A Note on the Inapproximability of Induced Disjoint Paths

We study the inapproximability of the induced disjoint paths problem on an arbitrary $n$-node $m$-edge undirected graph, which is to connect the maximum number of the $k$ source-sink pairs given in the graph via induced disjoint paths. It is known that the problem is NP-hard to approximate within $m^{{1\over 2}-\varepsilon}$ for a general $k$ and any $\varepsilon>0$. In this paper, we prove that the problem is NP-hard to approximate within $n^{1-\varepsilon}$ for a general $k$ and any $\varepsilon>0$ by giving a simple reduction from the independent set problem.

💡 Research Summary

The paper investigates the hardness of approximation for the Induced Disjoint Paths Problem (IDPP) on arbitrary undirected graphs. In IDPP we are given a graph G and a collection of k source‑sink vertex pairs; the goal is to connect as many pairs as possible with paths that are both induced (no chords) and mutually vertex‑disjoint and non‑adjacent. Prior work (Zhang et al., 2011) established that approximating IDPP within a factor of m^{1/2‑ε} is NP‑hard, where m is the number of edges. The authors improve this bound dramatically by proving that, for any ε > 0, achieving an approximation ratio of n^{1‑ε} (where n is the number of vertices) is also NP‑hard.

The main technical contribution consists of two lemmas that together yield the final theorem. Lemma 1 shows a scaling argument: if an algorithm could approximate IDPP within n^{1‑ε} on all instances, then one can construct an algorithm that approximates within (n³)^{1‑ε′} for some ε′ > 0. The construction simply runs a brute‑force optimal solver when the instance is tiny (n < 3^{1+1/ε}) and otherwise invokes the assumed n^{1‑ε} algorithm. By choosing ε′ = ε² the inequality (n³)^{1‑ε′} ≥ n^{1‑ε} holds, establishing the claim.

Lemma 2 provides a clean, approximation‑preserving reduction from the Maximum Independent Set problem to IDPP. Starting from an arbitrary graph G′ with n′ vertices, the authors add for each original vertex two new auxiliary vertices and connect each auxiliary vertex to its original counterpart. The resulting graph G has exactly 3n′ vertices and m′ + 2n′ edges. The k = n′ source‑sink pairs are taken to be the auxiliary vertex pairs. A set of t induced disjoint paths in G corresponds precisely to an independent set of size t in G′, because each path must use one auxiliary pair and cannot share the original vertex with another path. Consequently, any approximation algorithm for IDPP would yield an equally good approximation for Independent Set.

Since it is known (Zuckerman, 2007) that approximating Maximum Independent Set (or equivalently Maximum Clique) within n^{1‑ε} is NP‑hard on general graphs, the reduction implies that approximating IDPP within (3n′)^{1‑ε} = Θ(n^{1‑ε}) is also NP‑hard. Combining Lemma 1 and Lemma 2 gives Theorem 1: no polynomial‑time algorithm can achieve an n^{1‑ε} approximation for IDPP unless P = NP.

The authors emphasize that this result is strictly stronger than the earlier m^{1/2‑ε} bound because m^{1/2} ≤ n for any simple graph, and the new bound becomes especially tight for sparse graphs where m = O(n^{α}) with α < 2. They also note the relationship to the Node‑Disjoint Paths problem (DPP), whose best known hardness is logarithmic, highlighting that IDPP is intrinsically harder.

In summary, the paper delivers a concise yet powerful hardness proof for IDPP, establishing that even a polynomial‑factor approximation proportional to n^{1‑ε} is unlikely. The reduction is elementary, preserving approximation ratios, and leverages well‑studied inapproximability of Independent Set. This advances our understanding of the limits of efficient routing in networks where interference constraints force induced, non‑adjacent paths. Future work may explore tighter bounds for special graph families or investigate parameterized algorithms that circumvent the general hardness.