2 research outputs found
Single source unsplittable flows with arc-wise lower and upper bounds
In a digraph with a source and several destination nodes with associated demands, an unsplittable flow routes each demand along a single path from the common source to its destination. Given some flow x that is not necessarily unsplittable but satisfies all demands, it is a natural question to ask for an unsplittable flow y that does not deviate from x by too much, i.e., ya≈xa for all arcs a. Twenty years ago, in a landmark paper, Dinitz et al. (Combinatorica 19:17–41, 1999) proved that there exists an unsplittable flow y such that ya≤xa+dmax for all arcs a, where dmax denotes the maximum demand value. Our first contribution is a considerably simpler one-page proof for this classical result, based upon an entirely new approach. Secondly, using a subtle variant of this approach, we obtain a new result: There is an unsplittable flow y such that ya≥xa−dmax for all arcs a. Finally, building upon an iterative rounding technique previously introduced by Kolliopoulos and Stein (SIAM J Comput 31:919–946, 2002) and Skutella (Math Program 91:493–514, 2002), we prove existence of an unsplittable flow that simultaneously satisfies the upper and lower bounds for the special case when demands are integer multiples of each other. For arbitrary demand values, we prove the weaker simultaneous bounds xa/2−dmax≤ya≤2xa+dmax for all arcs a.TU Berlin, Open-Access-Mittel – 202
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Combinatorial Optimization (hybrid meeting)
Combinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years