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. Polynomial time algorithms on “simple” graphs: trees (greedy), graphs of bounded treewidth (dynamic programming).

Complexity on most graphs On random graphs, no polynomial time algorithm is known to find max-IS. Holds for all densities except for extremely sparse or extremely dense graphs. “Best” algorithmic lower bound – greedy. “Best” upper bound – theta function.

Planted models Model the case that a graph happens to have an exceptionally large IS. Random graph with edge probability d/n. All edges within a random set S of vertices are removed. When d|S| > n log n, the set S is likely to be the maximum IS.

Planted model

S

S

Some known results When d=n/2, can find S of size [Alon, Krivelevich, Sudakov 1998]. Can also certify maximality, and handle semirandom graphs [Feige, Krauthgamer 2000]. When d=log n, can find S of size, even in semirandom graphs, up to the point when it becomes NP-hard [Feige, Kilian 2001]

Our results 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.

S is not max-IS V-S S

Some related work Many of the techniques in this area were initiated in work of Alon and Kahale (1997) on coloring. Amin Coja-Oghlan (2005): finds a planted bisection in a sparse random graph. The min bisection is not the planted one. Amin’s algorithm is based on spectral techniques and certifies minimality.

Greedy algorithm Select vertex i to put in solution (e.g., vertex i may be vertex of degree 0, degree 1, or of lowest degree). Remove neighbors of i. Repeat on G – i – N(i).

Simplify analysis 2-stage greedy Select an independent set I. Remove neighbors of I. Finish off by exact algorithm. Last stage takes polynomial time if G-I-N(I) has “simple” structure.

Required properties of I Partition graph into Independent, Cover and Undecided. No edge within I. No edge between I and U. Every vertex of C must have at least one neighbor in I. Note: U is then precisely V(G) – I – N(I).

How we select I Initialization. Threshold t = d(1 - |S|/2n) < d. Put vertices of degree lower than t in I. Put vertices of degree higher than t in C. Iteratively, move to U: Vertices of I with neighbors in I or U. Vertices of C with < 4 neighbors in I.

End of first step I C U

Theorems for planted model 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.

Algorithm for G(U) Iteratively Move vertices of degree 0 to I. Move vertices of degree 1 to I, and their neighbors to C. Use exhaustive search to find maximum IS in each of the remaining connected components. Thm: CC of 2-core have size < O(log n).

Why did we consider 2-core? 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.

Conclusions Planted model in sparse graphs, in which planted solution is not optimal. Natural algorithm provably finds max-IS in planted model. (All difficulties are hidden in the analysis.) Improve tradeoff between d and |S|. Output matching upper bound on |max-IS|.