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Sorting by Strip Swaps is NP-Hard

Swapnoneel Roy, Asai Asaithambi, Debajyoti Mukhopadhyay

TL;DR

The paper proves that Sorting by Strip Swaps (SbSS) is NP-hard by a polynomial reduction from Block Sorting using a schedule-free construction. Central to the reduction are two gadgets: cages isolate each reversal boundary to ensure internal decreases only, and hinges couple neighboring cages to enforce global consistency, creating a bijection between exact SbSS schedules and perfect block schedules of length $R=rev(\pi)$. A relabeling step preserves increasing adjacencies while allowing the concatenation of gadgets into a single permutation $\pi^{\dagger}$ with exactly $2R$ decreases. Together, these components yield a robust reduction that transfers NP-hardness from Block Sorting to SbSS, resolving the complexity status of SbSS and aligning with existing hardness results for related genomic rearrangement problems.

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.

Sorting by Strip Swaps is NP-Hard

TL;DR

The paper proves that Sorting by Strip Swaps (SbSS) is NP-hard by a polynomial reduction from Block Sorting using a schedule-free construction. Central to the reduction are two gadgets: cages isolate each reversal boundary to ensure internal decreases only, and hinges couple neighboring cages to enforce global consistency, creating a bijection between exact SbSS schedules and perfect block schedules of length . A relabeling step preserves increasing adjacencies while allowing the concatenation of gadgets into a single permutation with exactly decreases. Together, these components yield a robust reduction that transfers NP-hardness from Block Sorting to SbSS, resolving the complexity status of SbSS and aligning with existing hardness results for related genomic rearrangement problems.

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 by a guarded triple enclosed by guards , 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.

Paper Structure

This paper contains 11 sections, 6 theorems, 6 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Sorting By Strip Swaps
  3. Overview of Results and Techniques
  4. Lower Bounds for Sorting by Strip Swaps
  5. The Reduction
  6. Cage gadget (local isolation)
  7. Hinge gadget (coupling shared elements)
  8. Relabeling
  9. Correctness
  10. Worked Examples
  11. Conclusion

Key Result

Lemma 1

$\textnormal{SSD}(\pi)\geq\left \lceil \textnormal{rev}(\pi)/2\right \rceil$, where $\textnormal{rev}(\pi)$ is the number of reversals in $\pi$.

Figures (4)

  • Figure 1: Cage gadget for a reversal boundary $(a_i,a_{i+1})$: only internal decreases exist.
  • Figure 2: Hinge gadget: two cages sharing $a_j$ coupled by $h_j^L,h_j^R$.
  • Figure 3: YES instance $\pi_Y=4\,1\,3\,2$: two independent cages; two local strip swaps (each $-2$) sort $\pi_Y^\dagger$ in $R=2$ moves.
  • Figure 4: NO instance $\pi_N=7\,2\,6\,5\,8\,3\,1\,4$: two cages share element $3$ and are coupled by a hinge. Any attempt to resolve one cage in isolation leaves a hinge penalty, so an exact $R$-swap schedule does not exist.

Theorems & Definitions (13)

  • Definition 1: Strip Swap Distance
  • Definition 2: Reversal
  • Lemma 1
  • Lemma 2: Forward (existence)
  • proof
  • Lemma 3: Compatibility of $-2$ moves
  • proof
  • Lemma 4: Projection
  • proof
  • Theorem 1
  • ...and 3 more