TY - GEN

T1 - Semirandom models as benchmarks for coloring algorithms

AU - Krivelevich, Michael

AU - Vilenchik, Dan

PY - 2006

Y1 - 2006

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

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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 (SIAM)

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 -