Finding a maximum independent set in a sparse random graph. Uriel Feige and Eran Ofek. Max Independent Set. Largest set of vertices that induce no edge. NP-hard, even to approximate. NP-hard on planar graphs.
Uriel Feige and Eran Ofek
Largest set of vertices that induce no edge.
Allow d to be (a sufficiently large) constant.
W.h.p., the random graph) has no independent set larger than n (log d)/d.
Plant S of size
We find max independent set in polynomial time.
New aspect: S is not the max-IS. Complicates analysis.
Last stage takes polynomial time if G-I-N(I) has “simple” structure.
Partition graph into Independent, Cover and Undecided.
Note: U is then precisely V(G) – I – N(I).
Initialization. Threshold t = d(1 - |S|/2n) < d.
Iteratively, move to U:
Lem: S highly correlated with max-IS.
Lem: Low degree highly correlated with S.
Thm: I is contained in max-IS.
(Difficulty in proof: max-IS is not known not only to the algorithm, but also in analysis.)
Thm: G(U) has simple structure.
Use exhaustive search to find maximum IS in each of the remaining connected components.
Thm: CC of 2-core have size < O(log n).
Asymmetry: vertices of S enter U more easily than vertices of V-S.
A tree might have most its vertices from S.
In a cycle, at least half the vertices must be from V-S.
Easier to show that U has no large cycles then to show that has no large trees.