[Paper Review] Approximation and Hardness for Token Swapping

Tillmann Miltzow, Lothar Narins+4 more|arXiv (Cornell University)|Jan 1, 2016

Genome Rearrangement Algorithms7 references21 citations

TL;DR

This paper establishes tight algorithmic bounds for the token swapping problem on graphs, proving it is NP-complete and APX-hard, and showing it cannot be solved in 2^o(n) time under the Exponential Time Hypothesis. It presents a 4-approximation algorithm for the colored version and a 2-approximation for trees, complemented by a 2^O(n log n) exact algorithm and matching lower bounds, resolving long-standing open questions about approximability and conditional hardness for general graphs.

ABSTRACT

Given a graph G=(V,E) with V={1,...,n}, we place on every vertex a token T_1,...,T_n. A swap is an exchange of tokens on adjacent vertices. We consider the algorithmic question of finding a shortest sequence of swaps such that token T_i is on vertex i. We are able to achieve essentially matching upper and lower bounds, for exact algorithms and approximation algorithms. For exact algorithms, we rule out any 2^{o(n)} algorithm under the ETH. This is matched with a simple 2^{O(n*log(n))} algorithm based on a breadth-first search in an auxiliary graph. We show one general 4-approximation and show APX-hardness. Thus, there is a small constant delta > 1 such that every polynomial time approximation algorithm has approximation factor at least delta. Our results also hold for a generalized version, where tokens and vertices are colored. In this generalized version each token must go to a vertex with the same color.

Motivation & Objective

• To resolve the open problem of whether the token swapping problem is polynomial-time solvable or NP-complete on general graphs.
• To determine the existence of a constant-factor polynomial-time approximation algorithm for general graphs.
• To establish tight conditional lower bounds on the complexity of exact and approximation algorithms.
• To extend results to the colored token swapping variant, where tokens and vertices have colors and tokens must reach vertices of matching color.

Proposed method

• Design a novel gadget called an even permutation network to simulate arbitrary even permutations of tokens between input and output vertices.
• Construct a linear reduction from 3SAT to a structured graph problem involving disjoint paths, then to the colored token swapping problem.
• Reduce the colored token swapping problem to the uncolored version by attaching even permutation networks to each color class, preserving linear blow-up.
• Use the Sparsification Lemma to ensure reductions preserve subexponential time complexity, enabling ETH-based lower bounds.
• Develop a 4-approximation algorithm for the colored token swapping problem using structural insights into token movement and path assignment.
• Present a 2-approximation for trees based on tree-specific properties and token path decomposition.

Experimental results

Research questions

• RQ1 Is the token swapping problem on general graphs NP-complete?
• RQ2 Does a constant-factor polynomial-time approximation algorithm exist for the general token swapping problem?
• RQ3 Can the problem be solved in subexponential time, specifically 2^o(n), under the Exponential Time Hypothesis?
• RQ4 Is the colored token swapping problem, where tokens must reach vertices of the same color, also NP-hard and APX-hard?
• RQ5 Can the hardness results be extended to bounded-degree graphs?

Key findings

• The token swapping problem is NP-complete, even for graphs of bounded degree.
• The problem cannot be solved in 2^o(n) time unless the Exponential Time Hypothesis fails, establishing a tight lower bound.
• The problem is APX-hard, implying no polynomial-time approximation scheme exists unless P=NP, and no constant-factor approximation better than some δ>1 exists.
• A 4-approximation algorithm is presented for the colored token swapping problem on general graphs.
• A 2-approximation algorithm is given for trees, improving upon prior results.
• An exact algorithm runs in 2^O(n log n) time, which is nearly optimal given the 2^o(n) lower bound under ETH.