1) Is this problem NP-hard? (2) If yes, how to design approximation algorithm?
$\begingroup$
$\endgroup$
2
-
$\begingroup$ By fail, you mean that one of your paths still exists in the smaller gragh graph $G'$ with those edges removed? $\endgroup$JoshuaZ– JoshuaZ2024-06-13 12:08:44 +00:00Commented Jun 13, 2024 at 12:08
-
$\begingroup$ @JoshuaZ By fail I mean that the path traversing the edge is broken. I want to ensure that no matter which $k$ edges in $E'$ are broken, there is a non-broken $st$-path. $\endgroup$lchen– lchen2024-06-14 03:26:47 +00:00Commented Jun 14, 2024 at 3:26
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
4
For $(i,j)\in A$, let $c_{ij}$ be the arc cost. Let $P$ be the set of all $(s,t)$ paths. Let binary decision variable $\lambda_p$ indicate whether path $p\in P$ is selected, with arc set $A_p$ and cost $c(P) = \sum_{(i,j)\in A_p} c_{ij}$. The problem is to minimize $$\sum_{p\in P} c(P) \lambda_p \tag0\label0$$ subject to \begin{align} \sum_{p\in P: A_p \cap S = \emptyset} \lambda_p &\ge 1 &&\text{for all $S\subseteq E'$ such that $|S|\le k$} \tag1\label1\\ \lambda_p &\in\{0,1\} &&\text{for all $p\in P$} \tag2\label2 \end{align} Equivalently, you can replace $|S|\le k$ with $|S|=k$ in \eqref{1} and relax \eqref{2} to $\lambda_p \in \mathbb{Z}^+$.
There are an exponential number of variables, so you might use column generation to solve the problem exactly.
I suspect that you can demonstrate NP-hardness by reduction from weighted set cover. In any case, this problem is a special case of weighted set cover, so you can use any approximation algorithms for that problem, such as https://en.wikipedia.org/wiki/Set_cover_problem#Greedy_algorithm.
-
$\begingroup$ Thank you Rob for your comments. I suppose by set cover, you probably mean mapping each set into a path. However, there is an exponential number of paths, so set cover algorithms may not lead to polynomial approximation algo in my case. $\endgroup$lchen– lchen2024-06-16 08:30:44 +00:00Commented Jun 16, 2024 at 8:30
-
$\begingroup$ To implement an approximation algorithm, you need not explicitly enumerate all paths. Where the algorithm specifies to “choose a set that…” you would instead “construct a path that…” $\endgroup$RobPratt– RobPratt2024-06-16 11:42:03 +00:00Commented Jun 16, 2024 at 11:42
-
$\begingroup$ I still cannot see how to map my problem to set cover. Could you pls provide more details on the mapping. Thank you anyway. $\endgroup$lchen– lchen2024-06-17 02:06:28 +00:00Commented Jun 17, 2024 at 2:06
-
$\begingroup$ In the notation of en.wikipedia.org/wiki/…, the universe $\mathcal{U}$ corresponds to the $k$-subsets of vulnerable edges, and each member of the family $\mathcal{S}$ corresponds to the set of $(s,t)$ paths that avoid a given $k$-subset. $\endgroup$RobPratt– RobPratt2024-06-17 02:25:36 +00:00Commented Jun 17, 2024 at 2:25