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A parallel algorithm for the odd two-face shortest k-disjoint path problem

Srijan Chakraborty, Samir Datta

TL;DR

This work addresses the Odd Two-Face planar SDPP, where $2k$ terminals are split across two faces with odd counts per face, by developing a parallel, randomized $NC^2$ algorithm. The method encodes path configurations via permanents modulo a power of two and reduces the resulting linear system to a triangular form using two involutions, interpreting the system as a DAG to prove invertibility. The key contributions include a fully parallelizable solution for the two-face odd case and an $NC^2$ algorithm for the generalized $(A+B,q)$-SDPP, extending prior one-face and two-face planar results. The approach blends algebraic generating-function techniques with combinatorial cancellations, enabling efficient computation and providing new insights into the structure of disjoint-path problems on planar graphs.

Abstract

The shortest Disjoint Path problem (SDPP) requires us to find pairwise vertex disjoint paths between k designated pairs of terminal vertices such that the sum of the path lengths is minimum. The focus here is on SDPP restricted to planar graphs where all terminals are arbitrarily partitioned over two distinct faces with the additional restriction that each face is required to contain an odd number of terminals. We call this problem the Odd two-face planar SDPP. It is shown that this problem is solvable in randomized polynomial time and even in RNC. This is the first parallel (or even polynomial time) solution for the problem. Our algorithm combines ideas from the randomized solution for 2-SDPP by Björklund and Huslfeldt with its parallelization by Datta and Jaiswal along with the deterministic algorithm for One-face planar SDPP by Datta, Iyer, Kulkarni and Mukherjee. The proof uses a combination of two involutions to reduce a system of linear equations modulo a power of 2 to a system of triangular form that is, therefore, invertible. This, in turn, is proved by showing that the matrix of the equations, can be interpreted as (the adjacency matrix of) a directed acyclic graph (DAG). While our algorithm is primarily algebraic the proof remains combinatorial. We also give a parallel algorithm for the (A + B)-SDPP introduced by Hirai and Namba.

A parallel algorithm for the odd two-face shortest k-disjoint path problem

TL;DR

This work addresses the Odd Two-Face planar SDPP, where terminals are split across two faces with odd counts per face, by developing a parallel, randomized algorithm. The method encodes path configurations via permanents modulo a power of two and reduces the resulting linear system to a triangular form using two involutions, interpreting the system as a DAG to prove invertibility. The key contributions include a fully parallelizable solution for the two-face odd case and an algorithm for the generalized -SDPP, extending prior one-face and two-face planar results. The approach blends algebraic generating-function techniques with combinatorial cancellations, enabling efficient computation and providing new insights into the structure of disjoint-path problems on planar graphs.

Abstract

The shortest Disjoint Path problem (SDPP) requires us to find pairwise vertex disjoint paths between k designated pairs of terminal vertices such that the sum of the path lengths is minimum. The focus here is on SDPP restricted to planar graphs where all terminals are arbitrarily partitioned over two distinct faces with the additional restriction that each face is required to contain an odd number of terminals. We call this problem the Odd two-face planar SDPP. It is shown that this problem is solvable in randomized polynomial time and even in RNC. This is the first parallel (or even polynomial time) solution for the problem. Our algorithm combines ideas from the randomized solution for 2-SDPP by Björklund and Huslfeldt with its parallelization by Datta and Jaiswal along with the deterministic algorithm for One-face planar SDPP by Datta, Iyer, Kulkarni and Mukherjee. The proof uses a combination of two involutions to reduce a system of linear equations modulo a power of 2 to a system of triangular form that is, therefore, invertible. This, in turn, is proved by showing that the matrix of the equations, can be interpreted as (the adjacency matrix of) a directed acyclic graph (DAG). While our algorithm is primarily algebraic the proof remains combinatorial. We also give a parallel algorithm for the (A + B)-SDPP introduced by Hirai and Namba.

Paper Structure

This paper contains 15 sections, 24 theorems, 18 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Disjoint Paths on Two Faces: The Odd Case
  4. Notation and Modification:
  5. Axis-Crossing Number
  6. The Algorithm
  7. Proof of Correctness
  8. Cancelling Unwanted Path Configurations
  9. Invertibility of $\boldsymbol{M}$
  10. Shortest $(A+B,q)$ paths
  11. Conclusion
  12. Appendix
  13. Left out proofs from \ref{['sec:3']}
  14. Example
  15. Extracting bivariate coefficients modulo $2^k$

Key Result

Theorem 1

The decision as well as the search version of the Odd Two-Face $k$-$\SDPP$ is in randomized $\NC^2$ for constant $k$.

Figures (5)

  • Figure 1: Clockwise edges get multiplied by $y$, and anti-clockwise edges by $y^{-1}$.
  • Figure 4: $\mathcal{A}^{(1)}=(z^{(1)}_5,z^{(1)}_8)$ in both $(a),(b)$. In $(a)$, $\mathcal{N}^{(1)}=(2,2)$, whereas in $(b)$, $\mathcal{N}^{(1)}=(3,1)$. $\alpha^{(1)}(\mathcal{C})=\{z^{(1)}_3,z^{(1)}_4,z^{(1)}_6,z^{(1)}_7\}$ and $\{z^{(1)}_2,z^{(1)}_3,z^{(1)}_4,z^{(1)}_7\}$ in $(a)$ and $(b)$ respectively. $\Rp^{(1)}$ in both figures is $\{z^{(1)}_8\}$.
  • Figure 5: The circled terminal is the sink which is switched from $J_1$ to $\widetilde{J}_1$.
  • Figure 6: The circled terminal is the source switched from $J_2$ to $\widetilde{J}_2$.
  • Figure 7: The configurations are $\mathcal{C}_1$,$\mathcal{C}_2$ and $\mathcal{C}_3$ respectively.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Lemma 4
  • Lemma 5: Robertson-SeymourRSvi Rephrased
  • Lemma 6: Isolation Lemma MVV
  • Lemma 7
  • Proposition 8
  • Proposition 9
  • Lemma 10
  • ...and 15 more