When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks

Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the same set of vertices. An assignment of capacities to the edges of $G$ and demands to the edges of $H$ is said to satisfy the \emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair $(G,H)$ is called \emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on $G$. We prove a previous conjecture, which states that when the supply graph $G$ is series-parallel, the pair $(G,H)$ is cut-sufficient if and only if $(G,H)$ does not contain an \emph{odd spindle} as a minor; that is, if it is impossible to contract edges of $G$ and delete edges of $G$ and $H$ so that $G$ becomes the complete bipartite graph $K_{2,p}$, with $p\geq 3$ odd, and $H$ is composed of a cycle connecting the $p$ vertices of degree 2, and an edge connecting the two vertices of degree $p$. We further prove that if the instance is \emph{Eulerian} --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.Comment: An extended abstract of this paper will be published at the 44th Symposium on Theory of Computing (STOC 2012

Approximating Maximum Integral Multiflows on Bounded Genus Graphs

We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles

Maximum Edge-Disjoint Paths in kk-sums of Graphs

We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The integrality gap is known to be $\Omega(\sqrt{n})$ even for planar graphs due to a simple topological obstruction and a major focus, following earlier work, has been understanding the gap if some constant congestion is allowed. In this context, it is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold. It is easy to deduce that for given constant bounds on the approximation and congestion, the class of "nice" graphs is nor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a constant factor, constant congestion bound relative to the LP relaxation? We conjecture that the answer is yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph of bounded treewidth and is able to integrally route a constant fraction of the LP solution's value. Note that we do not incur any edge congestion. Previously this was not known even for series parallel graphs which have treewidth 2. The algorithm is based on a more general argument that applies to $k$-sums of graphs in some graph family, as long as the graph family has a constant factor, constant congestion bound. We then use this to show that such bounds hold for the class of $k$-sums of bounded genus graphs

Approximating maximum integral multiflows on bounded genus graphs

We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations

An Approximation Algorithm for Fully Planar Edge-Disjoint Paths

We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem