TY - GEN
T1 - Semirandom models as benchmarks for coloring algorithms
AU - Krivelevich, Michael
AU - Vilenchik, Dan
PY - 2006/1/1
Y1 - 2006/1/1
N2 - Semirandom models generate problem instances by blending random and adversarial decisions, thus intermediating between the worst-case assumptions that may be overly pessimistic in many situations, and the easy pure random case. In the Gn,p,k random graph model, the n vertices are partitioned into k color classes each of size n/k. Then, every edge connecting two different color classes is included with probability p = p(n). In the semirandom variant, Gn,p,k,* an adversary may add edges as long as the planted coloring is respected. Feige and Killian prove that unless NP ⊆ BPP, no polynomial time algorithm works whp when np < (1 - ε)ln n, in particular when np is constant. Therefore, it seems like G n,p,k,* is not an interesting benchmark for polynomial time algorithms designed to work whp on sparse instances (np a constant). We suggest two new criteria, using semirandom models, to serve as benchmarks for such algorithms. We also suggest two new coloring heuristics and compare them with the coloring heuristics suggested by Alou and Kahale 1997 and by Böttcher 2005. We prove that in some explicit sense both our heuristics are preferable to the latter.
AB - Semirandom models generate problem instances by blending random and adversarial decisions, thus intermediating between the worst-case assumptions that may be overly pessimistic in many situations, and the easy pure random case. In the Gn,p,k random graph model, the n vertices are partitioned into k color classes each of size n/k. Then, every edge connecting two different color classes is included with probability p = p(n). In the semirandom variant, Gn,p,k,* an adversary may add edges as long as the planted coloring is respected. Feige and Killian prove that unless NP ⊆ BPP, no polynomial time algorithm works whp when np < (1 - ε)ln n, in particular when np is constant. Therefore, it seems like G n,p,k,* is not an interesting benchmark for polynomial time algorithms designed to work whp on sparse instances (np a constant). We suggest two new criteria, using semirandom models, to serve as benchmarks for such algorithms. We also suggest two new coloring heuristics and compare them with the coloring heuristics suggested by Alou and Kahale 1997 and by Böttcher 2005. We prove that in some explicit sense both our heuristics are preferable to the latter.
UR - http://www.scopus.com/inward/record.url?scp=33646838819&partnerID=8YFLogxK
U2 - 10.1137/1.9781611972962.4
DO - 10.1137/1.9781611972962.4
M3 - Conference contribution
AN - SCOPUS:33646838819
SN - 0898716101
SN - 9780898716108
T3 - Proceedings of the 8th Workshop on Algorithm Engineering and Experiments and the 3rd Workshop on Analytic Algorithms and Combinatorics
SP - 211
EP - 221
BT - Proceedings of the 8th Workshop on Algorithm Engineering and Experiments and the 3rd Workshop on Analytic Algorithms and Combinatorics
PB - Society for Industrial and Applied Mathematics Publications
T2 - 8th Workshop on Algorithm Engineering and Experiments and the 3rd Workshop on Analytic Algorithms and Combinatorics
Y2 - 21 January 2006 through 21 January 2006
ER -