# The stubborn list partition problem

**Problem**Does there exist a polynomial time algorithm which takes as input a graph and for every vertex a subset of , and decides if there exists a partition of into so that only if and so that are independent, is a clique, and there are no edges between and ?

List partition problems form a very general framework for graph partitioning algorithms. Let be a symmetric matrix over the alphabet . An -*partition* of a graph is a collection of pairwise disjoint sets with union with the following properties. For every , there are no edges between and if , and all possible edges between and if . Similarly, for every , the set is independent if and a clique if . This framework captures a number of interesting problems. For instance, 3-colorability and skew-partitions are represented by the following matrices. So, a graph has an -partition for the first matrix above if and only if it is 3-colorable, and it has a skew partition if and only if it has an -partition for the second matrix above with the added constraint that are all nonempty.

An even more general framework than -partitions is list -partitions. Here, in addition to a matrix , each vertex of the input graph comes with a list of permitted sets. A solution to this problem is an -partition with the added property that whenever we must have . List -partition problems arise naturally when studying -partitions, since if we decide to put a particular vertex into a particular set, this will generally place some restrictions on the neighbors and nonneighbors of which can be encoded by lists.

So now, for every possible symmetric matrix (over ) we have a decision problem. Namely, given a graph with a list for each vertex , does have a list -partition? This problem is quite well understood for matrices of size and smaller. Here, every such problem is known to either be polynomial solvable, or be NP-complete. In general, Feder and Hell have proven a kind of quasi-dichotomy result for list partition problems. Namely, every such problem is either NP-complete or has a quasi-polynomial time algorithm ( where is the number of vertices, and are constants depending only on the matrix). Cameron et all investigated the matrices and were able to classify all of the associated list partition problems as either polynomial solvable or NP-complete with the exception of two. One is the stubborn problem, specified by the matrix below (and given in the problem statement), the other is the complementary problem (replace 's by 's and 's by 's). Hell has mentioned to me (M. DeVos) that he believes this stubborn problem to be the tip of a very interesting iceberg, and he suggested the following problem which is closely related, but perhaps better behaved (it is certainly more symmetric).

**Problem:** Does there exist a polynomial time algorithm which takes as input a complete graph with a (not necessarily proper) edge-coloring and decides if there exists a vertex coloring so that no edge has the same color as both and ?

## Bibliography

[CEHS] K. Cameron, E. Eschen, C.T. Hoàng, R. Sritharan The list partition problem for graphs

* indicates original appearance(s) of problem.

## Solved

http://arxiv.org/abs/1004.5010