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Capacitated network design on undirected graphs
 In APPROXRANDOM
, 2013
"... In this paper, we study the approximability of the capacitated network design problem (CapNDP) on undirected graphs: GivenG = (V,E) with nonnegative costs c and capacities u on its edges, sourcesink pairs (si, ti) with demand ri, the goal is to find the minimum cost subgraph where the minimum (si ..."
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In this paper, we study the approximability of the capacitated network design problem (CapNDP) on undirected graphs: GivenG = (V,E) with nonnegative costs c and capacities u on its edges, sourcesink pairs (si, ti) with demand ri, the goal is to find the minimum cost subgraph where the minimum (si, ti) cut with ucapacities is at least ri. When u ≡ 1, we get usual SNDP for which Jain gave a 2approximation algorithm [9]. Prior to our work, the approximability of undirected CapNDP was not well understood even in the single sourcesink pair case. In this paper, we show that the singlesource pair CapNDP is labelcover hard in undirected graphs. An important special case of single sourcesink pair undirected CapNDP is the following source location problem. Given an undirected graph, a collection of sources S and a sink t, find the minimum cardinality subset S ′ ⊆ S such that flow(S′, t), the maximum flow from S ′ to t, equals flow(S, t). In general, the problem is known to be setcover hard. We give a O(ρ)approximation when flow(s, t) ≈ρ flow(s′, t) for s, s ′ ∈ S, that is, all sources have maxflow values to the sink within a multiplicative ρ factor of each other. The main technical ingredient of our algorithmic result is the following theorem which may have other application. Given a capacitated, undirected graph G with a dedicated sink t, call a subset X ⊆ V irreducible if the maximum flow f(X) from X to t is strictly greater than that from any strict subset X ′ ⊂ X, to t. We prove that for any irreducible set, X, the flow f(X) ≥ 12 i∈X fi, where fi is the maxflow from i to t. That is, undirected flows are quasiadditive on irreducible sets. 1
Cluster Before You Hallucinate: Approximating NodeCapacitated Network Design and Energy Efficient Routing
, 2013
"... We consider circuit routing with an objective of minimizing energy, in a network of routers that are speed scalable and that may be shutdown when idle. We consider both multicast routing and unicast routing. It is known that this energy minimization problem can be reduced to a capacitated flow netw ..."
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We consider circuit routing with an objective of minimizing energy, in a network of routers that are speed scalable and that may be shutdown when idle. We consider both multicast routing and unicast routing. It is known that this energy minimization problem can be reduced to a capacitated flow network design problem, where vertices have a common capacity but arbitrary costs, and the goal is to choose a minimum cost collection of vertices whose induced subgraph will support the specified flow requirements. For the multicast (singlesink) capacitated design problem we give a polynomialtime algorithm that is O(log3 n)approximate with O(log4 n) congestion. This translates back to a O(log4α+3 n)approximation for the multicast energyminimization routing problem, where α is the polynomial exponent in the dynamic power used by a router. For the unicast (multicommodity) capacitated design problem we give a polynomialtime algorithm that is O(log5 n)approximate with O(log12 n) congestion, which translates back to a O(log12α+5 n)approximation for the unicast energyminimization routing problem.