We apply a multi-color extension of the Beck-Fiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1 /2-approximable on directed graphs and randomized 2 /3-approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satisfiablilty problem is 1 /2-approximable.
We study the approximability and the hardness of combinatorial multi-objective NP optimization problems (multi-objective problems, for short). Our contributions are: – We define and compare several solution notions that capture reasonable algorithmic tasks for computing optimal solutions. – These solution notions induce corresponding NP-hardness notions for… (More)
For every list of integers x 1 ,. .. , x m there is some j such that x 1 + · · · + x j − x j+1 − · · · − x m ≈ 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the x i are k-dimensional integer vectors? Using results from topological degree theory we show that balancing is still… (More)
We study the autoreducibility and mitoticity of complete sets for NP and other complexity classes, where the main focus is on logspace reducibilities. In particular, we obtain:
Instances of optimization problems with multiple objectives can have several optimal solutions whose cost vectors are incomparable. This ambiguity leads to several reasonable notions for solving multiobjective problems. Each such notion defines a class of multivalued functions. We systematically investigate the computational complexity of these classes.… (More)
We improve and derandomize the best known approximation algorithm for the two-criteria metric traveling salesman problem (2-TSP). More precisely, we construct a deter-ministic 2-approximation which answers an open question by Manthey. Moreover, we show that 2-TSP is randomized (3 /2 + ε, 2)-approximable, and we give the first randomized approximations for… (More)