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On the Complexity of Secluded Path Problems

Tesshu Hanaka, Daisuke Tsuru

Abstract

This paper investigates the complexity of finding secluded paths in graphs. We focus on the \textsc{Short Secluded Path} problem and a natural new variant we introduce, \textsc{Shortest Secluded Path}. Formally, given an undirected graph $G=(V, E)$, two vertices $s,t\in V$, and two integers $k,l$, the \textsc{Short Secluded Path} problem asks whether there exists an $s$-$t$ path of length at most $k$ with at most $l$ neighbors. This problem is known to be computationally hard: it is W[1]-hard when parameterized by the path length $k$ or by cliquewidth, and para-NP-complete when parameterized by the number $l$ of neighbors. The fixed-parameter tractability is known for $k+l$ or treewidth. In this paper, we expand the parameterized complexity landscape by designing (1) an XP algorithm parameterized by cliquewidth and (2) fixed-parameter algorithms parameterized by neighborhood diversity and twin cover number, respectively. As a byproduct, our results also yield parameterized algorithms for the classic \textsc{$s$-$t$ $k$-Path} problem under the considered parameters. Furthermore, we introduce the \textsc{Shortest Secluded Path} problem, which seeks a shortest $s$-$t$ path with the minimum number of neighbors. In contrast to the hardness of the original problem, we reveal that this variant is solvable in polynomial time on unweighted graphs. We complete this by showing that for edge-weighted graphs, the problem becomes W[1]-hard yet remains in XP when parameterized by the shortest path distance between $s$ and $t$.

On the Complexity of Secluded Path Problems

Abstract

This paper investigates the complexity of finding secluded paths in graphs. We focus on the \textsc{Short Secluded Path} problem and a natural new variant we introduce, \textsc{Shortest Secluded Path}. Formally, given an undirected graph , two vertices , and two integers , the \textsc{Short Secluded Path} problem asks whether there exists an - path of length at most with at most neighbors. This problem is known to be computationally hard: it is W[1]-hard when parameterized by the path length or by cliquewidth, and para-NP-complete when parameterized by the number of neighbors. The fixed-parameter tractability is known for or treewidth. In this paper, we expand the parameterized complexity landscape by designing (1) an XP algorithm parameterized by cliquewidth and (2) fixed-parameter algorithms parameterized by neighborhood diversity and twin cover number, respectively. As a byproduct, our results also yield parameterized algorithms for the classic \textsc{- -Path} problem under the considered parameters. Furthermore, we introduce the \textsc{Shortest Secluded Path} problem, which seeks a shortest - path with the minimum number of neighbors. In contrast to the hardness of the original problem, we reveal that this variant is solvable in polynomial time on unweighted graphs. We complete this by showing that for edge-weighted graphs, the problem becomes W[1]-hard yet remains in XP when parameterized by the shortest path distance between and .
Paper Structure (16 sections, 15 theorems, 7 equations, 4 figures)

This paper contains 16 sections, 15 theorems, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Preliminaries
  5. Cliquewidth
  6. Neighborhood diversity and twin cover
  7. Integer linear programming
  8. XP Algorithm Parameterized by Cliquewidth
  9. FPT Algorithm Parameterized by Neighborhood Diversity
  10. FPT algorithm for $s$-$t$ $k$-Path
  11. FPT algorithm for Short Secluded Path
  12. FPT Algorithm Parameterized by Twin Cover Number
  13. Computational Complexity of Shortest Secluded Path
  14. Polynomial-time algorithm on unweighted graphs
  15. Shortest Secluded Path on weighted graphs
  16. ...and 1 more sections

Key Result

Theorem 1

$p$-Variable Integer Linear Programming Feasibility can be solved in $O(p^{2.5p+o(p)}\cdot L)$ time, where $L$ is the number of bits in the input.

Figures (4)

  • Figure 1: An illustration of a 5-secluded $s$-$t$ path $P$ of length $5$. The $s$-$t$ path $P$ consists of the red vertices. The blue vertices are neighbors of $P$.
  • Figure 2: Parameterized complexity of Short Secluded Path with respect to structural graph parameters. The connection between two parameters indicates that the upper parameter $p$ is bounded by some computable function $f(\cdot)$ of the lower parameter $q$; that is, $p \leq f(q)$. Parameters marked with an asterisk ($*$) represent our contributions in this paper. Double-bordered rectangles indicate that the parameterized problem belongs to XP but is W[1]-hard; rounded rectangles indicate that the parameterized problem is fixed-parameter tractable (FPT); and dotted rectangles represent cases that remain open.
  • Figure 3: An illustration of a partition of $V$ with respect to a path $P$ in Claim \ref{['tc_vertex']}.
  • Figure 4: The reduction from Multicolored Clique to Shortest Secluded Path in \ref{['thm:weighted:W[1]']}. The weight of each thin edge is $1$ and the weight of each bold edge is $k+1$.

Theorems & Definitions (36)

  • Definition 1: $k$-labeled graph
  • Definition 2: Cliquewidth
  • Definition 3: Neighborhood diversity
  • Definition 4: $p$-Variable Integer Linear Programming Feasibility ($p$-ILP)
  • Theorem 1: mor/Lenstra83combinatorica/FrankT87mor/Kannan87
  • Theorem 2
  • Lemma 1
  • proof
  • Corollary 1
  • Lemma 2
  • ...and 26 more