An On-line Competitive Algorithm for Coloring Bipartite Graphs Without Long Induced Paths
Algorithmica
An On-line Competitive Algorithm for Coloring Bipartite Graphs Without Long Induced Paths
Piotr Micek 0
Veit Wiechert 0
Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
0 Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University , Kraków , Poland
The existence of an on-line competitive algorithm for coloring bipartite graphs is a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs. We propose an on-line competitive coloring algorithm for P9-free bipartite graphs. Veit Wiechert
On-line algorithm; Graph coloring; Bipartite graphs
1 Introduction
A proper coloring of a graph is an assignment of colors to its vertices such that
adjacent vertices receive distinct colors. It is easy to devise a (linear time) algorithm
for 2-coloring bipartite graphs. Now, imagine that an algorithm receives vertices of a
graph one by one knowing only the adjacency status of the vertex to vertices presented
so far. The color of the current vertex must be fixed by the algorithm before the next
vertex is revealed and it cannot be changed afterwards. This kind of algorithm is called
an on-line coloring algorithm.
This paper is an extended version of [13] from the proceedings of ISAAC 2014.
B Piotr Micek
Formally, an on-line graph (G, π ) is a graph G with a permutation π of its vertices.
An on-line coloring algorithm A takes an on-line graph (G, π ), say π = (v1, . . . , vn),
as an input. It produces a proper coloring of the vertices of G where the color of a vertex
vi , for i = 1, . . . , n, depends only on the subgraph of G induced by v1, . . . , vi . It is
convenient to imagine that consecutive vertices along π are revealed by some adaptive
(malicious) adversary and the coloring process is a game between that adversary and
the on-line algorithm.
Still, it is an easy exercise to show that if an adversary presents a bipartite graph and
all the time the graph presented so far is connected then there is an on-line algorithm
2-coloring these graphs. But if an adversary can present a bipartite graph without any
additional constraints then it can trick out any on-line algorithm to use an arbitrary
number of colors!
Indeed, there is a strategy for the adversary forcing any on-line algorithm to use at
least log n + 1 colors on a forest of size n. On the other hand, the First-Fit algorithm
(that is an on-line algorithm coloring each incoming vertex with the least admissible
natural number) uses at most log n + 1 colors on forests of size n. When the game
is played on bipartite graphs, an adversary can easily force First-Fit to use n2 colors
on a bipartite graph of size n. Lovász, Saks and Trotter [12] proposed a simple on-line
algorithm (in fact as an exercise; see also [8]) using at most 2 log n + 1 colors on
bipartite graphs of size n. This is best possible up to an additive constant as Gutowski
et al. [4] showed that there is a strategy for the adversary forcing any on-line algorithm
to use at least 2 log n − 10 colors on a bipartite graph of size n.
For an on-line algorithm A by A(G, π ) we mean the number of colors that A uses
against an adversary presenting graph G with presentation order π .
An on-line coloring algorithm A is competitive on a class of graphs G if there
is a function f such that for every G ∈ G and permutation π of vertices of G we
have A(G, π ) f (χ (G)). As we have discussed, there is no competitive coloring
algorithm for forests. But there are reasonable classes of graphs admitting
competitive algorithms, e.g., interval graphs can be colored on-line with at most 3χ − 2
colors (where χ is the chromatic number of the presented graph; see [11]) and
cocomparability graphs can be colored on-line with a number of colors bounded
by a tower function in terms of χ (see [9]). Also classes of graphs defined in
terms of forbidden induced subgraphs were investigated in this context. For
example, P4-free graphs (also known as cographs) are colored by First-Fit optimally,
i.e. with χ colors, since any maximal independent set meets all maximal cliques
in a P4-free graph. Also P5-free graphs can be colored on-line with O(4χ )
colors (see [10]). And to complete the picture there is no competitive algorithm for
P6-free graphs as Gyárfás and Lehel [6] showed a strategy for the adversary
forcing any on-line algorithm to use an arbitrary number of colors on bipartite P6-free
graphs.
Confronted with so many negative results, it is not surprising that Gyárfás, Király
and Lehel [5] introduced a relaxed version of competitiveness for on-line algorithms.
The idea is to measure the efficiency of an on-line algorithm by comparing it to the
best on-line algorithm for a given input. Hence, the on-line chromatic number of a
graph G is defined as
where the infimum is taken over all on-line algorithms A and the maximum is taken over
all permutations π (...truncated)