Coloring graphs with locally few colors
Discrete Mathematics
On the fractional dimension of partially ordered sets
Discrete Mathematics - Special issue: trends in discrete mathematics
Scheduling to minimize average completion time: off-line and on-line approximation algorithms
Mathematics of Operations Research
Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
The Lovász Theta Function and a Semidefinite Programming Relaxation of Vertex Cover
SIAM Journal on Discrete Mathematics
Precedence constrained scheduling to minimize sum of weighted completion times on a single machine
Discrete Applied Mathematics
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs
SIAM Journal on Computing
Proceedings of the 5th International IPCO Conference on Integer Programming and Combinatorial Optimization
Local chromatic number and Sperner capacity
Journal of Combinatorial Theory Series B
Single-Machine Scheduling with Precedence Constraints
Mathematics of Operations Research
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Scheduling with Precedence Constraints of Low Fractional Dimension
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
A better approximation ratio for the vertex cover problem
ACM Transactions on Algorithms (TALG)
Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Optimal Long Code Test with One Free Bit
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Approximating precedence-constrained single machine scheduling by coloring
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Operations Research Letters
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Erdos et al. defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex. For any @D=2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than @D different colors (bounded local colorability), and (ii) adjacent vertices from two color classes induce a complete bipartite graph (biclique coloring). We investigate the weighted vertex cover problem in graphs when a locally bounded coloring is given. This generalizes the vertex cover problem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number. Assuming the Unique Game Conjecture (UGC), we provide a tight characterization. We prove that it is UGC-hard to improve the approximation ratio of 2-2/(@D+1) on (@D+1)-locally (but not necessarily biclique) colorable graphs. A matching upper bound is also provided. Vice versa, when properties (i) and (ii) hold, we present a randomized algorithm with approximation ratio of 2-@W(1)lnln@Dln@D. This matches known inapproximability results for the special case of bounded degree graphs. Moreover, we show that when both the above two properties (i) and (ii) hold, the obtained result finds a natural application in a classical scheduling problem, namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time, denoted as 1|prec|@?w"jC"j in standard scheduling notation. In a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs G"P of incomparable pairs defined in the dimension theory of partial orders. We show that G"P satisfies properties (i) and (ii) where @D-1 is the maximum number of predecessors (or successors) of each job.