1
Approximability Results for Induced Matchings in
Graphs
David ManloveUniversity of Glasgow
Joint work with Billy Duckworth Michele Zito
Macquarie University University of Liverpool
Supported by EPSRC grant GR/R84597/01,Nuffield Foundation award NUF-NAL-02, and RSE / SEETLLD Personal Research Fellowship
2
What is a matching?
u1
u2
u3
u4
w1
w2
w3
w4
Let G=(V,E) be a graph A matching M is a set of edges in E, such that
no pair of edges of M are adjacent in G
A matching of size 3
3
What is a matching?
u1
u2
u3
u4
w1
w2
w3
w4
Let G=(V,E) be a graph A matching M is a set of edges in E, such that
no pair of edges of M are adjacent in G
A matching of size 4 – a maximum matching
4
What is an induced matching?
Not an induced matching
An induced matching M is a matching such that no pair of edges of M are joined by an edge in G
u1
u2
u3
u4
w1
w2
w3
w4
5
What is an induced matching?
u1
u2
u3
u4
w1
w2
w3
w4
An induced matching of size 2
An induced matching M is a matching such that no pair of edges of M are joined by an edge in G
6
What is an induced matching?
u1
u2
u3
u4
w1
w2
w3
w4
An induced matching of size 3 – a maximum induced matching
An induced matching M is a matching such that no pair of edges of M are joined by an edge in G
7
Maximum induced matchings
Let MIM denote the problem of finding a maximum induced matching in a given graph
MIM has applications in: VLSI design Channel assignment problems Network flow
MIM is NP-hard (Stockmeyer and Vazirani, 1982) No polynomial-time algorithm exists unless P=NP
Consider restricted classes of graphs Some cases might be polynomial-time solvable Many cases remain NP-hard!
8
Restrictions on vertex degrees
The degree of a vertex v is the number of edges incident to v
A graph has maximum degree d if every vertex has degree ≤d
A graph is d-regular if each vertex has degree d
A 3-regular graph is also called a cubic graph
9
Complexity results
MIM is NP-hard even for: planar bipartite graphs of maximum degree 3
(Ko and Shepherd, 1994) 4k-regular graphs for each k ≥ 1 (Zito, 1999) r-regular graphs for each r ≥ 5 (Kobler and Rotics, 2003)
MIM is solvable in polynomial time for: chordal graphs (Cameron, 1989) trees (Fricke and Laskar, 1992; Zito, 1999) and many other classes of graphs
10
Maximisation problems
A maximisation problem consists of: a set of instances each instance has a (finite) set of feasible solutions each feasible solution has a value for an instance I, denote by OPT(I) the value of a maximum
feasible solution
An optimising algorithm determines the value of OPT(I) for every instance I
For many problems, the only available optimising algorithms may be of exponential time complexity
An approximation algorithm is a polynomial-time algorithm that returns a feasible solution for a given instance
11
Approximation algorithms
Let P be a maximisation problem and let A be an approximation algorithm for P
For an instance I of P, suppose A returns a feasible solution with value A(I)
A has a performance guarantee c 1 if
A(I) (1/c) OPT(I) for all instances I
We say that A is a c-approximation algorithm
A has asymptotic performance guarantee c if there is some N such that, for any instance I of P where OPT(I)N,
A(I) 1/c OPT(I)
12
Polynomial-time approximation schemes
Let P be a maximisation problem
Suppose that, for any instance I of P and for any > 0 there exists a (1+ )-approximation algorithm A for P
Complexity of A must be polynomial in |I|
The family of algorithms {A : > 0 } is called a polynomial-time approximation scheme (PTAS)
13
Our results
For any d-regular graph, where d 3:
MIM admits an approximation algorithm with asymptotic performance guarantee d - 1
MIM is APX-complete i.e. MIM does not admit a polynomial-time
approximation scheme unless P=NP
Duckworth, Manlove, Zito, to appear inJournal of Discrete Algorithms, 2004
14
Approximation algorithm for MIM
let M be the empty matching;
select an edge {u,v} from E;
add {u,v} to M;
delete each edge at distance ≤ 2 from {u,v};
delete each vertex adjacent to u or v;
while there is some edge in G loop
choose a vertex u of minimum degree;
choose a vertex v of minimum degree adjacent to u;
add {u,v} to M;
delete each edge at distance ≤ 2 from {u,v};
delete each vertex adjacent to u or v;
end loop
15
Execution of the algorithm (1)
16
Execution of the algorithm (1)
u
v
17
Execution of the algorithm (1)
u
v
18
Execution of the algorithm (1)
u
v
19
1
1 3
3 3
3
3
3
3
3
Execution of the algorithm (1)
20
Execution of the algorithm (1)
u v
21
Execution of the algorithm (1)
u v
22
Execution of the algorithm (1)
u v
23
3
3
1
2
Execution of the algorithm (1)
1
2
24
Execution of the algorithm (1)
u v
25
Execution of the algorithm (1)
u v
26
Execution of the algorithm (1)
u v
27
Execution of the algorithm (1)
28
Execution of the algorithm (1)
u
v
29
Execution of the algorithm (1)
u
v
30
Execution of the algorithm (1)
31
Execution of the algorithm (1)
Algorithm produces optimal solution (size 4)
32
Execution of the algorithm (2)
33
Execution of the algorithm (2)
u
v
34
Execution of the algorithm (2)
u
v
35
Execution of the algorithm (2)
u
v
36
Execution of the algorithm (2)
2
3
3
3
3
37
Execution of the algorithm (2)
u
v
38
Execution of the algorithm (2)
u
v
39
Execution of the algorithm (2)
u
v
40
Execution of the algorithm (2)
41
Execution of the algorithm (2)
Algorithm produces induced matching of size 2
42
A maximum induced matching
Maximum induced matching has size 3
43
Bounds for induced matchings
Let G=(V,E) be a d-regular graph, where n=|V|
Theorem The algorithm produces an induced matching M where
Theorem (Zito ’99) Any induced matching M* satisfies
)1)(12(2
)2(||
dd
ndM
)12(2|| *
d
dnM
44
Bounds for induced matchings
Corollary The algorithm has asymptotic performance guarantee d - 1.
Proof let M be an induced matching returned by A
Let M* be a maximum induced matching in G
)1(2
)1)(12(2)2()12(2
||
||
)(
)( *
d
n
n
ddndddn
M
M
GA
GOPT
45
APX-completeness (1)
Theorem MIM is APX-complete for cubic graphs
Proof By reduction from MIS in cubic graphs
MIS is the problem of finding a maximum independent set in a given graph G
A set of vertices S is independent if no two vertices in S are adjacent in G
MIS is APX-complete in cubic graphs (Alimonti and Kann, 2000)
46
APX-completeness (1)
Theorem MIM is APX-complete for cubic graphs
Proof By reduction from MIS in cubic graphs
MIS is the problem of finding a maximum independent set in a given graph G
A set of vertices S is independent if no two vertices in S are adjacent in G
MIS is APX-complete in cubic graphs (Alimonti and Kann, 2000)
47
APX-completeness (2)
Theorem MIM is APX-complete for 4-regular graphs
Proof By reduction from MIM in cubic graphs (which is APX-complete by the previous theorem)
Theorem MIM is APX-complete for d-regular graphs, for d 5
Proof By reduction from MIS in (d - 2)-regular graphs (Kobler and Rotics, 2003)
MIS is APX-complete for (d - 2)-regular graphs (Chlebík and Chlebíková, 2003)
48
Open problems
Constant factor approximation algorithm for general graphs?
Improved approximation algorithms for d-regular graphs
Improved lower bounds for d-regular graphs Is there a PTAS for planar graphs?