[Paper Review] On independent set on B1-EPG graphs

Marin Bougeret, Stéphane Bessy+2 more|arXiv (Cornell University)|Oct 2, 2015

Advanced Graph Theory Research7 references4 citations

TL;DR

This paper investigates the Maximum Independent Set (MIS) problem on B₁-EPG graphs—graphs represented by paths with at most one bend on a grid. It proves MIS is NP-complete and admits no PTAS even under strong restrictions (e.g., one path shape, bounded horizontal/vertical segments), but a PTAS exists when horizontal lengths are bounded. It also shows MIS is FPT for three-path shapes and W₁-hard for B₂-EPG, leaving the general B₁-EPG case open.

ABSTRACT

In this paper we consider the Maximum Independent Set problem (MIS) on $B_1$-EPG graphs. EPG (for Edge intersection graphs of Paths on a Grid) was introduced in ~\cite{edgeintersinglebend} as the class of graphs whose vertices can be represented as simple paths on a rectangular grid so that two vertices are adjacent if and only if the corresponding paths share at least one edge of the underlying grid. The restricted class $B_k$-EPG denotes EPG-graphs where every path has at most $k$ bends. The study of MIS on $B_1$-EPG graphs has been initiated in~\cite{wadsMIS} where authors prove that MIS is NP-complete on $B_1$-EPG graphs, and provide a polynomial $4$-approximation. In this article we study the approximability and the fixed parameter tractability of MIS on $B_1$-EPG. We show that there is no PTAS for MIS on $B_1$-EPG unless P$=$NP, even if there is only one shape of path, and even if each path has its vertical part or its horizontal part of length at most $3$. This is optimal, as we show that if all paths have their horizontal part bounded by a constant, then MIS admits a PTAS. Finally, we show that MIS is FPT in the standard parameterization on $B_1$-EPG restricted to only three shapes of path, and $W_1$-hard on $B_2$-EPG. The status for general $B_1$-EPG (with the four shapes) is left open.

Motivation & Objective

• To determine the approximability and fixed-parameter tractability of the Maximum Independent Set (MIS) problem on B₁-EPG graphs.
• To investigate whether a PTAS exists for MIS on B₁-EPG graphs under structural constraints such as a single path shape or bounded segment lengths.
• To explore parameterized complexity of MIS on B₁-EPG, particularly for restricted path shapes.
• To establish hardness results for MIS on B₂-EPG and compare with B₁-EPG.

Proposed method

• Reduction from the k-clique problem to prove MIS is NP-complete and W₁-hard on B₂-EPG graphs.
• Construction of edge and vertex gadgets using circular tracks and path intersections to simulate clique selection in the independent set.
• Use of 6-path and 3-path cliques per edge and vertex gadget to bound independent set size and enforce selection consistency.
• Design of a PTAS for B₁-EPG graphs when horizontal path lengths are bounded by a constant, leveraging structural constraints.
• Proof that no PTAS exists for MIS on B₁-EPG even with a single path shape or bounded horizontal/vertical segments.
• Analysis of path representations using corner coordinates, horizontal/vertical projections, and grid-edge sharing to model adjacency.

Experimental results

Research questions

• RQ1 Is there a PTAS for MIS on B₁-EPG graphs when all paths have bounded horizontal length?
• RQ2 Does MIS admit a PTAS on B₁-EPG graphs with only one path shape?
• RQ3 Can MIS be solved in FPT time on B₁-EPG graphs restricted to three path shapes?
• RQ4 Is MIS W₁-hard on B₂-EPG graphs?
• RQ5 What is the approximability status of MIS on general B₁-EPG graphs with all four path shapes?

Key findings

• There is no PTAS for MIS on B₁-EPG graphs unless P = NP, even when only one path shape is used or when horizontal/vertical path segments are bounded to length 3.
• A PTAS exists for MIS on B₁-EPG graphs when all horizontal path segments are bounded by a constant.
• MIS is FPT in the standard parameterization on B₁-EPG graphs restricted to only three path shapes.
• MIS is W₁-hard on B₂-EPG graphs, indicating strong intractability in the parameterized sense.
• The reduction from k-clique to MIS on B₁-EPG graphs uses edge and vertex gadgets with 6- and 3-path cliques to simulate clique selection via independent set size.
• The paper leaves open the parameterized complexity of MIS on general B₁-EPG graphs with all four path shapes.