[Paper Review] Maximum Weight Independent Set in lClaw-Free Graphs in Polynomial Time

Andreas Brandstädt, Raffaele Mosca|arXiv (Cornell University)|Feb 18, 2016

VLSI and FPGA Design Techniques6 references3 citations

TL;DR

This paper presents a polynomial-time algorithm for the Maximum Weight Independent Set (MWIS) problem in $l$-claw-free graphs for any fixed $l$, extending prior results on claw-free and $lK_2$-free graphs. The approach uses a recursive decomposition based on anti-neighborhoods and a novel algorithm, Algorithm Gamma($l$), that constructs a polynomial-sized family of subgraphs, each of which is claw-free and contains all maximal independent sets, enabling dynamic programming to solve MWIS efficiently.

ABSTRACT

The Maximum Weight Independent Set (MWIS) problem is a well-known NP-hard problem. For graphs $G_1, G_2$, $G_1+G_2$ denotes the disjoint union of $G_1$ and $G_2$, and for a constant $l \ge 2$, $lG$ denotes the disjoint union of $l$ copies of $G$. A {\em claw} has vertices $a,b,c,d$, and edges $ab,ac,ad$. MWIS can be solved for claw-free graphs in polynomial time; the first two polynomial time algorithms were introduced in 1980 by \cite{Minty1980,Sbihi1980}, then revisited by \cite{NakTam2001}, and recently improved by \cite{FaeOriSta2011,FaeOriSta2014}, and by \cite{NobSas2011,NobSas2015} with the best known time bound in \cite{NobSas2015}. Furthermore MWIS can be solved for the following extensions of claw-free graphs in polynomial time: fork-free graphs \cite{LozMil2008}, $K_2$+claw-free graphs \cite{LozMos2005}, and apple-free graphs \cite{BraLozMos2010,BraKleLozMos2008}. This manuscript shows that for any constant $l$, MWIS can be solved for $l$claw-free graphs in polynomial time. Our approach is based on Farber's approach showing that every $2K_2$-free graph has ${\cal O}(n^2)$ maximal independent sets \cite{Farbe1989}, which directly leads to a polynomial time algorithm for MWIS on $2K_2$-free graphs by dynamic programming. Solving MWIS for $l$claw-free graphs in polynomial time extends known results for claw-free graphs, for $lK_2$-free graphs for any constant $l$ \cite{Aleks1991,FarHujTuz1993,Prisn1995,TsuIdeAriShi1977}, for $K_2$+claw-free graphs, for $2P_3$-free graphs \cite{LozMos2012}, and solves the open questions for $2K_2+P_3$-free graphs and for $P_3$+claw-free graphs being two of the minimal graph classes, defined by forbidding one induced subgraph, for which the complexity of MWIS was an open problem.

Motivation & Objective

• To resolve the complexity of MWIS in $l$-claw-free graphs, a natural extension of claw-free graphs.
• To extend known polynomial-time results for MWIS in $lK_2$-free and $K_2$+claw-free graphs to a broader class.
• To close open problems for $2K_2+P_3$-free and $P_3$+claw-free graphs, which are minimal forbidden subgraph classes.
• To generalize Farber's approach on $2K_2$-free graphs to $l$-claw-free graphs via structural decomposition.

Proposed method

• Propose Algorithm Gamma($l$) to compute a good claw-free family of subgraphs, each containing all maximal independent sets of the original graph.
• Use recursive decomposition: for each induced $L_k$ (a specific 5-vertex subgraph), compute the anti-neighborhood $A_G(L_k)$, which is $(l-1)$-claw-free.
• Apply induction on $l$, assuming the method works for $l-1$, to ensure the anti-neighborhoods admit a good claw-free family.
• Construct a polynomial-sized family ${\cal S}$ of subsets of vertices, each inducing a claw-free subgraph, via iterative extension in $O(n)$ time per loop.
• Use dynamic programming on each member of ${\cal S}$, leveraging the known $O(n^2 \log n)$ MWIS algorithm for claw-free graphs (Theorem 1).
• Combine results from all members of ${\cal S}$ to return the maximum-weight independent set.

Experimental results

Research questions

• RQ1 Can the MWIS problem be solved in polynomial time for $l$-claw-free graphs for any fixed $l$?
• RQ2 Does the structural approach used for $2K_2$-free graphs extend to $l$-claw-free graphs?
• RQ3 Are the $2K_2+P_3$-free and $P_3$+claw-free graphs, which were open problems, now resolved by this framework?
• RQ4 Can a good claw-free family be computed in polynomial time for $l$-claw-free graphs?

Key findings

• The MWIS problem is solvable in polynomial time for $l$-claw-free graphs for any fixed $l$, resolving an open problem.
• Algorithm Gamma($l$) computes a good claw-free family in polynomial time, with the number of members bounded by a polynomial in $n$.
• The family ${\cal S}$ constructed by Algorithm Gamma($l$) contains all maximal independent sets of the input graph.
• Each member of ${\cal S}$ induces a claw-free subgraph, enabling the use of known $O(n^2 \log n)$ MWIS algorithms on each component.
• The overall time complexity of Algorithm MWIS($l$) is polynomial, matching the time bound of the underlying claw-free MWIS algorithm.
• The result generalizes prior results on $lK_2$-free, $K_2$+claw-free, and $2P_3$-free graphs, and resolves two minimal open cases: $2K_2+P_3$-free and $P_3$+claw-free graphs.