Frank Gurski

How to Solve NP-hard Graph Problems on Clique-Width Bounded Graphs in Polynomial Time

We show that many non-MSO1 NP-hard graph problems can be solved in polynomial time on clique-width and NLC-width bounded graphs using a very general and simple scheme. Our examples are partition into cliques, partition into triangles, partition into complete bipartite subgraphs, partition into perfect matchings, partition into forests, cubic subgraph, Hamiltonian path, minimum maximal matching, and vertex/edge separation problems.

The Tree-Width of Clique-Width Bounded Graphs Without Kn, n

We proof that every graph of clique-width k which does not contain the complete bipartite graph Kn,n for some n > 1 as a subgraph has tree-width at most 3k(n - 1) - 1. This immediately implies that a set of graphs of bounded clique-width has bounded tree-width if it is uniformly l-sparse, closed under subgraphs, of bounded degree, or planar.

On the relationship between NLC-width and linear NLC-width

In this paper, we consider NLC-width, NLCT-width, and linear NLC-width bounded graphs. We show that the set of all complete binary trees has unbounded linear NLC-width and that the set of all co-graphs has unbounded NLCT-width. Since trees have NLCT-width 3 and co-graphs have NLC-width 1, it follows that the family of linear NLC-width bounded graph classes is a proper subfamily of the family of NLCT-width bounded graph classes and that the family of NLCT-width bounded graph classes is a proper subfamily of the family of NLC-width bounded graph classes.

Vertex disjoint paths on clique-width bounded graphs

We show that the l vertex disjoint paths problem between l pairs of vertices can be solved in linear time for co-graphs but is NP-complete for graphs of clique-width at most 6 and NLC-width at most 4. The NP-completeness follows from the fact that the line graph of a graph of tree-width k has clique-width at most 2k + 2 and NLC-width at most k + 2, and a result by Nishizeki et al. [The edge-disjoint paths problem is NP-complete for series-parallel graphs, Discrete Appl. Math. 115 (2001) 177-186]. The vertex disjoint paths problem is the first graph problem shown to be NP-complete on graphs of bounded clique-width but solvable in linear time on co-graphs and graphs of bounded tree-width. Additionally, we show that the r vertex disjoint paths problem between each of l pairs of vertices can be solved in polynomial time for co-graphs, if l is given to the input, and for graphs of bounded clique-width, if l is fixed.

The Behavior of Clique-Width under Graph Operations and Graph Transformations

Clique-width is a well-known graph parameter. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded clique-width. The same holds for NLC-width. In this paper we study the behavior of clique-width and NLC-width under various graph operations and graph transformations. We give upper and lower bounds for the clique-width and NLC-width of the modified graphs in terms of the clique-width and NLC-width of the involved graphs.

Minimizing nLC-width is nP-complete

We show that a graph has tree-width at most 4k–1 if its line graph has nLC-width or clique-width at most k, and that an incidence graph has tree-width at most k if its line graph has nLC-width or clique-width at most k. In [9] it is shown that a line graph has nLC-width at most k+2 and clique-width at most 2k+2 if the root graph has tree-width k. Using these bounds we show by a reduction from tree-width minimization that nLC-width minimization is nP-complete.

Linear layouts measuring neighbourhoods in graphs

In this paper we introduce the graph layout parameter neighbourhood-width as a variation of the well-known cut-width. The cut-width of a graph G=(V,E) is the smallest integer k, such that there is a linear layout @f:V->{1,...,|V|}, such that for every 1=i. The neighbourhood-width of a graph is the smallest integer k, such that there is a linear layout @f, such that for every 1=i. We show that the neighbourhood-width of a graph differs from its linear clique-width or linear NLC-width at most by one. This relation is used to show that the minimization problem for neighbourhood-width is NP-complete. Furthermore, we prove that simple modifications of neighbourhood-width imply equivalent layout characterizations for linear clique-width and linear NLC-width. We also show that every graph of path-width k or cut-width k has neighbourhood-width at most k+2 and we give several conditions such that graphs of bounded neighbourhood-width have bounded path-width or bounded cut-width.

Moving Bins from Conveyor Belts onto Pallets Using FIFO Queues

We study the combinatorial FIFO stack-up problem. In delivery industry, bins have to be stacked-up from conveyor belts onto pallets. Given \(k\) sequences \(q_1, \ldots , q_k\) of labeled bins and a positive integer \(p\), the goal is to stack-up the bins by iteratively removing the first bin of one of the \(k\) sequences and put it onto a pallet located at one of \(p\) stack-up places. Each of these pallets has to contain bins of only one label, bins of different labels have to be placed on different pallets. After all bins of one label have been removed from the given sequences, the corresponding place becomes available for a pallet of bins of another label. The FIFO stack-up problem is NP-complete in general. In this paper we show that the problem can be solved in polynomial time, if the number \(k\) of given sequences is fixed.

Algorithms for Controlling Palletizers

Palletizers are widely used in delivery industry. We consider a large palletizer where each stacker crane grabs a bin from one of k conveyors and position it onto a pallet located at one of p stack-up places. All bins have the same size. Each pallet is destined for one customer. A completely stacked pallet will be removed automatically and a new empty pallet is placed at the palletizer. The FIFO Stack-Up problem is to decide whether the bins can be palletized by using at most p stack-up places. We introduce a digraph and a linear programming model for the problem. Since the FIFO Stack-Up problem is computational intractable and is defined on inputs of various informations, we study the parameterized complexity. Based on our models we give xp-algorithms and fpt-algorithms for various parameters, and approximation results for the problem.

A Practical Approach for the FIFO Stack-Up Problem

We consider the FIFO Stack-Up problem which arises in delivery industry, where bins have to be stacked-up from conveyor belts onto pallets. Given k sequences q 1, …, q k of labeled bins and a positive integer p. The goal is to stack-up the bins by iteratively removing the first bin of one of the k sequences and put it onto a pallet located at one of p stack-up places. Each of these pallets has to contain bins of only one label, bins of different labels have to be placed on different pallets. After all bins of one label have been removed from the given sequences, the corresponding stack-up place becomes available for a pallet of bins of another label. The FIFO Stack-Up problem is computational intractable [3]. In this paper we introduce a graph model for this problem, which allows us to show a breadth first search solution. Our experimental study of running times shows that our approach can be used to solve a lot of practical instances very efficiently.