Multi-Clique-Width

TL;DR

Multi-clique-width is a graph parameter that extends tree-width and often remains exponentially smaller than clique-width, enabling more efficient algorithms for problems like coloring and independent set computation.

Contribution

The paper introduces multi-clique-width as a natural extension of tree-width that avoids exponential blow-up compared to clique-width, allowing faster algorithms for certain graph problems.

Findings

Multi-clique-width is often exponentially smaller than clique-width.

Algorithms based on multi-clique-width are as efficient as the best known algorithms for clique-width.

Multi-clique-width enables exponential speed-ups for some graphs when generating expressions are provided.

Abstract

Multi-clique-width is obtained by a simple modification in the definition of clique-width. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the independent set polynomial or testing $c$-colorability. In particular, $c$-colorability can be tested in time linear in $n$ and singly exponential in $c$ and the width $k$ of a given multi-$k$-expression. For these tasks, the running time as a function of the multi-clique-width is the same as the running time of the fastest known algorithm as a function of the clique-width. This results in an exponential speed-up for some graphs, if the corresponding graph generating expressions are given. The reason is that the multi-clique-width is never bigger, but is exponentially smaller than the clique-width for many graphs. This gap shows up when the tree-width is basically equal to the multi-clique width as well as when the tree-width is not bounded by any function of the clique-width.