New results on the complexity of oriented colouring on restricted

Motivation and background Required notions Complexity results The reduction

New results on the complexity of orientedcolouring on restricted digraph classes

Robert Ganian, Petr Hlinený

Masaryk University, Brno

Sofsem 2010, 25 January 2010

Robert Ganian, Petr Hlinený Brno

Sofsem 2010


Outline

1 Motivation and background

2 Required notions

3 Complexity results

4 The reduction


Sofsem 2010


Outline


2 Required notions


4 The reduction


Sofsem 2010


Parameterized complexity

Directed and undirected graphs useful for modeling allkinds of systemsUnfortunately, most problems are NP-hard on graphs ingeneralSolution: Parameterized algorithms

In most cases we don’t need to solve problems on generalgraphs; some structure is presentThis structure can be characterized by some structuralparameter having bounded sizeWe can then design algorithms which are polynomial forany fixed value of the parameter


Sofsem 2010


Parameterized complexity

Directed and undirected graphs useful for modeling allkinds of systemsUnfortunately, most problems are NP-hard on graphs ingeneralSolution: Parameterized algorithms

In most cases we don’t need to solve problems on generalgraphs; some structure is presentThis structure can be characterized by some structuralparameter having bounded sizeWe can then design algorithms which are polynomial forany fixed value of the parameter


Sofsem 2010


Parameterized complexity - Overview

The concept was very successful in undirected graphsTree-width: extremely popular width parameter, verypowerful (solves many problems), quite restrictive (requiresthe graphs to be “tree-like”)Rank-width: less restrictive but slightly less powerful, usefulwhen working with dense graphs

Situation more complicated in directed graphs:Rank-width→ bi-rank-width.Tree-width→ directed tree-width, DAG-width, Kelly-width,D-width, Cycle rank . . . ?


Sofsem 2010


Parameterized complexity - Overview

The concept was very successful in undirected graphsTree-width: extremely popular width parameter, verypowerful (solves many problems), quite restrictive (requiresthe graphs to be “tree-like”)Rank-width: less restrictive but slightly less powerful, usefulwhen working with dense graphs

Situation more complicated in directed graphs:Rank-width→ bi-rank-width.Tree-width→ directed tree-width, DAG-width, Kelly-width,D-width, Cycle rank . . . ?


Sofsem 2010


Parameterized complexity - Overview II

Classical “tree-width-like” directed parameters – not veryuseful for algorithm designIn IWPEC’09 we introduced two proxy parametersDAG-depth and K-width:

each extending all of the classical directed parameterseach being much more restrictive

And...

many problems still remain NP-hard! (for somevalues of the parameters)


Sofsem 2010


Parameterized complexity - Overview II

Classical “tree-width-like” directed parameters – not veryuseful for algorithm designIn IWPEC’09 we introduced two proxy parametersDAG-depth and K-width:

each extending all of the classical directed parameterseach being much more restrictive

And... many problems still remain NP-hard! (for somevalues of the parameters)


Sofsem 2010


Motivation

So, how bad is the situation?

We take a look at the complexity of Oriented Colouring, awell-studied problem on digraphs, parameterized by DAG-depthand K-width.


Sofsem 2010


Outline


2 Required notions


4 The reduction


Sofsem 2010


Oriented Colouring

Introduced by Courcelle, studied e.g. by Nešetril, Raspaud,Sopena.Only makes sense on digraphs; cannot be extended dographsMotivation: Scheduling, networking modelsTwo equivalent definitions:

Homomorphism into a tournament (some orientation of acomplete graph)The orientation of arcs must be preserved

A B

C

D

E

A

B C=D

E


Sofsem 2010


Oriented Colouring


Vertices are assigned colours, neighbouring verticescannot have same coloursAll arcs between colours go in the same direction


A B

C

D

E

A

B C=D

E


Sofsem 2010


Oriented Colouring



A B

C

D

E

A

B C=D

ERobert Ganian, Petr Hlinený Brno

Sofsem 2010


The Oriented Colouring problem

The Oriented Colouring decision problem for fixed k (OCNk inshort) is the problem of deciding whether an input digraph isorientably colourable by k colours.Equivalently, we may ask whether there exists a tournament oforder k and a homomorphism into this tournament.

There exists a polynomial time algorithm for OCN3, howeverOCN4 is NP-hard in general. The problem of computing theminimum k such that OCNk is true (OCN in short) is alsoNP-hard in general.


Sofsem 2010


DAG-depth

Directed counterpart to the Tree-depth of Nešetril and deMendezVery restrictive, similar design principles as classical widthmeasures (DAG-width, Kelly-width etc.)Bounded DAG-depth =⇒ all classical width measures arebounded


Sofsem 2010


DAG-depth definition

Formal definition:1 For a digraph G and any v ∈ V (G), let Gv denote the

subdigraph of G induced by the vertices reachable from v2 The maximal elements of the poset {Gv : v ∈ V (G) } in the

digraph-inclusion order are then called reachable fragmentsof G

3 The DAG-depth ddp(G) of a digraph G is inductivelydefined as follows:If |V (G)| = 1, then ddp(G) = 1If G has a single reachable fragment, thenddp(G) = 1 + min{ddp(G − v) : v ∈ V (G)}. Otherwise,ddp(G) = max{ddp(F ) : F ∈ RF(G)}


Sofsem 2010



More intuitive definition – Cops and robber gameA digraph G has DAG-depth d if it is possible to catch arobber in G by placing d stationary cops.The robber can move on directed paths which are notblocked by a cop.


Sofsem 2010




Red – Robber Blue – Cops


Sofsem 2010






Sofsem 2010






Sofsem 2010






Sofsem 2010


K-width

A digraph G has K-width k iff the maximum number ofdirected paths between any pair of vertices in G is k .Note that these directed paths need not be pairwise vertexdisjoint.Again, bounded K-width =⇒ all classical widthparameters are bounded.


Sofsem 2010


K-width

A digraph G has K-width k iff the maximum number ofdirected paths between any pair of vertices in G is k .Note that these directed paths need not be pairwise vertexdisjoint.Again, bounded K-width =⇒ all classical widthparameters are bounded.


Sofsem 2010


Outline


2 Required notions


4 The reduction


Sofsem 2010


OCN on K-width

It is possible to compute the OCN of a digraph G withK-width 1 with a single source in polynomial time

However, already deciding OCN4 on general digraphs ofK-width 1 is NP-hard.


Sofsem 2010


OCN on K-width

It is possible to compute the OCN of a digraph G withK-width 1 with a single source in polynomial timeHowever, already deciding OCN4 on general digraphs ofK-width 1 is NP-hard.


Sofsem 2010


OCN on DAG-depth

Digraphs of DAG-depth 2 are always orientedly3-colourable.

However, deciding OCN4 is NP-hard already on digraphsof DAG-depth 3.


Sofsem 2010


OCN on DAG-depth

Digraphs of DAG-depth 2 are always orientedly3-colourable.However, deciding OCN4 is NP-hard already on digraphsof DAG-depth 3.


Sofsem 2010


Outline


2 Required notions


4 The reduction


Sofsem 2010


Hardness proofs I

We prove NP-hardness for DAG-depth and K-width with asingle reduction from 3-SAT.(x ∨ ¬y ∨ z) – clausex , y , z . . . – literals

First, for each literal in a given 3-SAT formula we create acopy of the following gadget L:

x

¬x


Sofsem 2010


Hardness proofs I

We prove NP-hardness for DAG-depth and K-width with asingle reduction from 3-SAT.First, for each literal in a given 3-SAT formula we create acopy of the following gadget L:

x

¬x


Sofsem 2010


Hardness proofs I


x

¬x


Sofsem 2010


Hardness proofs I


x

¬x

B

T F

A


Sofsem 2010


Hardness proofs II

For each clause, we create a copy of the following gadgetS:

s

l1l2

s′

l3

l1, l2, l3 are identified with the appropriate nodes in L.


Sofsem 2010


Hardness proofs III

OCN4 NP-hard on digraphs of K-width 1 and DAG-depth 3.it is also possible to use a different gadget L′ to improve theoriginal proof of NP-hardness on directed acyclic graphs:

Original reduction by Culus and Demange in Sofsem 2009.Original reduction required careful use of S, could createcycles otherwise – no longer neededNew reduction requires less vertices and, in our opinion, iseasier to understand.


Sofsem 2010


Hardness proofs III

OCN4 NP-hard on digraphs of K-width 1 and DAG-depth 3.it is also possible to use a different gadget L′ to improve theoriginal proof of NP-hardness on directed acyclic graphs:

Original reduction by Culus and Demange in Sofsem 2009.Original reduction required careful use of S, could createcycles otherwise – no longer neededNew reduction requires less vertices and, in our opinion, iseasier to understand.


Sofsem 2010


Hardness proofs IV

Gadget L′:

¬x

x


Sofsem 2010


Concluding notes

? We have introduced two new algorithms for computingOCN on restricted digraph classes

? The restrictions on these classes are tight – we prove thateven deciding OCN4 becomes NP-hard on slightly lessrestricted classes

? Classical “tree-width-like” parameters are not useful fordeciding and computing OCN, even after furtherrestrictions are applied.

? Other width parameters should be used – e.g.bi-rank-width


Sofsem 2010


ThankYouForYour

Attention!


Sofsem 2010

Documents

New results on the complexity of oriented colouring on restricted