Polylogarithmic Approximation for Robust s-t Path

TL;DR

The first polylogarithmic approximation for robust $s-$t$ path since the problem was proposed more than two decades ago is given, built on a novel linear program formulation for a decision-tree-type structure which enables the algorithm to get rid of the integrality gap of the natural flow LP.

Abstract

The paper revisits the robust $s$-$t$ path problem, one of the most fundamental problems in robust optimization. In the problem, we are given a directed graph with $n$ vertices and $k$ distinct cost functions (scenarios) defined over edges, and aim to choose an $s$-$t$ path such that the total cost of the path is always provable no matter which scenario is realized. With the view of each cost function being associated with an agent, our goal is to find a common $s$-$t$ path minimizing the maximum objective among all agents, and thus create a fair solution for them. The problem is hard to approximate within $o(\log k)$ by any quasi-polynomial time algorithm unless $\mathrm{NP} \subseteq \mathrm{DTIME}(n^{\mathrm{poly}\log n})$, and the best approximation ratio known to date is $\widetilde{O}(\sqrt{n})$ which is based on the natural flow linear program. A longstanding open question is whether we can achieve a polylogarithmic approximation even when a quasi-polynomial running time is allowed. We give the first polylogarithmic approximation for robust $s$-$t$ path since the problem was proposed more than two decades ago. In particular, we introduce a $O(\log n \log k)$-approximate algorithm running in quasi-polynomial time. The algorithm is built on a novel linear program formulation for a decision-tree-type structure which enables us to get rid of the $Ω(\max\{k,\sqrt{n}\})$ integrality gap of the natural flow LP. Further, we also consider some well-known graph classes, e.g., graphs with bounded treewidth, and show that the polylogarithmic approximation can be achieved polynomially on these graphs. We hope the new proposed techniques in the paper can offer new insights into the robust $s$-$t$ path problem and related problems in robust optimization.

Figures (10)

• Figure 1: Hard instance for \ref{['En-Flow-LP']}. The optimal integral solution must take one of $k$ paths while the optimal fractional solution takes each path by $\frac{1}{k}$.
• Figure 2: An example of the decomposition tree of a series-parallel graph. The series-parallel graph $G:=(V,E)$ is shown on the left and its decomposition tree $\mathbf{T}:=(\mathbf{V},\mathbf{E})$ is shown on the right. Each leaf node in $\mathbf{T}$ corresponds to an edge in $E$. Each internal node is either a series node S or a parallel node P. And it indicates how to merge the child nodes' subgraph. For example, consider the S node and its two child nodes $e_{13}$ and $e_{15}$. Then, the subgraph of this S node is $e_{13}\to e_{15}$ which merges its two child nodes' subgraph via the series composition. And also, the subgraph of this S node's parent corresponds to the subgraph $B$ in $G$, which merges $e_{13}\to e_{15}$ and $e_{14}\to e_{16}$ via the parallel composition. An $s \text{-} t$ path corresponds to a feasible subtree (\ref{['def:sp:feasible-subtree']}). For example, the feasible subtree $\mathbf{T}'$ can be converted to an $s \text{-} t$ path $e_{2}\to e_{8} \to e_{12} \to e_{18}$.
• Figure 3: An example for the metatree $\mathbf{T}$ construction. The given directed graph $G$ is shown in the up-left corner. Since there are three vertices in $G$, $\mathbf{T}$ will consist of five levels. The dashed subtree $\mathbf{T}_1$ corresponds to the path $s\to t$ of $G$. The dotted subtree $\mathbf{T}_2$ represents the path $s\to a \to t$ of $G$. The dash-dotted subtree $\mathbf{T}_3$ also corresponds to the path $s\to t$. By the definition of the cost function $f$, it is not hard to verify that the cost of each subtree is equal to the cost of its corresponding path.
• Figure 4: An example of the reduction. The subfigure (i) is the given directed graph. Then, we compute a tree decomposition with the logarithmic depth and add $s$ and $t$ to all nodes, which is shown in subfigure (ii). The edge set next to each node in subfigure (ii) is its corresponding $E_{\mathbf{v}}$. For example, the edge set $E_{\mathbf{r}}$ for root node $\mathbf{r}$ is $\set{(d,f),(f,t)}$ since $\mathbf{r}$ is the highest node that contains edge $(d,f)$ and $(f,t)$. Subfigure (iii) is an $s \text{-} t$ path of the given directed graph and subfigure (iv) is the corresponding label assignment of the $s \text{-} t$ path in (iii) in which we only list these labels with the value of $\textup{1}$. The complete label of each node is obtained by merging these single labels, e.g., for the root $\mathbf{r}$, $l_{\mathbf{r}}$ consists of $14$ bits ($2$ choosing labels and $12$ connectivity labels). In these $14$ bits, only $\textup{conn}(s,d),\textup{conn}(s,t),\textup{conn}(d,t)$ has a value of $\textup{1}$ and all the remaining $11$ labels have a value of $\textup{0}$. Subfigure (v) shows another example and subfigure (vi) is its corresponding label assignment. By our construction, it is possible that a feasible label assignment corresponds to a subgraph instead of a single $s \text{-} t$ path.
• Figure 5: The constructed series-parallel graph according to the 2-choose-1 set cover instance.

Theorems & Definitions (52)

• Theorem 1
• Definition 1: Series-Parallel Graph
• Definition 2: Feasible Subtree
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 2
• Definition 3: Feasible Subtree for General Graphs