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

2025-03-20

AI Summary

This paper studies the $k$-pair shortest vertex-disjoint paths problem (SDPP) on planar graphs where terminal vertices appear an odd number of times on exactly two faces—termed *odd two-face planar SDPP*. We present the first randomized polynomial-time and RNC parallel algorithms for this NP-hard problem, achieving exact solutions. Methodologically, we introduce a novel integration of algebraic elimination with combinatorial structure: we construct a *dual-involution proof framework* based on DAG matrices and employ modular $2^m$-linear equation solving. We prove that the associated algebraic model is both triangularizable and invertible. Furthermore, we generalize our approach to the broader $(A+B)$-SDPP model and provide its first RNC parallel algorithm. This work fills a fundamental theoretical gap in parallel algorithms for disjoint paths under odd-face distribution constraints and establishes a new paradigm for path optimization in planar graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 Bj""orklund 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.

Problem

• Solves the Odd two-face planar SDPP in randomized polynomial time.
• First parallel solution for the Odd two-face planar SDPP problem.
• Combines algebraic and combinatorial methods for parallel algorithm.

Innovation

• Parallel algorithm for Odd two-face SDPP
• Combines randomized and deterministic approaches
• Uses DAG interpretation for matrix inversion