Closely related to the maximum clique problem, the maximum independent set problem (MIS) was among the first problems proved NP-complete [5]. It is now known that the size of a maximum independent set cannot be approximated even within a factor of n’-O(l) in polynomial time [2, 1, 6, 31. Therefore it is important to find faster exact algorithms for MIS. Tarjan and Trojanowski [9] were the first to break through the 2” barrier: they gave a 2°.333n-time, polynomial-space algorithm for MIS. Tarjan and Trojanowski [9] observed that if G has a node of degree 0 or 1, then it suffices to solve MIS on an induced subgraph of G. If G has a higher-degree vertex, then they solve MIS on several induced subgraphs of G. The result of Tarjan and Trojanowski [9] was improved by Jian [4] and independently by Robson [7]. Jian gave a 2°.304n-time, polynomial-space algorithm. Robson gave a 2°.2g2n -time, polynomial-space algorithm, as well as a 2°.276n-time, exponential-space algorithm. All of those algorithms involve tedious case enumeration. Shindo and Tomita [8] gave a 2°.34gn-time, polynomial-space algorithm that is easy to describe but nontrivial to analyze. Our algorithm produces subproblems that need not be induced subgraphs of G. One immediate advantage is that if G has a node of degree 2 we solve only one subproblem, whereas previous algorithms solved two. On the other hand, the resulting recursion may produce sub*problems containing nodes of arbitrarily high degree, even though .the degree of G is bounded. By counting edges instead of vertices as our progress measure, we are nonetheless able to obtain good results for bounded-degree graphs. We obtain the fastest MIS algorithms for sparse graphs and for graphs whose degree is bounded by 3 or by 4. We also obtain the fastest MIS algorithm for
[1]
Piotr Berman,et al.
On the Complexity of Approximating the Independent Set Problem
,
1989,
Inf. Comput..
[2]
László Lovász,et al.
Approximating clique is almost NP-complete
,
1991,
[1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.
[3]
Carsten Lund,et al.
On the hardness of approximating minimization problems
,
1994,
JACM.
[4]
Etsuji Tomita,et al.
A Simple Algorithm for Finding a Maximum Clique and Its Worst-Case Time Complexity
,
1990,
Systems and Computers in Japan.
[5]
J. Håstad.
Clique is hard to approximate withinn1−ε
,
1999
.
[6]
John Michael Robson,et al.
Algorithms for Maximum Independent Sets
,
1986,
J. Algorithms.
[7]
Tang Jian,et al.
An O(20.304n) Algorithm for Solving Maximum Independent Set Problem
,
1986,
IEEE Trans. Computers.
[8]
Jonas Holmerin,et al.
Clique Is Hard to Approximate within n1-o(1)
,
2000,
ICALP.
[9]
Robert E. Tarjan,et al.
Finding a Maximum Independent Set
,
1976,
SIAM J. Comput..