Sorting by Strip Swaps is NP-Hard

TL;DR

The paper proves that the problem of sorting by strip swaps is NP-hard by reducing it from block sorting, using gadgets to establish a computational equivalence.

Contribution

It introduces a novel reduction from block sorting to sorting by strip swaps, demonstrating NP-hardness through a new gadget-based construction.

Findings

Sorting by Strip Swaps is NP-hard.

A polynomial reduction from Block Sorting is established.

Gadget constructions are used to prove computational complexity.

Abstract

We show that \emph{Sorting by Strip Swaps} (SbSS) is NP-hard by a polynomial reduction of \emph{Block Sorting}. The key idea is a local gadget, a \emph{cage}, that replaces every decreasing adjacency $(a_i,a_{i+1})$ by a guarded triple $a_i,m_i,a_{i+1}$ enclosed by guards $L_i,U_i$, so the only decreasing adjacencies are the two inside the cage. Small \emph{hinge} gadgets couple adjacent cages that share an element and enforce that a strip swap that removes exactly two adjacencies corresponds bijectively to a block move that removes exactly one decreasing adjacency in the source permutation. This yields a clean equivalence between exact SbSS schedules and perfect block schedules, establishing NP-hardness.