[Paper Review] Complexity of Sequence-to-Graph Alignment with Co-Linear Chaining

Xingfu Li|arXiv (Cornell University)|Feb 5, 2026

Genomics and Phylogenetic Studies0 citations

TL;DR

The paper shows Gap-sensitive Co-Linear Chaining is unlikely to admit sub-quadratic algorithms under SETH, and Edit-CLC is NP-hard with graph errors, implying co-linear chaining does not reduce inherent complexity in sequence-to-graph alignment.

ABSTRACT

Sequence alignment is a cornerstone technique in computational biology for assessing similarities and differences among biological sequences. A key variant, sequence-to-graph alignment, plays a crucial role in effectively capturing genetic variations. In this work, we introduce two novel formulations within this framework: the Gap-sensitive Co-Linear Chaining (Gap-CLC) problem and the Co-Linear Chaining with Errors based on Edit Distance (Edit-CLC) problem, and we investigate their computational complexity. We show that solving the Gap-CLC problem in sub-quadratic time is highly unlikely unless the Strong Exponential Time Hypothesis fails -- even when restricted to binary alphabets. Furthermore, we establish that the Edit-CLC problem is NP-hard in the presence of errors within the pan-genome graph. These findings emphasize that incorporating co-linear structures into sequence-to-graph alignment models fails to reduce computational complexity, highlighting that these models remain at least as computationally challenging to solve as those lacking such prior information.

Motivation & Objective

• Motivate sequence-to-graph alignment as a fundamental tool for analyzing genomic variation.
• Introduce Gap-CLC and Edit-CLC as co-linear chaining formulations within this framework.
• Analyze computational hardness of Gap-CLC and Edit-CLC to understand limits of co-linear chaining.
• Show that co-linear chaining does not simplify complexity compared to non-co-linear models.

Proposed method

• Define anchors as Cartesian products of occurrences in query and pan-genome graphs to form implicit anchor-sets.
• Formulate Gap-CLC and Edit-CLC with gap-cost functions and anchor-chains.
• Provide linear-time reductions from known problems (Single-Exa-SGM and Single-Err-SGM) to Gap-CLC and Edit-CLC to establish complexity.
• Prove sub-quadratic hardness for Gap-CLC under SETH via a reduction, even on binary alphabets.
• Prove NP-hardness of Edit-CLC when graph errors are allowed, via a linear-time reduction from Single-Err-SGM.

Experimental results

Research questions

• RQ1 Does Gap-CLC admit a sub-quadratic time algorithm on general pan-genome graphs or even binary alphabets under SETH?
• RQ2 Is Edit-CLC NP-hard when allowing errors on the pan-genome graph, and does this hardness persist on binary alphabets?
• RQ3 Do co-linear chaining formulations in sequence-to-graph alignment reduce computational complexity relative to non-co-linear models?

Key findings

• Gap-CLC is unlikely to be solvable in sub-quadratic time unless the Strong Exponential Time Hypothesis fails, even for binary alphabets.
• Edit-CLC is NP-hard when errors are allowed on pan-genome graphs, even over a binary alphabet.
• A linear-time reduction from Single-Exa-SGM to Gap-CLC demonstrates sub-quadratic hardness for Gap-CLC.
• A linear-time reduction from Single-Err-SGM to Edit-CLC establishes NP-hardness for Edit-CLC with graph errors.
• Co-linear chaining in sequence-to-graph alignment does not reduce computational complexity compared to models without co-linearity.