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

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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).

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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.

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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 dS gt n log n, the set S is likely to be
  the maximum IS.

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Planted model
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Planted model

• S

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Planted model

• S

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Some known results

• When dn/2, can find S of size Alon,
  Krivelevich, Sudakov 1998.
• Can also certify maximality, and handle
  semirandom graphs Feige, Krauthgamer 2000.
• When dlog n, can find S of size , even
  in semirandom graphs, up to the point when it
  becomes NP-hard Feige, Kilian 2001

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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.

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S is not max-IS

• V-S
  S

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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. Amins
  algorithm is based on spectral techniques and
  certifies minimality.

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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).

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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.

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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).

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How we select I

• Initialization. Threshold t d(1 - S/2n) lt 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 lt 4 neighbors in I.

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End of first step

• I
  C
• U

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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.

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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 lt O(log n).

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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.

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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.