Partition inequalities for survivable network design

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					      Partition inequalities for survivable network design using directed p-cycles

                                      ¨
                           Alper Atamturk, University of California at Berkeley.
                             Deepak Rajan, IBM T.J Watson Research Center.


Keywords: Survivable network design, partition inequalities, directed p-cycles, mixed-integer programming.


1.   Overview

We study the design of capacitated survivable networks using directed p-cycles (SNP). We first introduce this
methodology for designing survivable networks, and present a mixed-integer programming formulation. Next
we develop valid partition inequalities for this problem, and prove that they are facets to the polyhedron of
SNP under mild conditions. We study the separation problems for these inequalities and present polynomial
time algorithms (exact/heuristic, as the case may be). We compare these inequalities with those developed
for survivable network design using global rerouting. Finally, we present computational results that illustrate
the effectiveness of the new inequalities when incorporated in a branch-and-cut framework.


2.   Introduction and Related Literature

Given a directed network, flow costs, capacity installation costs for each edge, and a set of commodities (given
as origin-destination pairs and corresponding demands), the capacitated network design problem (NDP) is to
route the commodities so that the net flow on any arc is at most the capacity installed on that edge and all
demands are met, at minimum total cost.

Network design problems become significantly more difficult when the networks have to be designed to
survive failures. As a simultaneous failure of multiple network elements occurs very infrequently, here we
focus on single-edge failures. A network is said to be survivable if sufficient capacity exists on the edges of
the network so that disrupted flow can be rerouted in the event of an edge failure. To do so, requires installing
spare capacity to the network. Various approaches that attempt to minimize the capacity requirement of the
survivable networks have been developed.

The most capacity-efficient survivable networks can be designed by formulating the problem as a capacitated
network design problem (NDP) for every failure scenario, linked by common capacity variables across the
scenarios [2]. An optimal solution to such a scenario formulation may call for rerouting the flow of com-
modities unaffected by the failure. Because it is not practical to manipulate unaffected flow while restoring
affected flows, this approach, also referred to as global rerouting (GNP), is not popular. However, GNP
still serves an important purpose as it provides a lower bound on the capacity requirement of a survivable net-
work. Methodologies that are implemented in practice usually involve some form of local rerouting, either by
enforcing a ring-like topology (dedicated protection) on the network, or by shared local protection schemes.

We study a hybrid approach for designing survivable networks, where we use directed p-cycles for shared
protection of disrupted flow and do not impose any topology on the network. The idea of hybrid networks was
first introduced in [5]. In this approach, one imposes no restrictions on no-failure routing, but uses failure-
flow patterns for all failure flows. Using predefined undirected cycles as failure-flow patterns has been shown
to be capacity-efficient, and with fast reconfiguration times.

In [7] the authors present computational experiments comparing various frameworks for designing survivable
networks, and noted that using directed p-cycles yields networks with comparable capacity requirements as
global rerouting. In [8] the authors present polyhedral inequalities for designing survivable network using
directed cycles (SDC).
3.   Motivation

A directed p-cycle is a logical construct on a graph used to reroute flows disrupted by edge failures. In any
solution to SNP, this is done by reserving sufficient fractional capacity on directed p-cycles on the network. To
see exactly how this works, we present a small example that also emphasizes the distinction between directed
cycles and directed p-cycles, and shows why using directed cycles yields a more conservative framework.

A directed p-cycle provides coverage for flows in the reverse direction for the arc on the p-cycle, see arc (ba)
in Figure 1. A directed p-cycle also provides one recovery path for the flow on a chord, see arcs (cd) and
(dc) in Figure 1. Directed cycles (in SDC) provide recovery paths only for the arcs on the cycle, and not for
the chords. In the example in Figure 1, a directed cycle can be used only to reroute the disrupted flow on arc
(ba), and not on arcs (cd) or (dc).


                                              b



                                                            c
                                              a                                     d




                                                  Figure 1: A directed p-cycle



Next we compare the capacity-efficiency of the frameworks GNP, SDC and SNP using a small computational
study. In Figure 2, we report the ratio of installed capacity for the solutions provided by GNP, SDC, and
SNP to the capacity requirements of NDP. We see that survivable networks produced by SDC require 80%
more capacity than NDP, whereas SNP requires only about a 45% increase. Since GNP, which provisions the
lowest possible capacity for survivable networks, requires about 30% more capacity than NDP, we see that
SNP requires only an additional 18% capacity, whereas SDC provisions 38% excess capacity over GNP.
                               2.50


                                                                         GNP                 SNP         SDC



                               2.00
                       Ratio




                               1.50




                               1.00
                                      5   6       7     8       9       10     11       12         13   14     15

                                                                Problem Size


                    Figure 2: Comparing capacity efficiency of the survivability models



Finally we present a theoretical argument supporting the capacity-efficiency of SNP. The relative strength of
methodologies can also be stated in terms of the set of feasible capacities. For any non-empty 2-partition


                                                                    2
(A, B) of G, let [AB] be the set of edges with one end in A, the other in B. Let dA be the net demand with
source node in A and destination node in B. To illustrate our claim about the relative strength of SDC and
SNP when compared to GNP, we consider the partition inequality
                                                       xe ≥ f (η),                                          (1)
                                              e∈[AB]

where η = dA , and x is the capacity variable for each edge.

For an “apples-to-apples” comparison, we consider the strongest (facet-defining) inequality of the form (1)
for each framework. Inequality (1) is the cut-set inequality for NDP [1] when f (η) = η. For GNP [3], (1) is
the same as the cardinality-k cut-set inequality when f (η) = n ∗ η/(n − 1) , where n = |AB|. For SNP, we
show that (1) is a special case of the subset-Q inequalities, with f (η) = n ∗ η/(n − 1) , which is the same
for GNP. This comparison indicates that the difference between capacity-efficiency of SNP and GNP is quite
small. However, for SDC [8], inequality (1) is the cut-set inequality with f (η) = 2η. Therefore SDC yields
a network with significantly larger capacity installation than SNP does.


4.   Contributions

The focus of this paper is a polyhedral study of the survivable network design problem using directed p-cycles.
We present classes of strong valid inequalities for SNP. Interestingly, some of the inequalities presented in
this paper can also be derived as metric inequalities [6]. We compare our results with polyhedral results for
global rerouting given in [3, 4], and generalize some of the results therein.

The theoretical results in this work also provide valuable insight on why using directed p-cycles yields highly
capacity-efficient networks, and characterize when SNP yields almost as capacity-efficient networks as GNP.
Finally, we incorporate the partition inequalities in a branch-and-cut framework to solve survivable network
design problems using directed p-cycles. We present the results of these computational experiments, which
show that the inequalities often reduce computation times by an order of magnitude.


References
          u
 [1] Atamt¨ rk, A., “On capacitated network design cut-set polyhedra,” Mathematical Programming, 92, pp.
     425–437, 2002.
                       o                    a
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                             u
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                           u
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