Faster scaling algorithms for network problems
SIAM Journal on Computing
Approximation algorithms for scheduling unrelated parallel machines
Mathematical Programming: Series A and B
Planar orientations with low out-degree and compaction of adjacency matrices
Theoretical Computer Science
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Dynamic Representation of Sparse Graphs
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
Solving the Convex Cost Integer Dual Network Flow Problem
Management Science
Discrete Applied Mathematics
Upper degree-constrained partial orientations
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Complexity of approximating bounded variants of optimization problems
Theoretical Computer Science - Foundations of computation theory (FCT 2003)
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
Route-Enabling Graph Orientation Problems
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Journal of Computer and System Sciences
Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree
Journal of Combinatorial Optimization
Iterative Methods in Combinatorial Optimization
Iterative Methods in Combinatorial Optimization
Approximation scheme for lowest outdegree orientation and graph density measures
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
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Given an undirected graph G = (V, E), a graph orientation problem is to decide a direction for each edge so that the resulting directed graph G = (V, Λ (E)) satisfies a certain condition, where Λ(E) is a set of assignments of a direction to each edge {u, v} ε E. Among many conceivable types of conditions, we consider a degree constrained orientation: Given positive integers av and bv for each v (av ≤ bv), decide an orientation of G so that av ≤ |{(v, u) ε Λ(E)}| ≤ bv holds for every v ε V. However, such an orientation does not always exist. In this case, it is desirable to find an orientation that best fits the condition instead. In this paper, we consider the problem of finding an orientation that minimizes ΣvεV cv, where cv is a penalty incurred for v's violating the degree constraint. As penalty functions, several classes of functions can be considered, e. g., linear functions, convex functions and concave functions. We show that the degree-constrained orientation with any convex (including linear) penalty function can be solved in O(m1.5 min{Δ0.5, log(nC)}), where n = |V|, m = |E|, Δ and C are the maximum degree and the largest magnitude of a penalty, respectively. In contrast, it has no polynomial approximation algorithm whose approximation factor is better than 1.3606, for concave penalty functions, unless P=NP; it is APX-hard. This holds even for step functions, which are considered concave. For trees, the problem with any penalty functions can be solved exactly in O(n log Δ) time, and if the penalty function is convex, it is solvable in linear time.