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

We prove almost tight hardness results under <i>randomized reductions</i> 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 <i>n</i> is the number of vertices): • <i>NP-hardness</i> of finding an independent set of size larger than <i>O</i> (<i>n</i>[EQUATION]) in a 2-colorable <i>r</i>-uniform hypergraph for each fixed <i>r</i> ≥ 4. A simple algorithm is known to find independent sets of size Ω ([EQUATION]) in <i>any r</i>-uniform hypergraph of maximum degree Δ. Under a combinatorial conjecture on hypergraphs, the (log Δ)^{1/(<i>r</i>−1)} factor in our result is necessary. • Conditional hardness of finding an independent set with more than <i>O</i> ([EQUATION]) vertices in a <i>k</i>-colorable (with <i>k</i> ≥ 7) graph for some absolute constant <i>c</i> ≤ 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 <i>k</i>-colorable graphs. • Conditional hardness of finding independent sets of size <i>n</i>Δ^-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(<i>n</i>Δ^-1/8) in 2-colorable 3-uniform hypergraphs. • Conditional hardness of finding an independent set of size more than <i>O</i>(<i>n</i>Δ -1/r-1 log -1/r-1 Δ) in <i>r</i>-uniform hypergraphs that contain an independent set of size <i>n</i>(1 − <i>O</i>(log <i>r/r</i>)) assuming the Unique Games Conjecture.