Algorithms and Almost Tight Results for 3-Colorability of Small Diameter Graphs

72 Pages·2013·0.81 MB·English

Algorithms and Almost Tight Results
for 3-Colorability of Small Diameter Graphs
George B. Mertzios1 Paul G. Spirakis2
1DurhamUniversity,UK
2ComputerTechnologyInstitute(CTI)andUniversityofPatras,Greece
SOFSEM 2013
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 1/24
The 3-coloring problem is NP-complete, even if the given graph is:
the line graph of another graph [Holyer, 1981]
triangle-free with max. degree at most 4 [Maffray, Preissmann, 1996]
planar with max. degree at most 4 [Garey, Johnson, 1979]
but it becomes polynomial, if the given graph is:
perfect [Gr¨otschel et al., 1984]
a graph of bounded treewidth [Courcelle, 1990]
P -free graph [Ho`ang et al., 2010]
5
AT-free graph [Stacho, 2012]
The k-coloring problem
Problem (k-coloring problem)
Given a graph G, can we assign k colors to its vertices such that
neighboring vertices receive different colors?
The k-coloring problem is:
NP-complete for k 3
≥
polynomially solvable for k =2
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 2/24
The k-coloring problem
Problem (k-coloring problem)
Given a graph G, can we assign k colors to its vertices such that
neighboring vertices receive different colors?
The k-coloring problem is:
NP-complete for k 3
≥
polynomially solvable for k =2
The 3-coloring problem is NP-complete, even if the given graph is:
the line graph of another graph [Holyer, 1981]
triangle-free with max. degree at most 4 [Maffray, Preissmann, 1996]
planar with max. degree at most 4 [Garey, Johnson, 1979]
but it becomes polynomial, if the given graph is:
perfect [Gr¨otschel et al., 1984]
a graph of bounded treewidth [Courcelle, 1990]
P -free graph [Ho`ang et al., 2010]
5
AT-free graph [Stacho, 2012]
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 2/24
The radius of a graph G is rad(G) = min max d(u,v) : u,v V .
u V
∈ { ∈ }
Standard results:
Theorem
3-coloring is NP-complete for graphs G with diam(G) 4.
≤
Theorem
4-coloring is NP-complete for graphs G with diam(G) 2.
≤
Proof.
Reduce from 3-coloring on arbitrary graphs: add a universal vertex.
The 3-coloring problem
A central graph parameter: the distance d(u,v) between vertices u,v
Definition
The diameter of a graph G is diam(G) = max d(u,v) : u,v V .
{ ∈ }
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 3/24
Standard results:
Theorem
3-coloring is NP-complete for graphs G with diam(G) 4.
≤
Theorem
4-coloring is NP-complete for graphs G with diam(G) 2.
≤
Proof.
Reduce from 3-coloring on arbitrary graphs: add a universal vertex.
The 3-coloring problem
A central graph parameter: the distance d(u,v) between vertices u,v
Definition
The diameter of a graph G is diam(G) = max d(u,v) : u,v V .
{ ∈ }
The radius of a graph G is rad(G) = min max d(u,v) : u,v V .
u V
∈ { ∈ }
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 3/24
The 3-coloring problem
A central graph parameter: the distance d(u,v) between vertices u,v
Definition
The diameter of a graph G is diam(G) = max d(u,v) : u,v V .
{ ∈ }
The radius of a graph G is rad(G) = min max d(u,v) : u,v V .
u V
∈ { ∈ }
Standard results:
Theorem
3-coloring is NP-complete for graphs G with diam(G) 4.
≤
Theorem
4-coloring is NP-complete for graphs G with diam(G) 2.
≤
Proof.
Reduce from 3-coloring on arbitrary graphs: add a universal vertex.
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 3/24
These problems are also open if the given graph is triangle-free.
Observation
G is triangle-free with diam(G) 2 G is maximal triangle-free.
≤ ⇐⇒
Other known results do not help with 3-coloring of diam(G) 2 graphs:
≤
it is NP-complete for triangle-free graphs [Maffray et al., 1996]
(by this reduction nothing is implied for maximal triangle-free graphs)
almost all graphs G have diam(G) 2 [Bollob´as, 1981]
≤
The 3-coloring problem
Two longstanding open problems
Is 3-coloring tractable on graphs G with diam(G) 2?
≤
Is 3-coloring tractable on graphs G with diam(G) 3?
≤
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 4/24
Other known results do not help with 3-coloring of diam(G) 2 graphs:
≤
it is NP-complete for triangle-free graphs [Maffray et al., 1996]
(by this reduction nothing is implied for maximal triangle-free graphs)
almost all graphs G have diam(G) 2 [Bollob´as, 1981]
≤
The 3-coloring problem
Two longstanding open problems
Is 3-coloring tractable on graphs G with diam(G) 2?
≤
Is 3-coloring tractable on graphs G with diam(G) 3?
≤
These problems are also open if the given graph is triangle-free.
Observation
G is triangle-free with diam(G) 2 G is maximal triangle-free.
≤ ⇐⇒
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 4/24
The 3-coloring problem
Two longstanding open problems
Is 3-coloring tractable on graphs G with diam(G) 2?
≤
Is 3-coloring tractable on graphs G with diam(G) 3?
≤
These problems are also open if the given graph is triangle-free.
Observation
G is triangle-free with diam(G) 2 G is maximal triangle-free.
≤ ⇐⇒
Other known results do not help with 3-coloring of diam(G) 2 graphs:
≤
it is NP-complete for triangle-free graphs [Maffray et al., 1996]
(by this reduction nothing is implied for maximal triangle-free graphs)
almost all graphs G have diam(G) 2 [Bollob´as, 1981]
≤
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 4/24
For graphs G with diam(G) 3:
≤
for every ε [0,1), 3-coloring is NP-complete for triangle-free graphs
∈
G where diam(G) 3, rad(G) 2, and minimum degree δ = Θ(nε),
≤ ≤
a 2O(min{δ∆,nδlogδ}) subexponential algorithm, where ∆ = max. degree,
subexponential lower bounds (assuming Exp. Time Hypothesis):
δ(G) = Θ(nε): 0 ε < 1 1 ε < 1 1 ε < 1
≤ 3 3 ≤ 2 2 ≤
Lower bound: no 2o(n(1−2ε))-alg. no 2o(nε)-alg. no 2o(n1−ε)-alg.
for 1 ε < 1 this lower bound is almost tight.
2 ≤
Main results
For graphs G with diam(G) 2:
≤
a very simple, 2O(√nlogn) subexponential algorithm (worst-case),
a subclass of graphs with diameter at most 2 that admits a
polynomial algorithm
locally decomposable graphs.
GeorgeMertzios (DurhamUniversity) 3-ColorabilityofSmallDiameterGraphs SOFSEM2013 5/24