Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on , a model recently introduced bulk for addressing the need to model non-uniform failure patterns in systems. We significantly extend the techniques used in bulk to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.
@cite_18: We study a new two-stage version of an s-t path problem, which we call adaptable robust connection path (ARCP). Given an undirected graph G=(V,E), two vertices s,t@?V and two integers f,r@?Z"+, ARCP asks to find a set S@?E of minimum cardinality which connects s and t, such that for any 'failure set' F@?E with |F|@?f, the set of edges S@?F can be completed to a set which connects s and t by adding at most r edges from E@?F. We show the problem is NP-hard, and there is no polynomial-time @a-approximation algorithm for the problem for @a<2 unless P=NP. For f=r=1 we provide an exact polynomial algorithm, and for f=1 and arbitrary r we provide a polynomial 2-approximation algorithm. A characterization of the feasible set is provided for f=1 and several links are established between ARCP and other combinatorial optimization problems, including a new combinatorial optimization problem, the minimum reduced cost cycle problem whose complexity is open.
@cite_11: Following the well-studied two-stage optimization framework for stochastic optimization [15,8], we study approximation algorithms for robust two-stage optimization problems with an exponential number of scenarios. Prior to this work, [8] introduced approximation algorithms for two-stage robust optimization problems with explicitly given scenarios. In this paper, we assume the set of possible scenarios is given implicitly, for example by an upper bound on the number of active clients. In two-stage robust optimization, we need to pre-purchase some resources in the first stage before the adversary's action. In the second stage, after the adversary chooses the clients that need to be covered, we need to complement our solution by purchasing additional resources at an inflated price. The goal is to minimize the cost in the worst-case scenario. We give a general approach for solving such problems using LP rounding. Our approach uncovers an interesting connection between robust optimization and online competitive algorithms. We use this approach, together with known online algorithms, to develop approximation algorithms for several robust covering problems, such as set cover, vertex cover, and edge cover. We also study a simple buy-at-oncealgorithm that either covers all items in the first stage or does nothing in the first stage and waits to build the complete solution in the second stage. We show that this algorithm gives tight approximation factors for unweighted variants of these covering problems, but performs poorly for general weighted problems.
@cite_26: Robust network design takes the very successful framework of robust optimization and applies it to the area of network design, motivated by applications in communication networks. The main premise is that demands across the network are not fixed, but are variable or uncertain. However, they are known to fall within a prescribed uncertainty set. Our solution must have sufficient capacity to route any demand in this set; moreover, the routing must be oblivious, meaning it must be fixed up front, and not depend on the particular choice of demand from within the uncertainty set. A particular choice of uncertainty set within this framework yields the “hose model”, which has received particular attention due to applications to virtual private networks. A 2-approximation was known for the problem, using a solution template in the form of a tree. It was conjectured that this tree solution is actually always optimal; this became known as the VPN Conjecture. As one of the central results of this thesis, we prove this conjecture in full generality. In addition, we demonstrate a counterexample to a stronger multipath (fractional routing) version of the conjecture which had also been proposed. We initiate a study of the robust network design problem more generally, with a focus on approximability. In the general model, where the uncertainty set is given by an arbitrary separable polyhedron, we give a strong inapproximability result. We then consider a new and natural model generalizing the symmetric hose model, based on demands routable on a given tree, and provide a constant factor approximation algorithm. Lastly, we compare oblivious routing with the much more flexible (but also less practical) dynamic routing scheme where the routing may vary depending on the demand pattern. We show that in the worst case, the cost of an optimal oblivious routing solution can be much more expensive than the dynamic optimum, by up to a logarithmic factor.
@cite_7: An efficient heuristic is presented for the problem of finding a minimum-size k-connected spanning subgraph of a given (undirected or directed) graph G=(V,E). There are four versions of the problem, depending on whether G is undirected or directed, and whether the spanning subgraph is required to be k-node connected (k-NCSS) or k-edge connected (k-ECSS). The approximation guarantees are as follows: min-size k-NCSS of an undirected graph 1+[1 k], min-size k-NCSS of a directed graph 1+[1 k], min-size k-ECSS of an undirected graph 1+[7 k], & min-size k-ECSS of a directed graph 1+[4 spl radic k]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple, deterministic, and runs in time O(k|E| sup 2 ). For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or a-edge connected) spanning subgraph whose size is within a factor of (1.5+ spl epsiv ) of minimum, where spl epsiv >0 is a constant.
@cite_8: The smallest k-ECSS problem is, given a graph along with an integer k, find a spanning subgraph that is k-edge connected and contains the fewest possible number of edges. We examine a natural approximation algorithm based on rounding an LP solution. A tight bound on the approximation ratio is 1 + 3-k for undirected graphs with k > 1 odd, 1 + 2-k for undirected graphs with k even, and 1 + 2-k for directed graphs with k arbitrary. Using iterated rounding improves the first upper bound to 1 + 2-k. On the hardness side we show that for some absolute constant c > 0, for any integer k ≥ 2 (k ≥ 1), a polynomial-time algorithm approximating the smallest k-ECSS on undirected (directed) multigraphs to within ratio 1 + c-k would imply P = NP. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009
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@cite_1: We study a new robust path problem, the Online Replacement Path problem (ORP). Consider the problem of routing a physical package through a faulty network G = (V,E) from a source s ∈ V to a destination t ∈ V as quickly as possible. An adversary, whose objective is to maximize the latter routing time, can choose to remove a single edge in the network. In one setup, the identity of the edge is revealed to the routing mechanism (RM) while the package is in s. In this setup the best strategy is to route the package along the shortest path in the remaining network. The payoff maximization problem for the adversary becomes the Most Vital Arc problem (MVA), which amounts to choosing the edge in the network whose removal results in a maximal increase of the s-t distance. However, the assumption that the RM is informed about the failed edge when standing at s is unrealistic in many applications, in which failures occur online, and, in particular, after the routing has started. We therefore consider the setup in which the adversary can reveal the identity of the failed edge just before the RM attempts to use this edge, thus forcing it to use a different route to t, starting from the current node. The problem of choosing the nominal path minimizing the worst case arrival time at t in this setup is ORP. We show that ORP can be solved in polynomial time and study other models naturally providing middle grounds between MVA and ORP. Our results show that ORP comprises a highly flexible and tractable framework for dealing with robustness issues in the design of RM-s.
@cite_15: Demand-robust versions of common optimization problems were recently introduced by [4] motivated by the worst-case considerations of two-stage stochastic optimization models. We study the demand robust min-cut and shortest path problems, and exploit the nature of the robust objective to give improved approximation factors. Specifically, we give a @math approximation for robust min-cut and a 7.1 approximation for robust shortest path. Previously, the best approximation factors were O(log n) for robust min-cut and 16 for robust shortest paths, both due to [4]. Our main technique can be summarized as follows: We investigate each of the second stage scenarios individually, checking if it can be independently serviced in the second stage within an acceptable cost (namely, a guess of the optimal second stage costs). For the costly scenarios that cannot be serviced in this way (“rainy days”), we show that they can be fully taken care of in a near-optimal first stage solution (i.e., by ”paying today”). We also consider “hitting-set” extensions of the robust min-cut and shortest path problems and show that our techniques can be combined with algorithms for Steiner multicut and group Steiner tree problems to give similar approximation guarantees for the hitting-set versions of robust min-cut and shortest path problems respectively.
@cite_16: Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worst-case among the various uncertain scenarios in the model. While these approaches may be thought of defining data- or cost-robust problems, we formulate a new "demand-robust" model motivated by recent work on two-stage stochastic optimization problems. We propose this in the framework of general covering problems and prove a general structural lemma about special types of first-stage solutions for such problems: there exists a first-stage solution that is a minimal feasible solution for the union of the demands for some subset of the scenarios and its objective function value is no more than twice the optimal. We then provide approximation algorithms for a variety of standard discrete covering problems in this setting, including minimum cut, minimum multi-cut, shortest paths, Steiner trees, vertex cover and un-capacitated facility location. While many of our results draw from rounding approaches recently developed for stochastic programming problems, we also show new applications of old metric rounding techniques for cut problems in this demand-robust setting.
@cite_20: For the past few decades, combinatorial optimization techniques have been shown to be powerful tools for formulating and solving optimization problems arising from practical situations. In particular, many network design problems have been formulated as combinatorial optimization problems. With the advances of optical technologies and the explosive growth of the Internet, telecommunication networks have seen an important evolution and therefore designing survivable networks has become a major objective for telecommunication operators. Over the past years, much research has been carried out to devise efficient methods for survivable network models, and particularly cutting plane based algorithms. In this paper, we attempt to survey some of these models and the optimization methods used for solving them. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 1–21 2005
@cite_17: We study two-stage robustvariants of combinatorial optimization problems like Steiner tree, Steiner forest, and uncapacitated facility location. The robust optimization problems, previously studied by [1], [6], and [4], are two-stage planning problems in which the requirements are revealed after some decisions are taken in stage one. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust Steiner tree problem, for example, one buys some edges in stage one after which some terminals are revealed. In the second stage, one has to buy more edges, at a higher cost, to complete the stage one solution to build a Steiner tree on these terminals. The objective is to minimize the total cost under the worst-case scenario. In this paper, we focus on the case of exponentially manyscenarios given implicitly. A scenario consists of any subset of kterminals (for Steiner tree), or any subset of kterminal-pairs (for Steiner forest), or any subset of kclients (for facility location). We present the first constant-factor approximation algorithms for the robust Steiner tree and robust uncapacitated facility location problems. For the robust Steiner forest problem with uniform inflation, we present an O(logn)-approximation and show that the problem with two inflation factors is impossible to approximate within O(log1 2 i¾? i¾?n) factor, for any constant i¾?> 0, unless NP has randomized quasi-polynomial time algorithms. Finally, we show APX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question by [1] and [6].
An interesting class of problems with uncertainty in the feasible set was introduced by Dhamdhere, Goyal, Ravi and Singh @cite_16 . In this two-stage models the feasibility condition is only fully revealed in the second stage. While resources can be bought in both stages, they are cheaper in the first stage, in which only partial information about the feasible set is available. This model was subsequently studied by several other authors (see @cite_15 @cite_11 @cite_17 ). Different two-stage model was proposed in @cite_18 @cite_1 for the shortest path problem. Several other important network design problems are motivated by robust optimization. Such problems include the minimum @math -edge connected spanning subgraph problem @cite_7 @cite_8 and the survivable network design problem @cite_9 @cite_20 . Various other robust variants of classical combinatorial optimization problems were proposed. For a survey of these results we refer the reader to the theses of Adjiashvili @cite_21 and Olver @cite_26 .