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Finding a maximum independent set in a sparse random graph

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### 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.

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-SS

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.

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|.

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