Tight Parameterized (In)tractability of Layered Crossing Minimization: Subexponential Algorithms and Kernelization
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2025
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| _version_ | 1866912648632729600 |
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| author | Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Saurabh, Saket Zehavi, Meirav |
| author_facet | Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Saurabh, Saket Zehavi, Meirav |
| contents | The starting point of our work is a decade-old open question concerning the subexponential parameterized complexity of \textsc{2-Layer Crossing Minimization}. In this problem, the input is an $n$-vertex graph $G$ whose vertices are partitioned into two independent sets $V_1$ and $V_2$, and a non-negative integer $k$. The question is whether $G$ admits a 2-layered drawing with at most $k$ crossings, where each $V_i$ lies on a distinct line parallel to the $x$-axis, and all edges are straight lines. We resolve this open question by giving the first subexponential fixed-parameter algorithm for this problem, running in time $2^{O(\sqrt{k}\log k)} + n \cdot k^{O(1)}$.
We then ask whether the subexponential phenomenon extends beyond two layers. In the general $h$-Layer Crossing Minimization problem, the vertex set is partitioned into $h$ independent sets $V_1, \ldots, V_h$, and the goal is to decide whether an $h$-layered drawing with at most $k$ crossings exists. We present a subexponential FPT algorithm for three layers with running time $2^{O(k^{2/3}\log k)} + n \cdot k^{O(1)}$ for $h = 3$ layers. In contrast, we show that for all $h \ge 5$, no algorithm with running time $2^{o(k/\log k)} \cdot n^{O(1)}$ exists unless the Exponential-Time Hypothesis fails.
Finally, we address polynomial kernelization. While a polynomial kernel was already known for $h=2$, we design a new polynomial kernel for $h=3$. These kernels are essential ingredients in our subexponential algorithms. Finally, we rule out polynomial kernels for all $h \ge 4$ unless the polynomial hierarchy collapses. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_13335 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tight Parameterized (In)tractability of Layered Crossing Minimization: Subexponential Algorithms and Kernelization Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Saurabh, Saket Zehavi, Meirav Data Structures and Algorithms Computational Geometry The starting point of our work is a decade-old open question concerning the subexponential parameterized complexity of \textsc{2-Layer Crossing Minimization}. In this problem, the input is an $n$-vertex graph $G$ whose vertices are partitioned into two independent sets $V_1$ and $V_2$, and a non-negative integer $k$. The question is whether $G$ admits a 2-layered drawing with at most $k$ crossings, where each $V_i$ lies on a distinct line parallel to the $x$-axis, and all edges are straight lines. We resolve this open question by giving the first subexponential fixed-parameter algorithm for this problem, running in time $2^{O(\sqrt{k}\log k)} + n \cdot k^{O(1)}$. We then ask whether the subexponential phenomenon extends beyond two layers. In the general $h$-Layer Crossing Minimization problem, the vertex set is partitioned into $h$ independent sets $V_1, \ldots, V_h$, and the goal is to decide whether an $h$-layered drawing with at most $k$ crossings exists. We present a subexponential FPT algorithm for three layers with running time $2^{O(k^{2/3}\log k)} + n \cdot k^{O(1)}$ for $h = 3$ layers. In contrast, we show that for all $h \ge 5$, no algorithm with running time $2^{o(k/\log k)} \cdot n^{O(1)}$ exists unless the Exponential-Time Hypothesis fails. Finally, we address polynomial kernelization. While a polynomial kernel was already known for $h=2$, we design a new polynomial kernel for $h=3$. These kernels are essential ingredients in our subexponential algorithms. Finally, we rule out polynomial kernels for all $h \ge 4$ unless the polynomial hierarchy collapses. |
| title | Tight Parameterized (In)tractability of Layered Crossing Minimization: Subexponential Algorithms and Kernelization |
| topic | Data Structures and Algorithms Computational Geometry |
| url | https://arxiv.org/abs/2510.13335 |