[Paper Review] On the pathwidth of almost semicomplete digraphs

Kenta Kitsunai, Yasuaki Kobayashi+1 more|arXiv (Cornell University)|Jul 7, 2015

Advanced Graph Theory Research15 references4 citations

TL;DR

This paper presents a fixed-parameter tractable algorithm for computing the pathwidth of $h$-semicomplete digraphs—generalizations of semicomplete digraphs where each vertex has at most $h$ non-neighbors—in time $(h+2k+1)^{2k}n^{O(1)}$ for parameter $k$. It further proves that any $h$-semicomplete digraph with pathwidth at least $f(k,h)$ contains a semicomplete subgraph of pathwidth at least $k$, enabling polynomial-time algorithms for topological containment in this class for fixed $h$.

ABSTRACT

We call a digraph {\em $h$-semicomplete} if each vertex of the digraph has at most $h$ non-neighbors, where a non-neighbor of a vertex $v$ is a vertex $u eq v$ such that there is no edge between $u$ and $v$ in either direction. This notion generalizes that of semicomplete digraphs which are $0$-semicomplete and tournaments which are semicomplete and have no anti-parallel pairs of edges. Our results in this paper are as follows. (1) We give an algorithm which, given an $h$-semicomplete digraph $G$ on $n$ vertices and a positive integer $k$, in $(h + 2k + 1)^{2k} n^{O(1)}$ time either constructs a path-decomposition of $G$ of width at most $k$ or concludes correctly that the pathwidth of $G$ is larger than $k$. (2) We show that there is a function $f(k, h)$ such that every $h$-semicomplete digraph of pathwidth at least $f(k, h)$ has a semicomplete subgraph of pathwidth at least $k$. One consequence of these results is that the problem of deciding if a fixed digraph $H$ is topologically contained in a given $h$-semicomplete digraph $G$ admits a polynomial-time algorithm for fixed $h$.

Motivation & Objective

• To extend fixed-parameter tractable algorithms for pathwidth from semicomplete digraphs to the broader class of $h$-semicomplete digraphs, where each vertex has at most $h$ non-neighbors.
• To establish structural properties of $h$-semicomplete digraphs that ensure the existence of high-pathwidth semicomplete subgraphs when the overall pathwidth is large.
• To enable polynomial-time algorithms for topological containment and disjoint paths problems in $h$-semicomplete digraphs for fixed $h$, generalizing results previously known only for semicomplete digraphs.
• To provide a theoretical foundation for digraph analogues of Graph Minors-type results by extending the applicability of such results beyond semicomplete digraphs.

Proposed method

• Design a dynamic programming algorithm based on iterative bag construction and vertex ordering, leveraging the bounded non-neighborhood property of $h$-semicomplete digraphs.
• Use a parameterized approach with $k$ as the pathwidth target, constructing path-decompositions by exploring all possible combinations of $h+2k+1$ vertex sets in a bounded search space.
• Apply combinatorial bounds on the number of non-neighbors per vertex to limit the number of configurations to explore during path-decomposition construction.
• Use a martingale-based probabilistic argument to analyze the expected size of a random independent set in a subgraph, supporting structural lemmas about subgraph density and pathwidth.
• Establish a function $f(k,h)$ such that any $h$-semicomplete digraph with pathwidth $\geq f(k,h)$ must contain a semicomplete subgraph of pathwidth $\geq k$, using extremal combinatorics and structural decomposition.
• Integrate the algorithmic and structural results to show that topological containment of a fixed digraph $H$ in an $h$-semicomplete digraph $G$ is solvable in polynomial time for fixed $h$.

Experimental results

Research questions

• RQ1 Can the pathwidth of $h$-semicomplete digraphs be computed in fixed-parameter tractable time with respect to the pathwidth parameter $k$?
• RQ2 What structural properties ensure that high pathwidth in $h$-semicomplete digraphs implies the existence of a high-pathwidth semicomplete subgraph?
• RQ3 Does the class of $h$-semicomplete digraphs admit polynomial-time algorithms for problems like topological containment and disjoint paths, generalizing results from semicomplete digraphs?
• RQ4 How does the bounded non-neighborhood property ($h$ non-neighbors per vertex) affect the algorithmic tractability of pathwidth and related problems?
• RQ5 Can the pathwidth computation algorithm be extended to other digraph classes with bounded non-neighborhood or similar structural constraints?

Key findings

• An algorithm is developed that computes a path-decomposition of width at most $k$ in $(h+2k+1)^{2k}n^{O(1)}$ time for any $h$-semicomplete digraph $G$ on $n$ vertices, or correctly determines that $\text{pw}(G) > k$.
• The algorithm generalizes Pilipczuk's $k^{O(k)}n^2$ result for semicomplete digraphs to the broader class of $h$-semicomplete digraphs.
• There exists a function $f(k,h)$ such that every $h$-semicomplete digraph with pathwidth at least $f(k,h)$ contains a semicomplete subgraph of pathwidth at least $k$.
• As a consequence, the problem of deciding whether a fixed digraph $H$ is topologically contained in an $h$-semicomplete digraph $G$ is solvable in polynomial time for fixed $h$.
• The structural result implies that high pathwidth in $h$-semicomplete digraphs forces the presence of dense semicomplete subgraphs, enabling algorithmic reductions.
• The paper establishes a link between pathwidth and subgraph containment in $h$-semicomplete digraphs, extending known results from semicomplete digraphs to a wider class of digraphs.