The Complexity of the List Partition Problem for Graphs

Abstract

Abstract. The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A 1 , A 2 , . . . , A k , where it may be specified that A i induces a stable set, a clique, or an arbitrary subgraph, and pairs A i , A j (i = j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete.

Colouring, constraint satisfaction, and complexity

“…There are several variants of the basic M -partition problem: these include partitioning digraphs (the matrix M is not necessarily symmetric) [69,180], equipping the vertices of G with lists, or insisting that each part be nonempty [31,64,75,106], generalizing to certain constraint satisfac-tion problems [56], or restricting the input graphs to have special structure [57,65,70,100,106].…”

“…A relative of the compatible 3-colouring problem from [31], called the stubborn problem, is a list version of an M -partition problem where the matrix has size only four. For this problem there is an n O(log n) algorithm similar to the one described above for the compatible 3-colouring problem, but no polynomial algorithm is known.…”

Obstructions to partitions of chordal graphs

“…We will also discuss variants of this basic M -partition problem. In the list variant, the vertices of the input graph G have lists (of allowed parts), and an Mpartition must place each vertex of G in a part that is allowed for it [2,12]. *…”

“…(For recent progress on a related problem see [32] and [27].) The list variant of the M -partition problem for another matrix M of size 4 has been dubbed the "stubborn problem" [2] because its complexity was difficult to determine. This problem was also recently solved, and shown to be polynomial in [4].…”

“…Nevertheless, considerable effort has gone into classification of small matrices, for all the variants discussed: see [12] for the complexity of the basic problem and matrices of size m < 5, see [2,4,12] for the complexity of the list version and m < 5, see [6,32,27] for the complexity of the surjective variant and m < 5, see [15] for the complexity of the digraph variant and m < 4, and see [16] for the finiteness of the number of minimal obstructions in the basic variant and m < 4.…”