Approximation Algorithms for Demand-Robust Covering Problems∗

How to Pay, Come What May: Approximation Algorithms for Demand-Robust Covering Problems∗

Kedar Dhamdhere† Vineet Goyal R. Ravi Google Inc. Tepper School of Business Tepper School of Business Mountain View, CA 94043 Carnegie Mellon University Carnegie Mellon University [email protected] Pittsburgh, PA 15213 Pittsburgh, PA 15213 [email protected] [email protected] Mohit Singh Tepper School of Business Carnegie Mellon University Pittsburgh, PA 15213 [email protected]

Abstract 1 Introduction

Robust optimization has traditionally focused on uncer- Robust optimization has roots in both Decision The- tainty in data and costs in optimization problems to formu- ory [14, 16] as well as Mathematical Programming [4]. late models whose solutions will be optimal in the worst- While minmax regret approaches were advanced in the for- case among the various uncertain scenarios in the model. mer field as conservative decision rules, robust optimization While these approaches may be thought of defining data- or was discussed along with stochastic programming [3] as an cost-robust problems, we formulate a new “demand-robust” alternate way of tackling data uncertainty. model motivated by recent work on two-stage stochastic op- More recent attempts at capturing the concept of ro- timization problems. We propose this in the framework of bust solutions in optimization problems include the work of general covering problems and prove a general structural Rosenblatt and Lee [21] in facility design problem, Mulvey lemma about special types of first-stage solutions for such et al. [17] in mathematical programming, and most recently, problems: there exists a first-stage solution that is a mini- Kouvelis and Yu [12] in combinatorial optimization; here mal feasible solution for the union of the demands for some robust means “good in all or most versions”, a version be- subset of the scenarios and its objective function value is ing a plausible set of values for the data in the model. Even no more than twice the optimal. We then provide approxi- more recent work along similar lines is advocated by Bertsi- mation algorithms for a variety of standard discrete cover- mas et al. [1, 2]. A recent annotated bibliography available ing problems in this setting, including minimum cut, min- online summarizes this line of work [18]. imum multi-cut, shortest paths, Steiner trees, vertex cover and un-capacitated facility location. While many of our re- 1.1 A new model of “demand-robustness” sults draw from rounding approaches recently developed for stochastic programming problems, we also show new appli- In this paper, we take a different approach in our model cations of old metric rounding techniques for cut problems of uncertainty. We do not address uncertainty in the form in this demand-robust setting. of inaccuracy in the data itself; rather we address the uncertainty in a subset of the constraints that the problem is required to satisfy. As a simple example, consider the two alternate formulations of the shortest path problem from a r ∗ root node under the data-robust and the demand-robust Supported in part by NSF grants CCF-0430751 and ITR grant CCR- formulations. In the more traditional data-robust formula- 0122581 (The ALADDIN project). † t Work was done while the author was at Computer Science Depart- tion, the other terminal node to which the shortest path ment, Carnegie Mellon University, Pittsburgh, USA. from r must be built is specified in advance. However,

Proceedings of the 2005 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 0-7695-2468-0/05 $20.00 © 2005 IEEE the costs of the edges in the graph may change as stipu- ios is realized i.e. one of the subsets Si materializes and lated either in a set of discrete scenarios, or by intervals the corresponding requirements need to be satisfied. Now, within which each edge cost lies. The data-robust formula- a feasible solution specifies the elements X0 to be bought tion models the problem of finding a path P from r to t such in the first stage, and for each k, a set of elements Xk to be that over all possible settings of the data (edge-costs) among bought in the recourse stage if scenario k is realized, such the scenarios, the maximum value of the cost of P is mini- that X0 ∪ Xk contains a feasible solution for client set Sk. mized by taking this path. In the demand-robust model we The cost of covering scenario k is c0(X0)+ck(Xk).Inthe consider, the costs of edges are specified in advance. Each demand-robust two-stage problem, the objective is to mini- scenario now specifies which terminal tk must be connected mize the maximum cost over all scenarios. Note that we pay to r via the shortest path. Furthermore, in the scenario k for all the elements in X0 even though some of them may specified by terminal tk, all the edge costs are costlier by a not be required in the solution for any one fixed scenario. specified factor σk. The problem is now modelled as one of As an example, the demand-robust “rooted” min-cut choosing a few edges to buy today at the current specified problem has X = the edge set of an undirected graph, (non-inflated) cost and then, for each scenario k, completing a specified root and each Sk specified by a terminal tk. the current solution by adding more edges (at costs inflated sol(Sk) is the set of all edge sets that separate tk from r. by σk) to form a path from r to tk. The objective is to min- As another example, in the demand-robust “rooted” Steiner imize the maximum value of the first stage costs plus the tree problem, we have X = the edges of an undirected second stage completion costs over all possible scenarios k. graph, a specified root r and each scenario Sk specified by k k a set of terminals {t1 ,t2 ,...}. sol(Sk) is the set of all edge k k 1.2 Relation to Stochastic Programming sets that connect all terminals {t1 ,t2 ,...} to the root r.

The roots of our new model have strong links to the 1.4 Results class of two-stage stochastic programming problems with recourse, for which some approximable versions were stud- In this paper, we formulate demand-robust versions of ied in recent work [9, 5, 11, 19, 23]. These two-stage mod- commonly studied covering problems in optimization in- els (e.g., from [9]) have a very similar structure: costs are cluding minimum cut, minimum multi-cut, shortest paths, specified today and the demands occurring tomorrow (along Steiner trees, vertex cover and un-capacitated facility loca- with their respective inflation factors) are specified by a tion, and provide approximation algorithms for these prob- probability distribution. The goal is to purchase some an- lems. Our results are summarized in Figure 1. While ticipatory part of the solution in the first stage so that the many of our results draw from rounding approaches re- expected cost of the solution over all possible scenarios cently developed for stochastic programming problems, we is minimized. While the expected value minimization is also show new applications of old metric rounding tech- reasonable in a repeated decision-making framework, one niques for cut problems in this demand-robust setting. shortcoming of this approach is that it does not sufficiently guard against the worst case over all the possible scenarios. 1.5 Contributions Our demand-robust model for such problems is a natural way to overcome this shortcoming by postulating a model One of the main contributions of this paper is to frame that minimizes this worst-case cost. the demand-robust problems and show how this leads to interesting versions of well-studied problems in combinato- 1.3 Model and Notation rial optimization. In Section 2, we show how a natural LP formulation of the demand-robust version of the minimum- We define an abstract covering problem in the demand- cut problem can be rounded within a logarithmic factor us- robust two-stage model with finite number of scenarios. Let ing ideas for rounding multi-cut problems [8, 13]. In Sec- U be the universe of clients (or demand requirements), and tion 3, we also show how a demand-robust version of the let X be the set of elements that we can purchase. Ev- multi-cut problem can also be approximated using further ery scenario is a subset of clients and is explicitly speci- ideas by taking care of a constant fraction of the demands fied. Let S1,S2,...,Sm ⊂ U be all the scenarios. For per scenario in each iteration of an iterative method (also X every scenario Sk,letsol(Sk) denote the sets in 2 which used in [13] for the feedback arc set problem). One of are feasible to cover Si: the covering formulation require the unanticipated outcomes of these new algorithms for the that A ⊆ B and A ∈ sol(Sk) ⇒ B ∈ sol(Sk).The demand-robust versions of the cut problems is that we get cost of an element x ∈ X in the first stage is c0(x).In the same guarantees for the two-stage stochastic versions of th the k scenario, it becomes costlier by a factor σk i.e. these problems, thus giving first poly-logarithmic approxi- ck(x)=σkc0(x). In the second stage, one of the scenar- mations for them as well (See Section 3.2).

Proceedings of the 2005 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 0-7695-2468-0/05 $20.00 © 2005 IEEE Problem Deterministic Approximation Stochastic Approximation Demand-robust Approximation Steiner Tree 1.55 [20] 3.55 [9], 30 [10] 30∗ Vertex Cover 2 (Primal-dual) 2[19] 4∗ Facility Location 1.52 [15] 5 [19], 3.04 [23] 5∗ Min Cut 1 O(log m)∗ O(log m)∗ Min Multi-Cut O(log r) [8] O(log rm log log rm)∗ O(log rm log log rm)∗

Figure 1. Result Summary . ∗ denotes results in this paper. In the table, m, n and r denote the number of scenarios, number of nodes and maximum number of pairs per scenario respectively.

In Section 4, we prove a simple structural lemma about Integer Program Formulation We formulate an integer special classes of first-stage solutions to robust covering linear program for the problem as follows. problems: Informally, this states that there is a first-stage min z solution that is a minimal feasible solution for the union of 0 k z ≥ e c0(e)(xe + σkxe ) ∀ k demands for a subset of the scenarios in the specification 0 k (x + x )(P ) ≥ 1 ∀ r-tk path P, ∀ k of the problem whose total cost is no more than twice that xk ∈{0, 1}∀e, k of the optimal. This result holds for a large class of cover- e ing problems including vertex cover, minimum (multi)cut, k Relaxing the integrality constraints to xe ≥ 0 gives us Steiner trees and facility location. However, in that section the linear programming relaxation. While the LP formula- we mainly apply it to the robust Steiner tree problem to for- tion given here has an exponential number of constraints, it mulate a more structured LP relaxation which is the start- can be solved efficiently by the ellipsoid algorithm where ing point for applying the methods in [10], finally giving us the separation oracle is just a shortest path computation. the constant-factor approximation result for robust Steiner trees. 2.1 Algorithm In Section 5, we point out how techniques previously 0 k developed for two-stage stochastic problems that work by We start by solving the LP relaxation. Let x˜e and x˜e charging the first-stage and second-stage parts of the solu- denote the values of the variables in the fractional optimal tion independently to the corresponding lower bounds in the solution. Let LPopt denote the optimum value of the LP relaxation to arrive at the final performance guarantee, can relaxation. To round the fractional LP solution, we use the be used to derive analogous results for the robust versions region growing technique of Garg et al. [8]. We would like of such problems. This remark applies to all covering prob- to stress that the notion of volume used here is different from lems addressed by Shmoys and Swamy [23] such as vertex the LP volume used in [8]. Moreover, in our problem the LP cover and the rounding methods of Ravi and Sinha [19] for gives a different metric on the graph for each scenario. facility location. We start by making a copy of the graph G =(V,E) for Our paper opens up the investigation on a new class of the first stage and all the second stage scenarios. We denote robust models, and leaves much to be done in this area. the copies by G0,G1,...,Gm. Edge e costs c0(e) in the graph G0 and σkc0(e) in the graph Gk. First we give some 2 Robust Min-cut Problem notation to use in our algorithm description. Let distk be the shortest path metric defined by the following lengths on 0 k Problem Definition We are given an undirected graph the edges: lk(e)=˜xe +˜xe , ∀ e ∈ E.LetBk(tk,ρ) denote th G =(V,E) witharootr.Thek scenario consists of a ball of radius ρ around the terminal tk in the metric distk. a single terminal tk. Edge costs c0(e) in the first stage and We define the volume Vk(tk,ρ) of the ball as th σkc0(e) in the recourse stage if the k scenario is realized. th LPopt 0 k σ k Vk(tk,ρ):= + c0(e)(˜x +˜x ) Here k is the inflation factor for the scenario. m e e The objective is to find a set of edges E0 to be bought in e∈Bk(tk,ρ) the first stage and for each k,asetEk to be bought in the + c0(e)(ρ − distk(tk,e)) recourse stage if scenario k is realized, such that removing e∈δ(Bk(tk,ρ)) E0 ∪ Ek from the graph G disconnects r from the terminal tk. The objective is to minimize the maximum cost over all Here distk(tk,e) denotes the metric distance between scenarios. The complexity of robust min-cut (as formulated tk and the closer endpoint of edge e. Note that the vol- above) is still open. ume Vk(tk,ρ), for any ρ, is not same as the LP volume.

Proceedings of the 2005 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 0-7695-2468-0/05 $20.00 © 2005 IEEE Algorithm Robust-Min-cut Lemma 2.2 Cost of the edges E0 is at most 4logm · 0 (LPopt + e c0(e)˜xe). 1. Let G0,G1,...,Gm be copies of G =(V,E). Initialize E0,E1,...,Ek ← φ. Proof: In the algorithm, we include the edges δ(Bk(tk,ρ)) 0 in E0 when 2Vk (tk,ρ) ≥ Vk(tk,ρ). Therefore, the cost of 2. Repeat the following: the edges of δ(Bk(tk,ρ)) is bounded above by 4logm · 0 V (tk,ρ) (a) Find a terminal tk that is connected to r in k . In other words, each unit of volume inside Bk(tk,ρ) 4logm the graph Gk ∩ G0 =(V,E \ (E0 ∪ Ek)). gets a charge of . Since we remove the ball Bk(tk,ρ) from graph G0, each edge in G0 is charged (b) Find a radius ρ<1/2 for which at most once. Therefore the total cost of edges in E0 is V t ,ρ /C t ,ρ k( k ) k( k ) is minimum. bounded by 0 1 (c) If Vk (tk,ρ) ≥ 2 Vk(tk,ρ), c E ≤ m V 0 t ,ρ ( 0) 4log k k ( k ) • E0 ← E0 ∪ δ(Bk(tk,ρ)) 0 ≤ 4logm(LPopt + e c0(e)˜xe). • Remove δ(Bk(tk,ρ)) from G0. 1 1 Else Vk (tk,ρ) > 2 Vk(tk,ρ), • Ek ← Ek ∪ δ(Bk(tk,ρ)) Lemma 2.3 Cost of the edges Ek is at most 4logm · • Remove δ(Bk(tk,ρ)) from Gk. k e σkc0(e)˜xe . Until all the terminals are separated from r. Proof: Note that the only time we include edges in Ek is 1 1 when Vk (tk,ρ) > 2 Vk(tk,ρ). Buying edge e in Gk costs σk times higher. Therefore the costs of the edges in Ek can LPopt However, it is bounded above by , which facilitates be bounded as follows: Claim 2.1. We split the volume among ﬁrst and recourse c E ≤ σ C t ,ρ ≤ m · σ V 1 t ,ρ stage contributions as being the part of the volume con- ( k) k k( k ) 4log k k ( k ) k tributed by ﬁrst-stage and second-stage variables respec- ≤ 4logm e σkc0(e)˜xe . tively.

0 LPopt 0 V (tk,ρ)= + c0(e)˜x Theorem 2.4 The Algorithm Robust-Min-Cut produces an k m e O(log m)-approximate solution to the robust min-cut prob- e∈Bk(tk,ρ) 0 lem. + c0(e)(min{ρ − distk(tk,e), x˜e}) e∈δ(Bk(tk,ρ)) Proof: Using Lemmas 2.2 and 2.3 the total cost of any and scenario k can be bounded as follows: 1 k Vk (tk,ρ)= c0(e)˜xe 0 k c(E0)+c(Ek) ≤ 4logm LPopt + c0(e)(˜xe + σkx˜e ) e∈B (t ,ρ) k k e 0 + c0(e)(max{0,ρ− distk(tk,e) − x˜ }) e ≤ 8logm · LPopt e∈δ(Bk(tk,ρ)) 0 1 Therefore the maximum cost over all scenarios is Observe that V (tk,ρ)+V (tk,ρ)=Vk(tk,ρ). We deﬁne k k O(log m)LPopt as well. the cost of the edges crossing the boundary of the ball as Ck(tk,ρ):= c0(e). e∈δ(Bk(tk,ρ)) 2.2 Robust Min-cut in Trees

Claim 2.1 The analysis technique of Garg et. al [8] can be G used to show that there exists a radius ρ<1/2 such that In the special case when the input graph is a tree, we the following holds in the step 2b of the above algorithm: give a polynomial time exact algorithm for the robust min- cut problem. The algorithm uses the following fact cru- t r Ck(tk,ρ) ≤ 2logm · Vk(tk,ρ). (2.1) cially: if a terminal k is not separated from the root by the ﬁrst stage solution, then we need to buy only one edge th We will show that the total cost paid in any scenario is at in the k scenario in the recourse stage. most 4logm · LPopt. We argue about the cost of the ﬁrst stage solution and the cost in the recourse stage respectively Theorem 2.5 There is a polynomial-time exact algorithm in the next two lemmas. for the robust min-cut problem on a tree.

Proceedings of the 2005 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 0-7695-2468-0/05 $20.00 © 2005 IEEE Proof: The algorithm for robust min-cut on trees is as removing the boundary. We give a sketch of the algorithm follows. “Guess” c to be the maximum second-stage cost of here. k k an edge to be cut in recourse stage. Since for each terminal, We find disjoint balls Bk(si ,ρ) and Bk(ti ,ρ) around k k we need to remove only a single edge to separate it from si and ti respectively. The radii ρ, ρ ≤ 1/4 are cho- the root, there are m choices for this maximum cost (m is sen such that the cost of the edges crossing the bound- the number of scenarios). All terminals tk, that have first- ary of a ball is within O(log rm) factor of the volume in- c 1 k 1 k stage min-cut cost less than σ are cut in the recourse stage. side the ball. If Vk (si ,ρ) ≥ 2 Vk(si ,ρ) then we include k k k The rest of the terminals are separated from the root by a δ(Bk(si ,ρ)) in the edge set Ek and remove δ(Bk(si ,ρ)) 1 k 1 k minimum cost cut in the first stage. from the graph Gk. Else if Vk (ti ,ρ) ≥ 2 Vk(ti ,ρ) in- k k One of the guesses of c is the correct one, for which we clude δ(Bk(ti ,ρ)) in Ek and remove δ(Bk(ti ,ρ)) from will find a solution that pays at most c in the recourse stage. the graph Gk. Furthermore, the first stage min-cut cost for every terminal Otherwise, consider the number of unseparated terminal t k k k that is cut in the first stage by this solution is greater than pairs (from all scenarios) in Bk(si ,ρ) and Bk(ti ,ρ).As- c k tk B s ,ρ σk . Thus, any optimal solution separates from the root sume without loss of generality, k( i ) is the ball with in the first stage. Hence, the algorithm returns an optimal smaller number of unseparated terminal pairs. We include k k solution. the edges δ(Bk(si ,ρ)) in E0 and remove δ(Bk(si ,ρ)) from G0. We run the algorithm recursively inside each of 3 Robust Multi-cut the components formed. This algorithm is similar to the divide-and-conquer al- The robust multi-cut problem is a generalization of the gorithm for Feedback Arc Set problem due to Leighton and robust min-cut problem. The problem is defined on a graph Rao [13]. It divides the graph G0 in various components and G =(V,E). Here the kth scenario consists of pairs of recurses inside each component. In order to bound the ap- k k k k terminals {(s1 ,t1 ), (s2 ,t2 ),...}. We want to find a set of proximation factor of the algorithm, we need to prove that edges E0 to buy in the first stage and Ek tobuyinthere- the depth of the recursion tree is small and the algorithm course stage if scenario k is materialized such that E0 ∪ Ek pays only a small cost at each level of the recursion. k k k k separates each of the pairs {(s1 ,t1 ), (s2 ,t2 ),...}. Edge e costs c0(e) in the first stage and σkc0(k) in the scenario k Lemma 3.1 Depth of the recursion of the above algorithm of the recourse stage. The objective is to minimize the max- is bounded by log (rm). imum cost over all scenarios. 2 We first describe an O(log rm) algorithm for robust Proof: Each time our algorithm makes a recursive call, the multi-cut problem, where r is the maximum number of pairs number of terminal pairs inside the ball is at most half as in any scenario. The algorithm is similar to the one for ro- many as the total number of terminal pairs in all scenarios. bust min-cut. Since the total number of terminal pairs we started with is We formulate an integer linear program for the robust bounded by rm, the recursion depth is at most log2 rm. multi-cut problem as follows. Using an argument similar to that of Lemma 2.2 we can bound the cost of the algorithm paid for edges in G0 as fol- min z 0 k lows. z ≥ e(c0(e)xe + σkc0(e)xe ) ∀ k 0 k k k (x + x )(P ) ≥ 1 ∀ si -ti paths P, ∀ k, i k Lemma 3.2 In each level of recursion, each unit of volume xe ∈{0, 1}∀e, k in the graph G0 gets a charge of O(log rm). k Relaxing the integrality constraints to xe ≥ 0 gives us 0 k 2 the LP relaxation. Let x˜e and x˜e denote the optimal frac- Theorem 3.3 There is a polynomial-time O(log rm)- tional solution. The rounding procedure is similar to the approximation algorithm for the robust multi-cut problem. rounding procedure for robust min-cut. As before, we maintain m graphs G1,G2,...,Gm, one for each scenario. We Proof: Each unit of volume in the graph Gk is charged also maintain G0 for the first stage solution. Each of these at most once and receives a charge of O(log rm).On graphs are initialized as copies of G =(V,E). However, the other hand, each unit of volume in the graph G0 gets we need to modify the ball growing procedure. In robust a charge of O(log rm) for O(log rm) levels of recursion. min-cut problem, when a boundary of a ball Bk(tk,ρ) is re- Therefore the total cost paid by the algorithm for edges 2 moved from the graph G0, there are no terminal pairs left in G0 is O(log rm · OPT), where OPT is the optimum inside the ball. This property no longer holds for the robust value of the LP relaxation. Hence, the total cost paid in 2 multi-cut problem. Therefore, we recursively apply the al- any scenario is O(log rm · OPT)+O(log rm · OPT)= 2 gorithm inside each component of the graph formed after O(log rm · OPT).

Proceedings of the 2005 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 0-7695-2468-0/05 $20.00 © 2005 IEEE 3.1 Improved approximation multi-cut. In the case of robust min-cut or multi-cut, the volume is bounded by cost of the LP solution. Thus, we ρ ≤ 1 We now show how to improve the approximation fac- could claim that there exists a radius 2 such that the tor to O(log rm log log rm) using the ideas from [6, 7, 22]. cost of the cut Ck(tk,ρ) is at most O(log m) · Vk(tk,ρ) in We modify our divide-and-conquer algorithm as follows. the robust min-cut problem. But this is not true in the case k k For a terminal si , we find a ball Bk(si ,ρ) such that of stochastic min-cut problem. Here, we do some prepro- C sk,ρ ≤ V sk,ρ · V /V sk,ρ V k( i ) k( i ) 4log( k k( i )) log log k, cessing before applying the region growing argument. 0 0 k S {i | σ p ≤ 1 } where Vk = e ce(xe + xe ) is the total volume. The anal- For all scenarios in := i i m2 ,weintro- ysis technique from [6] shows that such a radius ρ exists. duce the constraint in the LP that cut for these scenarios To bound the total cost of the algorithm, we note that will be completely a recourse stage solution. We claim each unit of volume in the recourse stage graph Gk gets a that this transformation does not affect the optimum solu- charge of O(log rm log log rm) at most once. On the other tion by a large factor: in an optimum solution if we buy all hand each unit of volume in graph G0 gets charged multiple the first stage edges helping scenarios in S during the re- times. Given a graph G and solution to robust multi-cut course stage as well, the extra edges bought incur a cost of x0,x1,...,xm σ p · OPT ≤ |S| OPT ≤ OPT problem ( ), let the total first stage volume of at most i∈S i i m2 m . Hence, we 0 0 graph G, V0 = e ce ·xe. Also, let cost(V0) denote the cost can ignore these scenarios while constructing our first stage of the first stage solution constructed by the above recursive solution. solution with total first stage volume V0. We bound the cost Now, when we apply the region growing algorithm for paid using the following recurrence relation: the rest of the scenarios. The total volume in the graph is cost V ≤ cost V 0 sk,ρ cost V − V 0 sk,ρ 0 k ( 0) ( k ( i )) + ( 0 k ( i )) + at most V = k e c0(e)(xe + xe ). The cost of the LP k 0 k 4log(Vk/Vk(si ,ρ)) log log Vk · Vk (si ,ρ). solution is at least Solving this recurrence, we get that the cost paid for the c e x0 σ p xk ≥ c e x0 1 xk first stage edges is O(V0 · log rm log log rm). Hence k,e 0( )( e + k k e ) k,e 0( )( e + m2 e ) 1 0 k the total cost paid by the algorithm is bounded by ≥ m2 k,e c0(e)(xe + xe ) O(log rm log log rm) · LPopt which proves the following theorem. Hence, V ≤ m2 · c(LP ). Now, we can show using the 1 techniques of Garg et al. [8] that there exists a radius ρ ≤ 2 Theorem 3.4 There exists a O(log rm log log rm)- such that Ck(tk,ρ) ≤ 4logm·Vk(tk,ρ). Hence, by running approximation to the robust multi-cut problem. the same algorithm described above for the robust min-cut losing an extra factor of 2, we obtain the following theorem. 3.2 Stochastic Min-Cut and Multi-Cut Theorem 3.5 There exists a polynomial time algorithm The stochastic min-cut problem is defined as follows: which returns an O(log m) approximate solution to the We are given a graph G =(V,E) with a cost function c0 stochastic min-cut problem. on the edges and a root node r. We are also given a col- lection S1,...,Sm of m scenarios with pk being the prob- A similar transformation for the stochastic multi-cut ability of occurrence of scenario Sk. For each scenario Sk, problem will yield the following theorem. there exists a node tk and we demand that r and tk must S be separated if scenario k appears in the recourse stage. Theorem 3.6 There exists a polynomial time algorithm e c e σ c e Edge costs 0( ) in the first stage and k 0( ) if scenario which returns an O(log rm log log rm) approximate solu- S k appears in the recourse stage. The objective function to tion to the stochastic multi-cut problem. minimize the sum of the first-stage cost and the expected recourse stage cost. Stochastic multi-cut is similarly defined to be the stochastic counterpart of robust multi-cut problem. 4 Special types of first-stage solutions and We show that a simple modification to the approximation Steiner trees algorithms for robust min-cut and multi-cut yields approximation algorithms for the stochastic version of the problems In this section, we prove that for any robust two-stage with same performance guarantees. problem there is an approximate first stage solution with The region growing argument is not directly applicable a special structure: it is a minimal feasible solution for a to the stochastic min-cut problem for the following reason: subset of scenarios and can be without much cost overhead V the “volume” of a ball defined in the proof of robust min- extended to a complete solution in the second stage. We use LP cut is different from the cost of the solution in the ball this structural result to obtain a constant factor approxima- unlike the algorithm of Garg et al. [8] for the deterministic tion for the robust Steiner tree problem.

Proceedings of the 2005 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 0-7695-2468-0/05 $20.00 © 2005 IEEE 4.1 A Structural Lemma for the First Stage Solu- The above structural result about the first-stage solution tion of a covering problem in the robust two-stage model also Lemma 4.1 Given any problem Π in the robust two-stage holds for the problem in the stochastic two-stage model. Starting with an integral optimum solution to the stochastic model, there exists a first stage solution X˜0 and a subset version of the problem (say X∗,X∗,...,X∗ ), the special S⊆{S1,...,Sm} of scenarios, such that X˜0 is a minimal 0 1 m feasible solution for scenarios in S. Furthermore, it can solution can be constructed as in the procedure described X ,X ,...,X be extended to a solution for the remaining scenarios in the above. Let the constructed solution be 0 1 m. c X ≤ second stage and the cost of the final solution is at most From the proof of Lemma 4.1, we have that i( i) · c X∗ ,i , ,...,m 2 · OPT. 2 i( i ) =01 . Thus, the stochastic objective for the new solution is, Proof: Consider an optimal integral solution to the robust X∗ X∗ problem : let 0 be the first stage solution and i be the m m ∗ recourse stage solution in scenario i. Also, let X0i be the c0(X0)+ pici(Xi)=c0(X0)+ piσic0(Xi) i part of first stage solution used in scenario i.e. it is a min- i=1 i=1 ∗ ∗ ∗ imal subset of X0 such that X0i ∪ Xi is a feasible solution for scenario i. We construct an alternate first stage solution m X˜ ∗ ∗ 0, such that it is a union of feasible solutions for a sub- ≤ 2(c0(X0 )+ piσic0(Xi )) set of scenarios. X˜0 will contain elements from the optimal i=1 ∗ first stage solution X0 , and also from the optimal recourse X∗,...,X∗ A stage solutions 1 m.Let denote the elements of Thus, the above lemma gives an alternate proof for a sim- X∗ X˜ X˜ 0 in 0. We construct 0 as follows. ilar lemma in [10] that proves that there is a connected first- stage solution for the stochastic Steiner tree problem which A ← φ B ← φ 1. Initialize and . costs at most thrice the optimal. 2. For each scenario i =1, 2,...,m, repeat the following 4.2 Robust Steiner Tree ∗ (a) X0i = X0i \ A. ∗ (b) If c0(X0i) ≥ c0(Xi ), then A ← A ∪ X0i We use the structural lemma proved above to give a con- ∗ and B ← B ∪ Xi . stant factor approximation for the robust Steiner tree problem. The problem is defined on a graph G =(V,E) with ˜ 3. X0 ← A ∪ B. a root vertex r and a cost function c on the edges. In the second stage one of the m scenarios materializes. The kth Our new first stage solution X˜0 = A ∪ B. Note that ∗ ∗ scenario consists of a set Sk ⊆ V of terminals and an infla- A ⊆ X0 . Therefore, c0(A) ≤ c0(X0 ). Also, all elements tion factor σk. An edge e costs c0(e) in the first stage and in B are charged to disjoint parts of A. Thus, by construc- th ∗ ck(e)=σkc0(e) in the k scenario of the second stage. A tion c0(B) ≤ c0(A) which implies c0(X˜0) ≤ 2 · c0(X0 ). solution to the problem is a set of edges E0 to be bought Clearly, X˜0 is a feasible solution for a subset of scenarios ∗ ∗ ∗ in the first stage and a set Ek in the recourse stage for each and it is minimal due to optimality of X0 ,X1 ,...,Xm and ∗ scenario k. The solution is feasible if E0 ∪ Ek contains a the minimality of X0i for each i. Furthermore, we claim th Steiner tree connecting Sk ∪{r}. The cost paid in the k that X˜0 can be extended to a feasible solution for all scenar- scenario is c0(E0)+σk · c0(Ek). The objective is to mini- ios in the second stage such that the cost of final solution is mize the maximum cost over all scenarios. at most 2 · OPT. Consider some scenario which is not covered in the first The structural lemma (Lemma 4.1) shows that there is stage by X˜0,sayi. This implies that when scenario i a first stage solution which is feasible for some subset of ∗ was considered in the above sequence, c0(X0i)

Proceedings of the 2005 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 0-7695-2468-0/05 $20.00 © 2005 IEEE Theorem 4.2 The Robust Steiner Tree Problem can be approximated within a factor of 30 in polynomial time. min z (4.2) Proof: Lemma 4.1 shows that the optimum fractional solution of the LP relaxation can be rounded to an integral so- 0 k ∀ k, z ≥ c0(e) · (xe + σk · xe ) (4.3) lution such that cost of each scenario is increased by at most e∈E z ≤ · z ≤ · OPT a factor of 15. Thus, IP 15 ˜ 30 . Hence, 0 k we obtain a 30-approximation for the Robust Steiner Tree ∀ t ∈ Sk, ∀ k, (re (t)+r0 (t)) ≥ 1 (4.4) problem. e∈δ+(t) 5 Other Robust Optimization Problems ∀ v/∈{t, r}, ∀ t ∈ Sk, ∀ k, In this section, we consider some other combinatorial 0 k 0 k re (t)+re (t)= re (t)+re (t) (4.5) problems in the two-stage robust model and give approxi- e∈δ (v) e∈δ (v) + − mation algorithms for them. 0 0 re (t) ≤ re (t) (4.6) e∈δ (v) e∈δ (v) 5.1 Covering Problems of Shmoys and − + Swamy [23]

∀ e, ∀ t ∈ Sk, ∀ k, Two-stage stochastic set covering problems were studied k k in a general setting by Shmoys and Swamy in [23], where re (t) ≤ xe (4.7) they showed how a ρ-approximation algorithm for the sin- rk t ,xk ∈{, } e ( ) e 0 1 (4.8) gle stage problem gives a 2ρ-approximation for the corresponding two stage stochastic version. The key idea is to This formulation is similar to one used by Gupta et al. observe that every element will be at least half-covered by in [10], where they give a constant factor approximation for the first- or second-stage sets that contain it. By scaling up the stochastic Steiner tree problem. The x0 variables are in- both first- and second-stage by a factor of two, and using the dicators for the edges in the first stage, and, x1,x2,...,xk rounding algorithm on both scaled solutions, one obtains a are the indicators for recourse stage edges. For a termi- solution with the promised guarantee. A major contribution nal t in scenario k, the variable rk(t) indicates whether e of [23] is a polynomial-time approximation scheme to solve edge e is used in the recourse portion of t’s path to the the two-stage stochastic programs even though the underly- root, and r0(t) indicates whether it is used in the first-stage e ing problem may be #P -complete. portion of the path. These flow variables are directed; for A simple application of the above method to polynomial- e =(u, v), the variable rk (t) denotes the flow of com- uv sized robust problems gives a simple 2ρ- approximation al- modity t along a recourse edge in the direction u to v. gorithm for covering problems allowing a ρ-approximate Given these directed flow variables, the cut-sets are de- single stage rounding method. fined as δ+(S)={e =(u, v):u ∈ S, v∈ / S} and Consider the demand-robust version of minimum vertex δ−(S)=δ+(V \ S). Note, however, that the edge installa- cover: nodes have different costs in the first stage and in tion variables xk refer to undirected edges. e each of the scenarios in the second stage, while each sce- Consider the LP relaxation of the above IP formulation nario consists of a subgraph of the complete graph on the obtained by dropping the integrality constraints. Let zIP be nodes. The goal is to choose some vertices in the first the cost of the optimum IP solution, z˜ be the optimum LP stage and for every scenario, augment the chosen set at the solution and OPT be the optimum solution of the original second-stage costs to form a vertex cover of the edges in instance. From Lemma 4.1, we know that zIP ≤ 2OPT. this scenario. A simple corollary of the above observation The fractional LP solution can be rounded using the same along with the classical 2-approximation rounding result for rounding scheme as that of Gupta et al. [10]. Thus, the fol- regular vertex cover gives the following simple result. lowing lemma can be derived from [10]. Theorem 5.1 The demand-robust vertex cover problem can z,x0, x1,...,xk Lemma 4.1 ([10]) Let ˜ ˜ ˜ ˜ be a fractional so- be approximated within a factor of 4. lution to the linear relaxation of the IP in (4.2)-(4.8). It can be rounded to obtain an integral solution T 0,T1,...,Tk, 0 i 5.2 Robust Facility Location such that T ∪ T connects Si ∪{r}, ∀ i. Furthermore, c T 0 ≤ · c e · x0 ∀i, c T i ≤ · 0( ) 15 e∈E 0( ) e and i( ) 15 F i In this problem we are given a set of facilities and a ci(e) · x e∈E e. set of clients S1,S2,...,Sm for each scenario. A metric

Proceedings of the 2005 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 0-7695-2468-0/05 $20.00 © 2005 IEEE cij specifies the distances between every client and every [6] Guy Even, Joseph Naor, Satish Rao, and Baruch 0 facility. Facility i has a first-stage opening cost of fi , and Schieber. Divide-and-conquer approximation algo- k a recourse cost of fi in scenario k. Note that in this case rithms via spreading metrics. In IEEE Symposium on we can handle general second stage costs unlike the model Foundations of Computer Science, pages 62–71, 1995. stated earlier where the second stage costs change by certain inflation factors σ1,σ2,...,σm. [7] Guy Even, Joseph Naor, Baruch Schieber, and Madhu Our approximation algorithm proceeds along the lines of Sudan. Approximating minimum feedback sets the LP-rounding algorithm due to Ravi and Sinha [19]. The and multicuts in directed graphs. Algorithmica, algorithm in [19] rounds a fractional solution such that the 20(2):151–174, 1998. cost of each scenario in the integral solution is bounded by 5 times its cost in the fractional solution. Thus, the same [8] Naveen Garg, Vijay V. Vazirani, and Mihalis Yan- techniques give a 5-approximation for robust facility loca- nakakis. Approximate max-flow min-(multi)cut the- tion.1 orems and their applications. SIAM J. Comput., 25(2):235–251, 1996. Theorem 5.2 The demand-robust facility location problem can be approximated within a factor of 5. [9] Anupam Gupta, Martin Pal,´ R. Ravi, and Amitabh Sinha. Boosted sampling: approximation algorithms for stochastic optimization. In Proceedings of the 6 Conclusion and Open Problems 36th annual ACM symposium on Theory of computing (STOC), pages 417–426, 2004. In this paper, we introduce a new model called demand- robustness and give approximation algorithms for some [10] Anupam Gupta, R. Ravi, and Amitabh Sinha. An edge combinatorial problems in this model. There seems to be in time saves nine: LP rounding approximation algo- an interesting parallel between stochastic and robust set- rithms for stochastic network design. In Proceedings tings. For example, the rounding techniques for the stochas- of the 45th Annual IEEE Symposium on Foundations tic Steiner tree problem can be adapted to the robust version of Computer Science (FOCS), 2004. of the same problem. Similarly, the rounding technique used for robust min-cut and multi-cut can be adapted to [11] Nicole Immorlica, David Karger, Maria Minkoff, and stochastic min-cut and multi-cut with a slight modification. Vahab Mirrokni. On the costs and benefits of pro- It would be interesting to prove a general result showing crastination: Approximation algorithms for stochastic that a ρ-approximation for a stochastic optimization prob- combinatorial optimization problems. lem leads to a O(ρ)-approximation for the robust version of the problem and vice-versa. [12] P. Kouvelis and G. Yu. Robust Discrete Optimisation and Its Applications. Kluwer Academic Publishers, References Netherlands, 1997. [13] F. T. Leighton and S. Rao. An approximate max-flow [1] Dimitris Bertsimas and Melvyn Sim. The price of ro- min-cut theorem for uniform multicommodity flow bustness. Operation Research, 52(2):35–53. problems with applications to approximation algorithms. In Proceedings of the 29th Annual IEEE Sym- [2] Dimitris Bertsimas and Melvyn Sim. Robust discrete posium on Foundations of Computer Science (FOCS), optimization and network flows. Mathematical Pro- pages 422–431, 1988. gramming Series B, 98:49–71.

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