The complexity of finding independent sets in bounded degree (hyper)graphs of low chromatic number

  • Authors:

  • Venkatesan Guruswami;Ali Kemal Sinop

  • Affiliations:

  • Carnegie Mellon University, Pittsburgh, PA;Carnegie Mellon University, Pittsburgh, PA

  • Venue:
  • Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
  • Year:
  • 2011

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Abstract

We prove almost tight hardness results under randomized reductions for finding independent sets in bounded degree graphs and hypergraphs that admit a good coloring. Our specific results include the following (where Δ, a constant, is a bound on the degree, and n is the number of vertices): • NP-hardness of finding an independent set of size larger than O (n[EQUATION]) in a 2-colorable r-uniform hypergraph for each fixed r ≥ 4. A simple algorithm is known to find independent sets of size Ω ([EQUATION]) in any r-uniform hypergraph of maximum degree Δ. Under a combinatorial conjecture on hypergraphs, the (log Δ)1/(r−1) factor in our result is necessary. • Conditional hardness of finding an independent set with more than O ([EQUATION]) vertices in a k-colorable (with k ≥ 7) graph for some absolute constant c ≤ 4, under Khot's 2-to-1 Conjecture. This suggests the near-optimality of Karger, Motwani and Sudan's graph coloring algorithm which finds an independent set of size Ω ([EQUATION]) in k-colorable graphs. • Conditional hardness of finding independent sets of size nΔ-1/8+oΔ(1) in almost 2-colorable 3-uniform hypergraphs, under Khot's Unique Games Conjecture. This suggests the optimality of the known algorithms to find an independent set of size Ω(nΔ-1/8) in 2-colorable 3-uniform hypergraphs. • Conditional hardness of finding an independent set of size more than O(nΔ -1/r-1 log -1/r-1 Δ) in r-uniform hypergraphs that contain an independent set of size n(1 − O(log r/r)) assuming the Unique Games Conjecture.