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
Motivation and background Required notions Complexity results The reduction
Outline
1 Motivation and background
2 Required notions
3 Complexity results
4 The reduction
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
Outline
1 Motivation and background
2 Required notions
3 Complexity results
4 The reduction
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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 . . . ?
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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 . . . ?
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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)
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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)
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
Outline
1 Motivation and background
2 Required notions
3 Complexity results
4 The reduction
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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:
Vertices are assigned colours, neighbouring verticescannot have same coloursAll arcs between colours go in the same direction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
ERobert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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)}
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
DAG-depth definition
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
DAG-depth definition
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.
Red – Robber Blue – Cops
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
DAG-depth definition
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.
Red – Robber Blue – Cops
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
DAG-depth definition
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.
Red – Robber Blue – Cops
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
DAG-depth definition
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.
Red – Robber Blue – Cops
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
Outline
1 Motivation and background
2 Required notions
3 Complexity results
4 The reduction
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
Outline
1 Motivation and background
2 Required notions
3 Complexity results
4 The reduction
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
B
T F
A
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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.
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
Hardness proofs IV
Gadget L′:
¬x
x
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
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
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
Motivation and background Required notions Complexity results The reduction
ThankYouForYour
Attention!
Robert Ganian, Petr Hlinený Brno
Sofsem 2010
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