Integer multicommodity flow in seriesparallel graphs Naveen Garg

Integer multicommodity flow in series-parallel graphs Naveen Garg, IITD Joint work with Hannaneh Akrami (Sharif U), Nikhil Kumar (IITD)

Multicommodity flow • [integral routing] Find a path between end points of every demand s. t. #paths through an edge does not exceed its capacity. [fractional routing] 1 unit of flow is sent between end points of each demand. 1/2

Cut condition For demands to be (fractionally) routable the capacity of every cut should exceed demand across it. The cut condition is necessary but not sufficient. (2, 2) (3, 1) (3, 3) (capacity, demand)

Sufficiency of the cut condition •

Series-parallel graphs [parallel-paths graph] Graph obtained by identifying endpoints of 2 or more simple paths. [series-parallel graph] start with a pp-graph, replace edges by other pp-graphs and repeat. s t

Multicommodity flow in sp-graphs •

Integral flow in sp-graphs •

Algorithm for pp-graphs •

Phase 1 • 1 1 s 2 3 4 1 t 2 3 d a a a 4 3 b b c c b

At the end of Phase 1 • 1 1 s 2 3 4 1 4 3 2 3 e t

Phase 2 [segment] part of a path between two consecutive tight edges. A non-tight edge has residual capacity at least 2. A demand is incident to a segment if the path along which it is routed contains a vertex of the segment

Alternating sequence a sequence in which alternate elements are segments and (fully routed) demands (endpoints on different paths). All segments and demands are distinct. Each demand is incident to its adjacent segments in the sequence. Consecutive demands are routed through different ends. If the first and last demands are partially routed then we call it an augmenting sequence.

Flipping augmenting sequences •

Finding an augmenting sequence •

End of Phase 2 •

How to pick W? •

Why should no flow path go over more than 2 edges of W? We need to drop some edges picked in W. choice of these edges depends on the blossoms encountered in the algorithm for finding an alternating path in X.

Analysis •

Key results •

Open Questions Proving the conjecture for sp-graphs. A tight max multicommodity flow min multicut result for sp-graphs. [BIG ONE] Is relaxing cut condition by a constant factor sufficient to route flow (fractionally) in planar graphs.

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