Exploiting restricted linear structure to cope with the hardness of clique-width (original) (raw)

A new representation of proper interval graphs with an application to clique-width

Electronic Notes in Discrete Mathematics, 2009

We introduce a new representation of proper interval graphs that can be computed in linear time and stored in O(n) space. This representation is a 2-dimensional vertex partition. It is particularly interesting with respect to clique-width. Based on this representation, we prove new upper bounds on the clique-width of proper interval graphs.

The Maximum Clique Problem in Multiple Interval Graphs

Algorithmica, 2013

Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for t-interval graphs when t ≥ 3 and polynomial-time solvable when t = 1. The problem is also known to be NP-complete in t-track graphs when t ≥ 4 and polynomialtime solvable when t ≤ 2. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APXcomplete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called t-circular interval graphs and t-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time t-approximation algorithm for WEIGHTED MAXIMUM CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t. 2 Preliminaries Consider a circle C of length l with a distinguished point O. The coordinate of a point p ∈ C is the length of the arc going clockwise from O to p. Given two reals p and q, [p, q] is the arc of C going clockwise from the point with coordinate p to the one with coordinate q. In the following, coordinates are understood modulo l.

Graphs of linear clique-width at most 3

Theoretical Computer Science, 2011

A graph has linear clique-width at most k if it has a clique-width expression using at most k labels such that every disjoint union operation has an operand which is a single vertex graph. We give the first characterisation of graphs of linear clique-width at most 3, and we give the first polynomial-time recognition algorithm for graphs of linear clique-width at most 3. In addition, we present new characterisations of graphs of linear clique-width at most 2. We also give a layout characterisation of graphs of bounded linear clique-width; a similar characterisation was independently shown by Gurski and by Lozin and Rautenbach.

Proving NP-hardness for clique-width I: non-approximability of sequential clique-width

2005

Clique-width is a graph parameter, defined by a composition mechanism for vertexlabeled graphs, which measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic, that includes NPhard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in spite of considerable efforts, no NP-hardness proof has been found so far. In this paper we show a non-approximability result for restricted form of cliquewidth, termed "r-sequential clique-width", considering only such clique-width constructions where one of any two graphs put together by disjoint union must have r or fewer vertices. In particular, we show that for every positive integer r, the r-sequential cliquewidth cannot be absolutely approximated in polynomial time unless P = NP, and that given G and k the question of whether the r-sequential clique-width of G is at most k is NP-complete. We show further that this non-approximability result holds even for graphs of a very particular structure: for graphs obtained from cobipartite graphs by replacing edges with induced paths. In part II of this series of papers we use this strengthened result to show that, unless P = NP, there is no polynomial-time absolute approximation algorithm for (unrestricted) clique-width, and that, given a graph G and an integer k, deciding whether the the clique-width G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.

Proving NP-hardness for clique-width II: non-approximability of clique-width

2005

Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic with second-order quantification on vertex sets, that includes NP-hard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in spite of considerable efforts, no NP-hardness proof has been found so far. We give the first hardness proof. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless P = NP. We also show that, given a graph G and an integer k, deciding whether the clique-width of G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.

Polynomial-time recognition of clique-width ≤3 graphs

Discrete Applied Mathematics, 2012

Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time. This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently. One of the major open questions regarding clique-width is whether graphs of cliquewidth at most k, for fixed k, can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm (O(n 2 m)) to recognize graphs of clique-width at most 3.

Characterising the linear clique-width of a class of graphs by forbidden induced subgraphs

Discrete Applied Mathematics, 2012

We study the linear clique-width of graphs that are obtained from paths by disjoint union and adding true twins. We show that these graphs have linear clique-width at most 4, and we give a complete characterisation of their linear clique-width by forbidden induced subgraphs. As a consequence, we obtain a linear-time algorithm for computing the linear clique-width of the considered graphs. Our results extend the previously known set of forbidden induced subgraphs for graphs of linear clique-width at most 3.

Two-Way and Multiway Partitioning of a Set of Intervals for Clique-Width Maximization

Algorithmica, 1999

For a set S of intervals, the clique-interval I S is de ned as the interval obtained from the intersection of all the intervals in S , and the clique-width quantity w S is de ned as the length of I S. G i v en a set S of intervals, it is straightforward to compute its clique-interval and clique-width. In this paper we study the problem of partitioning a set of intervals in order to maximize the sum of the clique-widths of the partitions. We present a n Onlogn time algorithm for the balanced bi-partitioning problem, and an Okn 2 time algorithm for the k-way u n balanced partitioning problem.

A sequential and parallel algorithm for disjoint cliques problem on interval graphs

2018

Using DAG approach,A sequential algorithm is presented to solve disjoint cliques problem on interval graph G which takes O(n^2) time where n is the number of vertices of the graph. For the same problem a O(log2n) time parallel algorithm is presented which takes processors on an EREW PRAM model. Also, on a CREW model it takes O(logn) time with O(n^(3+e) ),e>0 processors.