arXiv:2611.14786 · v1

Source Assignment: Hardness, Approximation, and Parameterized Algorithms

首次提交 2026-11-11 06:17

版本更新 2026-11-11 06:17

雷达观测 2026-11-13 10:09

摘要

In Source Assignment, each pointer represents a substring of a common reference string and may have several possible source occurrences. The goal is to choose one source for each pointer while minimizing the number of distinct reference boundaries used by the selected sources. The problem was introduced by Qiu and Kingsford (Bioinformatics 2021) in the context of compact genome-graph construction. We distinguish \emph{Explicit Source Assignment}, where admissible source sets are given explicitly, from \emph{Complete Source Assignment}, where all occurrences of each pointer string are available. We show that Explicit Source Assignment has an exact value-preserving formulation as cardinality Set Cover with Pairs. For every fixed $\varepsilon>0$, it cannot be approximated within $(1-\varepsilon)\ln p$ unless $\mathrm{P}=\mathrm{NP}$, even over a binary alphabet when all pointers have length one, while known algorithms for Minimum-Weight Multicolored Subgraph give an $O(\sqrt{q\log p})$-approximation for arbitrary pointer lengths. Our main hardness result proves that Complete Source Assignment is APX-hard even over a binary alphabet, with equal pointer lengths and at most two sources per pointer. For edge-disjoint equal-length instances, we give a $3/2$-approximation and a $1.12585$-approximation when each pointer has at most two sources. We also introduce the rainbow-path length $λ$: the problem is polynomial-time solvable for $λ\le2$, and for $λ\le3$ we obtain a $(9/8+\varepsilon)$-approximation. We further give a polynomial-time test for deciding whether abstract source sets can be realized exactly as complete occurrence sets over an unrestricted alphabet. Finally, we give exact fixed-parameter algorithms for several structural parameters, and show that Complete Source Assignment is fixed-parameter tractable in the alphabet size and maximum pointer length.