Consensus models: Computational complexity aspects in modern approaches to the list coloring problem

Abstract

In the paper we study new approaches to the problem of list coloring of graphs. In the problem we are given a simple graph G = ( V , E ) and, for every v ∈ V , a nonempty set of integers S ( v ) ; we ask if there is a coloring c of G such that c ( v ) ∈ S ( v ) for every v ∈ V . Modern approaches, connected with applications, change the question—we now ask if S can be changed, using only some elementary transformations, to ensure that there is such a coloring and, if the answer is yes, what is the minimal number of changes. In the paper for studying the adding, the trading and the exchange models of list coloring, we use the following transformations: • adding of colors (the adding model): select two vertices u , v and a color c ∈ S ( u ) ; add c to S ( v ) , i.e. set S ( v ) : = S ( v ) ∪ { c } ; • trading of colors (the trading model): select two vertices u , v and a color c ∈ S ( u ) ; move c from S ( u ) to S ( v ) , i.e. set S ( u ) : = S ( u ) ∖ { c } and S ( v ) : = S ( v ) ∪ { c } ; • exchange of colors (the exchange model): select two vertices u , v and two colors c ∈ S ( u ) , d ∈ S ( v ) ; exchange c with d , i.e. set S ( u ) : = ( S ( u ) ∖ { c } ) ∪ { d } and S ( v ) : = ( S ( v ) ∖ { d } ) ∪ { c } . Our study focuses on computational complexity of the above models and their edge versions. We consider these problems on complete graphs, graphs with bounded cyclicity and partial k -trees, receiving in all cases polynomial algorithms or proofs of NP-hardness.