Parameterized complexity of the weighted independent set problem beyond graphs of bounded clique number

The maximum independent set problem is known to be NP-hard for graphs in general, but is solvable in polynomial time for graphs in many special classes. It is also known that the problem is generally intractable from a parameterized point of view. A simple Ramsey argument implies the fixed-parameter tractability of the maximum independent set problem in classes of graphs of bounded clique number. Beyond this observation very little is known about the parameterized complexity of the problem in restricted graph families. In the present paper we develop fpt-algorithms for graphs in some classes extending graphs of bounded clique number.     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.     

The weighted maximum independent set problem in permutation graphs

In this paper, sequential and parallel algorithms are presented to find a maximum independent set with largest weight in a weighted permutation graph. The sequential algorithm, which is designed based on dynamic programming, runs in timeO(nlogn) and requiresO(n) space. The parallel algorithm runs inO(log² n) time usingO(n ³/logn) processors on the CREW PRAM, orO(logn) time usingO(n ³) processors on the CRCW PRAM.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

On directed graphs with an independent covering set

We prove that if a directed graph,D, contains no odd directed cycle and, for all but finitely many vertices, EITHER the in-degrees are finite OR the out-degrees are at most one, thenD contains an independent covering set (i.e. there is a kernel). We also give an example of a countable directed graph which has no directed cycle, each vertex has out-degree at most two, and which has no independent covering set.Research supported by N.S.E.R.C. grants #69-0982 and #69-0259.     

PTAS for the minimum k-path connected vertex cover problem in unit disk graphs

In the Minimum k-Path Connected Vertex Cover Problem (MkPCVCP), we are given a connected graph G and an integer k ≥ 2, and are required to find a subset C of vertices with minimum cardinality such that each path with length k ? 1 has a vertex in C, and moreover, the induced subgraph G[C] is connected. MkPCVCP is a generalization of the minimum connected vertex cover problem and has applications in many areas such as security communications in wireless sensor networks. MkPCVCP is proved to be NP-complete. In this paper, we give the first polynomial time approximation scheme (PTAS) for MkPCVCP in unit disk graphs, for every fixed k ≥ 2.     

On hitting all maximum cliques with an independent set

We prove that every graph G for which has an independent set I such that ω(G?I)<ω(G). It follows that a minimum counterexample G to Reed's conjecture satisfies and hence also . This also applies to restrictions of Reed's conjecture to hereditary graph classes, and in particular generalizes and simplifies King, Reed and Vetta's proof of Reed's conjecture for line graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 32–37, 2010     

Rational Chebyshev approximation on the unit disk

Summary In a recent paper we showed that error curves in polynomial Chebyshev approximation of analytic functions on the unit disk tend to approximate perfect circles about the origin [23]. Making use of a theorem of Carathéodory and Fejér, we derived in the process a method for calculating near-best approximations rapidly by finding the principal singular value and corresponding singular vector of a complex Hankel matrix. This paper extends these developments to the problem of Chebyshev approximation by rational functions, where non-principal singular values and vectors of the same matrix turn out to be required. The theory is based on certain extensions of the Carathéodory-Fejér result which are also currently finding application in the fields of digital signal processing and linear systems theory.It is shown among other things that iff(z) is approximated by a rational function of type (m, n) for >0, then under weak assumptions the corresponding error curves deviate from perfect circles of winding numberm+n+1 by a relative magnitudeO( ^{m + n + 2} as 0. The CF approximation that our method computes approximates the true best approximation to the same high relative order. A numerical procedure for computing such approximations is described and shown to give results that confirm the asymptotic theory. Approximation ofe ^z on the unit disk is taken as a central computational example.     

A parallel maximum clique algorithm for large and massive sparse graphs

This paper describes BBMCPara, a new parallel exact maximum clique algorithm tailored for large and massive sparse graphs. The paper first presents a sequential algorithm BBMCSP, which builds on ideas from a leading bit-parallel published algorithm for middle-size graphs. It employs heavy pre-processing and a new sparse bitset encoding to outperform other state-of-the-art algorithms by up to several orders of magnitude over a set of real networks. BBMCPara parallelizes BBMCSP by splitting according to a preprocessing step of the latter. On a 20-core computer, it averages speedups close to an order of magnitude over real graphs of up to 3 million vertices. According to the reported results, BBMCPara appears to be the current fastest algorithm for large and massive real networks to the best of our knowledge.     

On clique covers and independence numbers of graphs

Relationships between the following graph invariants are studied: The node clique cover number, θ₀(>G); the clique cover number θ₁(G); node independence number, β₀(G); and an edge independence number, β′₁(G), We extend a theorem of Choudum, Parthasarathy and Ravindra with further statements equivalent to θ₀(G) = θ₁(G). More general results regarding the inequality θ₀(G) ? 0₁(G) are presented.     

A note on the complexity of minimum dominating set

The currently (asymptotically) fastest algorithm for minimum dominating set on graphs of n nodes is the trivial Ω(2ⁿ) algorithm which enumerates and checks all the subsets of nodes. In this paper we present a simple algorithm which solves this problem in O(1.81ⁿ) time.     

Approximating the minimum clique cover and other hard problems in subtree filament graphs

Subtree filament graphs are the intersection graphs of subtree filaments in a tree. This class of graphs contains subtree overlap graphs, interval filament graphs, chordal graphs, circle graphs, circular-arc graphs, cocomparability graphs, and polygon-circle graphs. In this paper we show that, for circle graphs, the clique cover problem is NP-complete and the h-clique cover problem for fixed h is solvable in polynomial time. We then present a general scheme for developing approximation algorithms for subtree filament graphs, and give approximation algorithms developed from the scheme for the following problems which are NP-complete on circle graphs and therefore on subtree filament graphs: clique cover, vertex colouring, maximum k-colourable subgraph, and maximum h-coverable subgraph.     

A polynomial algorithm for the minimum weighted clique cover problem on claw-free perfect graphs

Let G = (V, E) be a graph with a positive number wt(v) assigned to each v ? v. A weighted clique cover of the vertices of G is a collection of cliques with a non-negative weight y_c assigned to each clique C in the collection such that σ_C:v?CY_C?wt(v) for all v ? V. The problem considered is to minimize σ_Cy_C over all weighted clique covers. A polynomial time algorithm for this problem is presented for graphs that are claw-free and perfect.     

On the linear extension complexity of stable set polytopes for perfect graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph $G$ depending linearly on the size of $G$ and involving the depth of a decomposition tree of $G$ in terms of basic perfect graphs.     

Perfectness and imperfectness of unit disk graphs on triangular lattice points

Given a finite set of 2-dimensional points P⊆R² and a positive real d, a unit disk graph, denoted by (P,d), is an undirected graph with vertex set P such that two vertices are adjacent if and only if the Euclidean distance between the pair is less than or equal to d. Given a pair of non-negative integers m and n, P(m,n) denotes a subset of 2-dimensional triangular lattice points defined by where . Let T_m,n(d) be a unit disk graph defined on a vertex set P(m,n) and a positive real d. Let be the kth power of T_m,n(1).In this paper, we show necessary and sufficient conditions that [ is perfect] and/or [ is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (T_m,n(d),w) and .