Improved Hardness-of-Approximation for Token Swapping

TL;DR

It is proved that 0/1-weighted token swapping is NP-hard to approximate with ratio better than $(1-\varepsilon) \ln(n)$ for any constant $\epsilon>0$.

Abstract

We study the token swapping problem, in which we are given a graph with an initial assignment of one distinct token to each vertex, and a final desired assignment (again with one token per vertex). The goal is to find the minimum length sequence of swaps of adjacent tokens required to get from the initial to final assignment. The token swapping problem is known to be NP-complete. It is also known to have a polynomial-time 4-approximation algorithm. From the hardness-of-approximation side, it is known to be NP-hard to approximate with ratio better than 1001/1000. Our main result is an improvement of the approximation ratio of the lower bound: We show that it is NP-hard to approximate with ratio better than 14/13. We then turn our attention to the 0/1-weighted version, in which every token has a weight of either 0 or 1, and the cost of a swap is the sum of the weights of the two participating tokens. Unlike standard token swapping, no constant-factor approximation is known for this version, and we provide an explanation. We prove that 0/1-weighted token swapping is NP-hard to approximate with ratio better than $(1-\varepsilon) \ln(n)$ for any constant $ε>0$. Lastly, we prove two barrier results for the standard (unweighted) token swapping problem. We show that one cannot beat the current best known approximation ratio of 4 using a large class of algorithms which includes all known algorithms, nor can one beat it using a common analysis framework.

Figures (2)

• Figure 2: The graph for a weighted token swapping instance constructed from a set cover instance with $U = \{u,w,x,y\}$ and $S_1 = \{u,w\}$, $S_2 = \{w,x\}$, and $S_3 = \{w,x,y\}$. Pairs of black vertices correspond to elements of $U$, while blue vertices correspond to sets. Two tokens from blue vertices must be displaced for tokens on black vertices to be moved to their destinations.
• Figure 3: An (incomplete) graph for a token swapping instance where $p = 8$, $q = 4$. The diagram depicts the outer cycle and one inner cycle, but leaves out the remaining inner cycles for legibility. In the outer cycle, vertices in even segments are colored blue, while those in odd segments are colored black. The token starting on $v_0$ has target $v_8$, the token starting on $v_8$ has target $v_{16}$, the token starting on $v_{16}$ has target $v_{24}$, and the token starting on $v_{24}$ has target $v_{0}$.

Theorems & Definitions (21)

• Theorem 4.1
• Definition 4.1: Set Cover
• Theorem 4.2
• Claim 4.3
• proof
• Definition 5.1: Local Optimality
• Theorem 5.1
• Claim 5.2
• proof
• Claim 5.3