Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths

TL;DR

This work studies Maximum Vertex-Disjoint Shortest Paths in edge-weighted graphs with terminal pairs, seeking to maximize the number of pairs connected by vertex-disjoint shortest paths. Building on Lochet's result for fixed $k$, the authors establish tight approximation boundaries under gap-ETH, including no $o(k)$-approximation in time $f(k)\poly(n)$ and no $ceil(sqrt(ell))$-approximation improvements beyond this bound, while providing a simple $ceil(sqrt(ell))$-approximation and trivial $k$- or $ck$-approximations. They also present a fixed-parameter tractable algorithm with running time $2^{O(\ell)}\poly(n)$ for parameter $\ell$, along with strong kernelization lower bounds (no polynomial kernel under standard assumptions) and ETH-based time lower bounds, as well as planarity-specific hardness. The results together delineate a sharp boundary between approximation, parameterized complexity, and kernelization for this problem, and point to lossy kernel approaches as a promising direction for future work.

Abstract

We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) $n$-vertex graph $G$ along with $k$ terminal pairs $(s_1,t_1),(s_2,t_2),\ldots,(s_k,t_k)$. The task is to connect as many terminal pairs as possible by pairwise vertex-disjoint paths such that each path is a shortest path between the respective terminals. Our work is anchored in the recent breakthrough by Lochet [SODA '21], which demonstrates the polynomial-time solvability of the problem for a fixed value of $k$. Lochet's result implies the existence of a polynomial-time $ck$-approximation for Maximum Vertex-Disjoint Shortest Paths, where $c \leq 1$ is a constant. Our first result suggests that this approximation algorithm is, in a sense, the best we can hope for. More precisely, assuming the gap-ETH, we exclude the existence of an $o(k)$-approximations within $f(k)$poly($n$) time for any function $f$ that only depends on $k$. Our second result demonstrates the infeasibility of achieving an approximation ratio of $n^{\frac{1}{2}-\varepsilon}$ in polynomial time, unless P = NP. It is not difficult to show that a greedy algorithm selecting a path with the minimum number of arcs results in a $\sqrt{\ell}$-approximation, where $\ell$ is the number of edges in all the paths of an optimal solution. Since $\ell \leq n$, this underscores the tightness of the $n^{\frac{1}{2}-\varepsilon}$-inapproximability bound. Additionally, we establish that the problem can be solved in $2^{O(\ell)}$poly($n$) time, but does not admit a polynomial kernel in $\ell$. Moreover, it cannot be solved in $2^{o(\ell)}$poly($n$) time unless the ETH fails. Our hardness results hold for undirected graphs with unit weights, while our positive results extend to scenarios where the input graph is directed and features arbitrary (non-negative) edge weights.

Figures (7)

• Figure 1: An illustration of the reduction from Multicolored Clique to Maximum Vertex-Disjoint Shortest Paths. Top right: Example instance for Multicolored Clique with $k=4$ colors and $n=4$ vertices per color. A multicolored clique is highlighted (by thick edges). Bottom left: The constructed instance with the four shortest paths corresponding to the vertices of the clique highlighted. Note that these paths are pairwise disjoint. The dotted edges (incident to $s_i$ and $t_i$ vertices) indicate binary trees (where all leaves have distance $\lceil\log \nu \rceil$ from the root). Red edges indicate paths of length $2\nu$ and blue edges indicate paths of length $2$.
• Figure 2: An illustration of the reduction from Clique to Maximum Vertex-Disjoint Shortest Paths. Left side: Example instance for Clique with a highlighted solution (by thick edges). Right side: The constructed instance with the four shortest paths corresponding to the solution on the left side highlighted. Note that each shortest $s_i$-$t_i$-path uses exactly two of the diagonal edges.
• Figure 3: The graph $G_\phi$ associated with the formula $\phi = (x_1 \lor x_2 \lor x_3) \land (\neg x_1 \lor \neg x_3 \lor \neg x_4) \land (\neg x_1 \lor \neg x_4 \lor \neg x_5).$
• Figure 4: The gadget constructed for vertex $v_1$ in \ref{['fig:examplepm3']}. One of the two highest vertices is a dummy vertex.
• Figure 5: The gadget constructed for a clause with three literals. The vertices $a,b,$ and $c$ are the three corresponding neighbors in variable gadgets of vertices in the clause gadget. Each edge represents a path of length $5$ and the thick edges show the shortest paths between the terminals that are used if vertex $c$ is free (not used in a solution path in the corresponding variable gadget).

Theorems & Definitions (18)

• Theorem 1
• proof
• Corollary 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Proposition 1: folklore