[Paper Review] On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width

M. Rajaati, Hooshmandasl+2 more|arXiv (Cornell University)|Dec 25, 2016

Advanced Graph Theory Research4 citations

TL;DR

This paper presents a fixed-parameter tractable algorithm for the mixed domination problem in graphs with bounded tree-width, using a novel dynamic programming approach that assigns power values to vertices to track domination states. The method achieves an improved time complexity of $ O^*(6^w) $, proving that the mixed domination problem is fixed-parameter tractable with respect to tree-width.

ABSTRACT

A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$ such that every element $x \in (V \cup E) \backslash S$ is either adjacent or incident to an element of $S$. The mixed domination number of a graph $G$, denoted by $γ_m(G)$, is the minimum cardinality of mixed dominating sets of $G$. Any mixed dominating set with the cardinality of $γ_m(G)$ is called a minimum mixed dominating set. The mixed domination set (MDS) problem is to find a minimum mixed dominating set for a graph $G$ and is known to be an NP-complete problem. In this paper, we present a novel approach to find all of the mixed dominating sets, called the AMDS problem, of a graph with bounded tree-width $tw$. Our new technique of assigning power values to edges and vertices, and combining with dynamic programming, leads to a fixed-parameter algorithm of time $O(3^{tw^{2}} imes tw^2 imes |V|)$. This shows that MDS is fixed-parameter tractable with respect to tree-width. In addition, we theoretically improve the proposed algorithm to solve the MDS problem in $O(6^{tw} imes |V|)$ time.

Motivation & Objective

• To address the NP-complete mixed domination set (MDS) problem in graphs with bounded tree-width.
• To develop a fixed-parameter tractable algorithm for MDS by exploiting structural properties of tree-decompositions.
• To improve upon prior exponential-time bounds for MDS on bounded tree-width graphs through a novel state representation technique.
• To formally prove correctness and optimality of the proposed dynamic programming algorithm.
• To explore the potential of power-value assignment to vertices as a general technique for solving other graph problems.

Proposed method

• Introduces a power-value assignment scheme to vertices, where each vertex is assigned a value (1–9) indicating its domination status and the status of its incident edges.
• Uses dynamic programming over a tree decomposition of the graph, maintaining a stable table ($ Stable_i $) for each bag in the decomposition.
• For each bag, the algorithm tracks all valid partial solutions (domination states) using a coding scheme that encodes vertex and edge domination status.
• Applies three key rules during tree decomposition traversal: (1) handle leaf bags with initial states, (2) extend states via join operations, and (3) ensure dominated elements are properly accounted for.
• Implements pruning rules to eliminate invalid states—e.g., vertices not dominated or edges not dominated by any neighbor in the current subtree.
• Optimizes the algorithm by reducing the state space using structural constraints, leading to a theoretical improvement from $ O^*(3^{w^2}) $ to $ O^*(6^w) $.

Experimental results

Research questions

• RQ1 Can the mixed domination problem be solved in fixed-parameter tractable time with respect to tree-width?
• RQ2 What is the tightest possible time complexity for solving MDS on graphs of bounded tree-width?
• RQ3 How can vertex and edge domination be efficiently encoded and propagated in a dynamic programming framework over tree decompositions?
• RQ4 Can the state space of dynamic programming for MDS be significantly reduced through structural constraints and power-value encoding?
• RQ5 Is the proposed technique of power-value assignment generalizable to other NP-hard graph problems?

Key findings

• The mixed domination problem is fixed-parameter tractable with respect to tree-width, as demonstrated by a dynamic programming algorithm with time complexity $ O^*(3^{w^2}) $.
• A theoretical improvement reduces the time complexity to $ O^*(6^w) $, significantly improving upon the initial bound.
• The algorithm correctly computes the minimum mixed dominating set by maintaining and propagating valid domination states through all bags of the tree decomposition.
• The correctness of the algorithm is formally proven via lemmas and theorems, showing that all valid partial solutions are preserved and extended correctly.
• The root bag's stable table contains all possible minimum-cost states, from which the mixed domination number $ eta_m(G) $ can be extracted.
• The power-value technique enables efficient pruning of invalid states and provides a structured way to encode domination relationships between vertices and edges.