Transactions on Algorithms. Two-stage Robust Network Design with Exponential Scenarios

Transcription

1 Transactions on Algorithms Two-stage Robust Networ Design with Exponential Scenarios Journal: Transactions on Algorithms Manuscript ID: draft Manuscript Type: Original Article Date Submitted by the Author: Complete List of Authors: Khandear, Rohit; IBM, IBM T.J. Watson research center Kortsarz, Guy; Rutgers University-Camden, Computer Science Mirroni, Vahab; Google, Google research Salavatipour, Mohammad; University of Alberta, Department of Computing Science Computing Classification Systems: Approximation Algorithms, Hardness of Approximation, Robust Networ Design, Robust Steiner tree, Robust Steiner forest, Robust Facility location

2 Page of Transactions on Algorithms Two-stage Robust Networ Design with Exponential Scenarios Rohit Khandear Guy Kortsarz Vahab Mirroni Mohammad R. Salavatipour Abstract We study two-stage robust variants of combinatorial optimization problems lie Steiner tree, Steiner forest, and uncapacitated facility location. The robust optimization problems, previously studied by Dhamdhere et al. [], Golovin et al. [], and Feige et al. [], are two-stage planning problems in which the requirements are revealed after some decisions are taen 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 many scenarios given implicitly. A scenario consists of any subset of terminals (for Steiner tree), or any subset of terminal-pairs (for Steiner forest), or any subset of clients (for facility location). Feige et al. [] give an LP-based framewor for approximation algorithms for for a class of covering problems two stage robust problems, but their framewor does not wor for networ design problems lie Steiner tree, and gives only a logarithmic approximation for the robust facility location problem. We present the first constant-factor approximation algorithms for the robust Steiner tree (with exponential number of scenarios) and robust uncapacitated facility location problems. Our algorithms are combinatorial and are based on guessing the optimum cost and clustering to aggregate near-by vertices. For the robust Steiner forest problem on trees and with uniform inflation, we present a constant approximation and show that the problem on general graphs and with two inflation factors is hard to approximate within O(log / ǫ n) factor, for any constant ǫ > 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 [] and []. Introduction In a classical optimization problem, we are usually given a system with some nown parameters and constraints and the goal is to find a feasible solution of minimum cost (or maximum profit) with respect to the constraints. Often these parameters and constraints, which heavily influence the optimum solution, are assumed to be precisely nown. However, in reality, often it is very costly (or maybe impossible) to have an accurate picture A preliminary version of this paper appeared in the Proceedings of th Annual European Symposium on Algorithms (ESA) 00. IBM T.J.Watson research center. rhandear@gmail.com. Department of Computer Science, Rutgers University-Camden. Currently visiting IBM Research at Yortown Heights. Partially supported by NSF Award Grant number 0. guy@crab.rutgers.edu. Google Research, New Yor, USA. mirroni@gmail.com Department of Computing Science, University of Alberta, Edmonton, Alberta TG E, Canada. mreza@cs.ualberta.ca. Supported by NSERC and an Alberta Ingenuity New Faculty award.

3 Transactions on Algorithms Page of about the values of the parameters or even the constraints of the optimization problem at hand at the time of planning. Two of the common approaches studied in the literature to address this uncertainty about future are referred to as robust optimization and stochastic optimization. Robust optimization. This has been studied in both decision theory [] and Mathematical Programming [], deal with the uncertainty in data. In a typical data-robust model, we have a finite set of scenarios that can materialize and each scenario contains one possible set of data values. The goal is to find a solution that is good with respect to all or most scenarios. One example in this category is absolute robustness or min-max, where the goal is to find a solution such that the maximum cost over all possible scenarios is minimized. Another example is min-max regret, in which the goal is to minimize the maximum regret over all possible scenarios, where the regret of a scenario is defined as the difference between the cost of the solution in that scenario with respect to optimal solution for that scenario. Stochastic optimization. In this model, we are provided with a probability distribution on the possible scenarios and the goal is then to find a solution that minimizes the expected cost over this distribution. This approach is useful if we have a good idea about the probability distribution (which may be a strong requirement), and we have a repeated decision maing framewor. One particular version of stochastic optimization, that has attracted much attention in the last decade, is two-stage (or multi-stage) stochastic optimization, where the solution is built in two stages: in the first stage we have to decide to build a partial solution based on the probability distribution of possible scenarios. In stage two, once the actual scenario is revealed, we have to complete our partial solution to a feasible solution for the given scenario. There has been considerable amount of research focused on two-stage (or multi-stage) stochastic version of classical optimization problems such as set-cover, Steiner tree, vertex cover, facility location, cut problems, and other networ design problems [0,,, ] and efficient approximation algorithms have been developed for many of these problem. In some cases, the set of possible scenarios and the corresponding probabilities are given explicitly [0, ], and some papers study a more general model in which the sets of possible scenarios are given implicitly (rather than explicitly) as the product of a set of independent trials, or by an oracle [,,, ]. Demand-robust optimization. More recently, a new notion of robustness has been introduced by Dhamdhere et al. [] which can be viewed as the worst-case analogue of the (two-stage) stochastic optimization problem. This model, called demand-robust optimization (and we simply call robust optimization in the rest of the paper), deals with both uncertainty in data as well as constraints of the problem. In a two-stage robust optimization problem, similar to a two-stage stochastic optimization, we are given a set of possible scenarios (which can be explicit or implicit) and the goal is to compute a solution in two stages while minimizing the maximum cost over all possible scenarios. The major difference of this model w.r.t. data-robust model is that each possible scenario might have a different set of constraints to be satisfied. For example, in the robust Steiner tree problem, we are given a graph G = (V,E) with a cost function c : E R + on the edges. In the second stage one of m possible scenarios materializes; scenario i consists of a set S i V of terminals that need to be connected to each other. We also have an inflation factor λ i for edge costs. Each edge e costs c e in the first stage and λ i c e in the second stage if scenario i materializes. Our goal is to select a subset of edges E E in the first stage, and a set E (i) E in the second stage if scenario i is revealed, so that E E (i) is a feasible solution for the Steiner tree problem with terminal set S i ; the overall cost paid in scenario i is c(e ) + λ i c(e (i)). The objective function is to minimize this cost over all possible scenarios, i.e., to minimize c(e ) + max i λ i c(e (i)). Since in this model of robustness, each scenario has a different requirement, it allows one to handle uncertainty in the set of constraints. It also provides a worst-case guarantee unlie only the expected-cost guarantee

4 Page of Transactions on Algorithms as in the two-stage stochastic optimization model. In a very recent wor [] authors consider a more general model of (two-stage) robust optimization in which scenarios are given implicitly, instead of explicitly as in the original model introduced in []. The difference is that in the more general version, one can have an exponentially large set of possible scenarios. For example, for the robust Steiner tree problem (with exponential scenarios), instead of having a total of m possible scenarios S,...,S m (S i V ) given explicitly, we can have each set S V of size as one possible scenario and the inflation factor λ i be uniformly equal to λ. In other words, in stage, an arbitrary subset of size of vertices is revealed as the terminals that need to be connected. Clearly the number of scenarios is exponential in the input ; and for this reason, (demand) robust optimization problems typically become more difficult in this more general setting.. Previous wor We are aware of only three other papers [,, ] which have studied (demand) robust optimization. In [, ], authors consider the robust problems with polynomially-many explicitly given scenarios. They present constant factor approximation algorithms for robust versions of Steiner tree, vertex cover, and Facility location problems. They also give polylogarithmic approximation for robust min-cut and multi-cut. In the problem of robust min-cut, we are given a edge-weighted graph G = (V,E), a source s V, and inflation factor λ i (e); scenario i consist of a terminal t i V. The goal is to find a subset E E for stage one and E (i) E for stage two (if scenario i arrives) so that E E (i) is a s,t i -cut. In [], the authors present an O(log m)- approximation where m is the number of scenarios. This was improved to a ( + )-approximation in []. In the robust multi-cut, each scenario consists of a set of pairs of nodes that form a multi-cut problem. For this problem [] present an O(log(rm) log log(rm)) where m is the number of scenarios and r is the maximum number of pairs in any scenario. Feige et al. [] consider covering problems (such as set cover) where the set of possible scenarios is given implicitly, and therefore can be exponentially large. For example, in the robust set cover problem, we are given a universe of elements U = {e,...,e m } and a collection of sets S = {S,...,S m }, where S i U and has cost c(s i ), an inflation factor λ, and an integer. Each scenario can be a subset U U of elements with size that need to be covered. We have to purchase a collection of sets S S in stage one. Once a set U of elements is given in stage two, we have to purchase some (possibly none) other sets S (U ) S where the cost of each set now is inflated by λ, so that S S (U ) is a set cover for the given set U. The goal is to minimize the maximum total cost over all possible scenarios. Using an LP rounding method, they give a general framewor for designing approximation algorithms for a class of robust covering problems using competitive algorithms for online variants of the problems. This framewor gives an O(log m log n)-approximation algorithm for twostage robust set cover problem, a constant-factor approximation for the robust vertex cover problem. They also prove a hardness of factor Ω( log m log log m ) for the two-stage robust set cover under some plausible complexity assumptions. This framewor does not apply to robust networ design problems lie the robust Steiner tree problem, and gives logarithmic approximation for robust uncapacitated metric facility location problem.. Our results The model we study in this paper is the (demand) robust optimization model with (possibly) implicit sets of scenarios (which can be exponentially large). We study robust Steiner tree (in which a scenario consists of any terminals out of given terminals), robust Steiner forest (in which a scenario consists of any terminal-pairs out

5 Transactions on Algorithms Page of of given terminal-pairs), uncapacitated facility location (in which a scenario consists of any clients possibly location at the same location out of given clients), and min-cut (in which a scenario consists of any single terminal from a given set of terminals) problems under this model and present some approximation algorithms and hardness results. Specifically, we provide the first constant factor approximation algorithms for the (exponential scenarios) robust Steiner tree and robust facility location problems. Our algorithms are combinatorial in nature and are based on nice structural properties of the stage one solution of a near-optimum algorithm. Theorem. There exists a polynomial-time.-approximation algorithm for two-stage robust Steiner tree problem with a uniform inflation factor. Here a scenario consists of any terminals out of given terminals. Theorem. There exists a polynomial-time 0-approximation algorithm for robust uncapacitated facility location problem in which the inflation factor may depend on the facility. Here a scenario consists of any clients (perhaps co-located) out of given clients. We then present a constant approximation for robust Steiner forest problem when the underlying graph is a tree. Our algorithm is based on first solving a standard LP relaxation using a dynamic-programming based separation oracle and then rounding it to a near-optimum integral solution. Theorem. There exists a polynomial-time -approximation algorithm for two-stage robust Steiner forest problem on trees with a uniform inflation factor. Here a scenario consists of any terminal-pairs out of given terminal-pairs. We next show that the robust Steiner forest problem on general graphs on n vertices and with two inflation factors is impossible to approximate within a factor of O(log ǫ n), for any constant ǫ > 0, unless NP has randomized quasi-polynomial time algorithms. We emphasize that our -approximation algorithm of Theorem. holds for exponentially many scenarios (i.e., source-sin pairs need to be connected in stage ), and our hardness result holds for even polynomially many scenarios, in which we have to connect only one source-sin pair in the second stage. However in the hardness result, we wor with general graphs and allow two possible values for the inflation factors on the edge-costs. Theorem. For any constant ǫ > 0, there is no O(log ǫ n)-approximation for robust Steiner forest in which only one pair arrives in stage two, we have only two (distinct) edge costs and two (distinct) inflation factors, unless N P has randomized quasi-polynomial time algorithms. Finally, we resolve an open question posed by K. Dhamdhere, V. Goyal, R. Ravi, and M. Singh [] and D. Golovin, V. Goyal, and R. Ravi [] about the hardness of the two-stage robust min-cut problem by proving Theorem.. The hardness open question in [, ] was even for a much more difficult question where you have many sins and a different inflation factor for every sin. We show that a much simpler problem is already APX-hard. Theorem. The two-stage robust min-cut problem is APX-hard even with a uniform inflation factor and in which a scenario consists of a single source and only thins. Organization. The remainder of the paper is organized as follows. In the next section, we present our.- approximation algorithm for robust Steiner tree. In section we present our hardness of approximation for robust Steiner forest.

6 Page of Transactions on Algorithms A constant approximation for robust Steiner tree problem In this section we prove Theorem.. Recall that the input to the Steiner tree problem is an undirected graph G = (V,E), a cost function c : E R +, and a subset T V called terminals. The objective is to find a connected subgraph H that includes all the terminals T and has minimum cost c(h) := e H c e. In the robust version of the Steiner tree problem, the input also contains an integer and a real number λ. There are two stages. In the first stage the algorithm has to identify a subset E E of edges to buy. In the second stage, the cost of each edge in E \ E increases by a factor of λ and a subset T T of at most terminals is revealed. We refer to T as a scenario. The algorithm, in the second stage, has to augment the solution E by buying edges E (T ) so that the resulting graph E E (T ) includes a Steiner tree on terminals T. The choice of edges E (T ) is allowed to depend on the subset T. The overall cost of this solution is thus e E c e + λ e E (T ) c e. The objective is to minimize the maximum overall cost over all scenarios, i.e., to minimize c e + max λ c e. T e E T, T e E (T ) The edge-costs c e induce a shortest-path metric on the vertices V : for any two vertices u,v V, we use d G (u,v) to denote the length of the shortest path between u and v, under costs c e in graph G.. The algorithm Let E and E (T ) be the set of edges the optimum buys in the first stage and the second stage for scenario T respectively. Let OPT = OPT + λ OPT be the overall cost of the optimum, where OPT = e E c e is its cost in the first stage and OPT = max T T, T e E (T ) c e is the maximum cost in the second stage divided by λ. First stage. Our algorithm, in the first stage, guesses an upper bound on the value of OPT and then outputs some Steiner tree T. This means that we will search for a tight upper bound on OPT in powers of + ǫ. Let µ be a guess for OPT. later we show that if µ OPT we are certain to output a solution whose cost is at most c(t ) + r µ () with r a universal constant to be chosen later. We also show that c(t ) = O(OPT). This implies that the overall cost of the solution is small iff µ is closed to OPT. Indeed, say that we fail to output a solution for µ = ( + ǫ) i but output a solution for µ = ( + ǫ) i+. By the above discussion it is clear that µ < OPT and thus µ ( + ǫ)opt hence µ is a tight estimate of OPT and a constant ratio follows From Equation. To simplify the presentation, we assume that the guess on OPT is exact. Given the assumed fact that we now OPT exactly, our algorithm computes a subset of terminals C = {c,c,...,c p } T called centers and an assignment π : T C that satisfy: The centers are far apart: d G (c i,c j ) > r OPT for all i j, and Each terminal is close to its assigned center: d G (t,π(t)) r OPT for all t T, where r > is the above constant to be determined later. Such a clustering can be computed as follows. Pic any terminal and name it c. Assign all terminals within a distance of r OPT from c to c and remove these terminals. Pic any one of the remaining terminals and name it c, and so on.

7 Transactions on Algorithms Page of The algorithm then computes an approximate minimum-cost Steiner tree T in G on the centers C under the costs c e. Currently, the best nown polynomial-time algorithm for the Steiner tree problem is γ-approximate, where γ <. []. The algorithm buys the edges in the Steiner tree in the first stage. Second stage. In the second stage, a subset T of at most terminals is revealed. The algorithm, in the second stage, buys the shortest path from each terminal t T to its assigned center π(t). It is easy to see that the algorithm computes a feasible solution to the problem.. The analysis We first introduce the notion of a ball of certain radius around a vertex in a graph. Consider the graph G = (V,E) with edge-costs c e. We thin of each edge e as a continuous interval of length c e. For a vertex v and a radius R > 0, let B G (v,r) denote, intuitively speaing, the moat of radius R around v. More precisely, B(v, R) contains all the vertices u such that d G (u,v) R, all edges e = (u,w) such that d G (u,v) R and d G (w,v) R, and for the edges e = (u,w) such that d G (u,v) R and d G (w,v) > R, the sub-interval of edge e of length R d G (u,v) adjacent to vertex u. Note that since d G (c i,c j ) > r OPT for any two distinct centers in C, the balls B G (c i, r OPT ) and B G (c j, r OPT ) are disjoint. It is easy to see that the algorithm pays at most λ r OPT in the second stage. This holds since the distance of any terminal to its assigned center is at most r OPT. Since at most terminals need to be connected to their centers, the total cost of these connections is at most λ r OPT. We now bound the cost of the algorithm in stage one using the following lemma. Lemma. Assuming r >, there exists a Steiner tree on centers C in G that has cost at most OPT. r r OPT + Proof: Recall that E is the set of edges optimum buys in stage one and OPT = e E c e. Let H be a graph obtained from G by shrining the edges in E. We now perform another clustering of the centers C in the shortest-path metric on C induced by the graph H. For centers c i,c j C, let d H (c i,c j ) denote the shortestpath length under lengths c e in H. We identify a subset of centers L = {l,l,...,l t } called leaders and a mapping φ : C L such that The leaders are far apart: d H (l i,l j ) > OPT / for all i j, and Each center is close to its mapped leader: d H (c,φ(c)) OPT / for all centers c C. Such a clustering can be computed as follows. Pic any center and name it l. For all centers c C with d H (c,l ) OPT /, define φ(c) = l. Remove all such centers from C and repeat. Analogous to B G (v,r), we use B H (v,r) to denote the ball of radius R centered at v in the graph H with length c e for e H. Note that the balls of radii OPT around the leaders in L are disjoint in H. Claim. We have L <.

8 Page of Transactions on Algorithms Proof: Assume on the contrary that L and let T L be any subset of size. Consider the scenario T. Since even after shrining the edges in E that optimum bought in the first stage, the balls of radii OPT centered at the centers in T in the graph H are disjoint. Therefore the minimum Steiner tree on T in H has cost more than OPT. This is a contradiction since the optimum pays at most λ OPT in the second phase to connect all the centers in T after shrining the edges in E. Thus the claim holds. Since L <, we now consider scenario L. There exists a Steiner tree EL on L in H with cost at most OPT. Thus E E L has cost at most OPT + OPT and contains a Steiner tree on L in G. We now show how to extend this into a subgraph with low cost and which contains a Steiner tree on C in G. r OPT Now recall that the balls of radii around the centers C are disjoint in G. Note however that d H (c,φ(c)) OPT for all centers c C. Thus at least r OPT OPT = ( r ) OPT cost of E must lie inside the ball of radius r OPT around each center c C. We can thus extend the subgraph E E L by adding shortest paths from each c to φ(c) in H and charge this additional cost to the contribution of E in the respective balls around centers c C. The resulting subgraph clearly contains a Steiner tree on C in G. The overall cost of this subgraph is thus at most OPT + OPT + r OPT = r r OPT + OPT. Hence the proof. Since we use a γ-approximation algorithm to compute a Steiner tree in stage one, the overall cost of stage one is at most γ r r OPT +γ OPT. Combining this with the second stage cost, the overall cost of our solution is: γ r r OPT + (γ + λr) OPT. () A trivial strategy for solving the robust Steiner tree is to select nothing in stage and mae all the selections in stage. Given that every edge is inflated by λ and we use a γ-approximation for Steiner tree, this strategy will have an approximation factor of λ γ. Using the best nown approximation algorithm for Steiner tree [], which has approximation., we get a.λ-approximation. For values of λ. we use this trivial strategy which gives an approximation factor of.. For values of λ >. we use the above algorithm with parameter r defined below. Let r = r γ r to be the solution of: r = γ λ +r. Then at r = r = γλ γ+λ+ γ λ γ λ+γλ +γ +γλ+λ λ, the two factors in front of OPT and OPT in the ratio of our algorithm calculated in Equation () become equal. Therefore, for r = r and with γ =., the ratio of our algorithm becomes:.λ.+ 0.0λ +.λ+.0 λ. It can be verified that this expression is upper bounded by. (it has a limit of.). Thus, for values of λ >., by choosing r = r, the ratio of our algorithm presented will be at most. and for smaller values of λ we use the trivial strategy which has ratio at most. as well. This completes the proof of Theorem.. Hardness of Robust Steiner Forest In this section, we prove a lower bound on approximability of the robust Steiner forest even if only one pair of terminals are revealed in stage to be connected (so clearly a polynomial number of scenarios) and even if every edge has one of the two costs of L r or L a and we have two different inflation factors λ r and λ a. More precisely, we are given an undirected graph G = (V,E), each edge has either cost L a or L r, a set of source-sin pairs s i,t i, and two inflation factors λ a and λ r. The i th scenario consists of a single pair s i,t i. In stage, each edge with first stage cost L r will have cost λ r L r and each edge with first stage cost L a will have cost λ a L a.

9 Transactions on Algorithms Page of The objective is to find a set of edges E to be bought in stage one and for each i, a set E (i) to be bought in stage two (if scenario i is realized), such that E E (i) contains a path from s i to t i. Our goal is to minimize the total cost of E and E (i) over all possible scenarios i. Our main goal in this section is to prove Theorem.. We restate it here: Theorem. For any constant ǫ > 0, there is no O(log ǫ n)-approximation for robust Steiner forest in which only one pair arrives in stage, we have only two edge costs and two inflation factors, unless NP has randomized quasi-polynomial time algorithms. Our hardness result uses some ideas from the earlier wor on hardness of Buy-at-Bul networ design [] and the later wor [,, ]. The main idea of the hardness result is to start from a -prover -round proof system of Max-SAT() and build a randomized layered graph. This type of random construction has shown to be quite useful in proving hardness of approximation for uniform and non-uniform Buy-at-Bul problem [] and edge-disjoint paths or minimizing congestion and other networ design problems [,, ]. As can be seen below, our hardness result shows some similarities between the robust Steiner forest and multicommodity buy-at-bul networ design (see []). However, unlie for buy-at-bul, for which there is nown logarithmic hardness results for both uniform and non-uniform case [], we can only prove a hardness result if we have non-uniform inflation over the edges. The main reason is that in our problem the constraints of the problem are not nown completely in advance and the solution is constructed in two stages. Our proof uses a reduction from a two-prover one-round proof system for Max-SAT() with parallel repetition. A SAT() formula is a boolean formula φ with n variables and n/ clauses where each variable appears in exactly five variables and each clause contains exactly three literals. The goal in Max-SAT() problem is to find truth assignment to variables which maximizes the number of clauses that are satisfied simultaneously. It is well-nown by PCP theorem [, ] that there is a universal constant ǫ 0 > 0 such that it is NP-hard to distinguish between instance of Max-SAT() in which all clauses are satisfied (called a yesinstance) from those instances which no more than ǫ 0 fraction of clauses can be satisfied (called a noinstance). In a two-prover one-round (PR) proof system with l parallel repetition (a..a. Raz verifier), a verifier interacts with two provers. For a given Max-SAT() formula φ, using a source of random bits to which the verifier has access to, the verifier selects l random clauses and one random variable from each. Then sends the index of l randomly chosen clauses to one of the provers (referred to as the c-prover) and the index of l variables chosen from these clauses to the other prover (called v-prover). The c-prover returns the truth assignment to the variables of each clause in the query and the v-prover returns the truth assignment of each of the variables sent to it. The verifier accepts φ if all the answers are consistent and each clause in the query is satisfied; otherwise it rejects the clause. Raz parallel repetition theorem, together with PCP implies the following: Theorem. [] There is a universal constant δ > 0 s.t. if φ is a yes-instance then there is a PR proof system in which the verifier always accepts and if φ is a no-instance then the verifier accepts with probability at most δl. In a Raz verifier with l parallel repetitions we have Q c = (n/) l clause-queries each with A c = l possible answers and Q v = n l variable-queries each with l possible answers. The number of query pairs is R = (n) l ; so each c-query and v-query belongs to l and l query pairs, respectively. Each query pair defines l accepting interactions. Given an instance of Max-SAT(), φ and a Raz verifier with l parallel repetition for it, we start by building a basic instance of robust Steiner forest, called I 0 as follows.

10 Page 0 of Transactions on Algorithms s v l(a v) a v q v r(a v) l(b c) r(b c) Figure : Construction of the basic instance: only some of the edges are shown; answer edges are shown bolder. A canonical path for demand d v,c is: s v l(a v ) r(a v ) l(b c ) r(b c ) t c. For each v-query q v and each answer a v to q v we have a corresponding answer edge. Similarly, we have an answer edge for each answer b c to each c-query q c. We group together all the l answer edges corresponding to each c-query q c and call it the super-node corresponding to q c. We create super-nodes by grouping the l answer edges of each v-query in a similar way. We abuse the notation slightly by using q c and q v to refer to both the queries and the super-nodes corresponding them. Similarly, we use a v (or b c ) to refer to both an answer to a query and its corresponding answer edge in a super-node. We use l(a v ) and r(a v ) to refer to the left-node and right-node of answer edge a v, respectively. Similar notation is used for answer edges b c. For each v-query q v we create a source node s v and connect it to left node of the answer edges in the super-node of q v ; these edges are called regular edges; similarly for each c-query q c we create a sin node t c and connect it to the right node of the answer edges in the super-node of q c using regular edges. For each random string r and the corresponding c-v-query pair q v,q c, for each answer a v for q v and b c for q c that form an accepting interactions, we add a regular edge between r(a v ) and l(b c ) (see Fig ). Also, we define a demand pair d v,c with source s v and sin t c (corresponding to this c-v pair). So we have a total of R demand pairs. Each answer edge has c(e) = L (for a constant L ) and each regular edge has c(e) = ; we let the inflation factor be uniformly λ = (Q v + Q c )L on every edge. Note that for each random bit string corresponding to a query pair q v,q c, if a v and b c are two answer edges of the super-nodes of q v and q c, respectively, then the demand d v,c with the source s v (corresponding to q v ) and sin node t c (corresponding to q c ) has a canonical path with edges: s v l(a v ) r(a v ) l(b c ) r(b c ) t c. It is easy to see that every canonical path has exactly two answer edges and three regular edges and every non-canonical path has at least five regular edges. This completes our construction of the basic instance I 0 with R demand pairs. First assume that φ is a yes instance; so there is a truth assignment which satisfies all clauses. This corresponds to choosing one answer edge for each super-node (i.e. query) such that for each c-v-query pair the chosen answer edges correspond to an accepting interaction and therefore are connected by an edge. We can buy these answer edges (one from each super-node) in stage for a total price of (Q v +Q c )L. In stage, when the actual demand (source-sin pair) is revealed, say d v,c which is between source s v and sin t c corresponding to query pair q v and q c, where answer edges a v for q v and b c for q c are bought in stage, we buy three regular edges that connect s v to a v, a v to b c, and b c to t c, for a total price of λ = (Q v + Q c )L. So the total cost is at most λ. Now assume that φ is a no instance. By Theorem., if we purchase less than (Q v +Q c ) δl answer edges in stage, then there is at least one random string r which corresponds to a demand d v,c (query pair q v,q c ) such b c q c t c

11 Transactions on Algorithms Page of that the chosen answer edges from the super-nodes of q v and q c do not contain any pair of answers which form an accepting interactions. Thus, if in stage, demand d v,c is revealed, we cannot form a canonical path from s v to t c by buying only regular edges. So we have to either buy at least one answer edge and regular edges (to form a canonical path), or buy at least regular edges to form a non-canonical path. In either case, the cost in stage will be at least min{λl,λ} = λ. So the total cost in both stages is at least λ. Therefore, we get a gap of ratio at least / between yes-instances and no-instances of φ. The main instance is built using several copies of the basic instance. The overall idea is the following. To get a larger gap, we have to increase the cost of non-canonical paths that can be purchased (in stage ) in the case of a no-instance. For that, we permute the regular edges that run between the answer edges between different copies of the basic instance so that we get an almost random graph. This in turn will ensure that the length of a non-canonical path will be proportional to the girth of the graph, which in turn will be about the logarithm of the instance size. As a side effect of this construction, we have to also increase the length of canonical paths from constant to about square-root of the logarithm of instance size. Therefore, we get a gap (hardness) of about log n at the end. Next we describe the structure of the main construction.. Main construction (b x,y c Our main construction for robust Steiner forest consists of a graph G which contains Y levels, where each odd-level contains X copies of each super-node q v and each even-level contains X copies of each super-node q c. We use qv x,y to refer to the x th copy of super-node q v in level y. So for each answer edge a v for a variable query, we have X copies at every odd-level, for a total of XY copies. Similarly we have a total of XY copies of each answer edge for a clause query. We denote the x th copy of answer edge a v (b c ) at level y by a x,y v ). We will have a total of XR demand pairs. Now we define how the regular edges between the supernodes are added. Consider an arbitrary random string r and suppose it corresponds to query-pair q v,q c. We assume we have an arbitrary (but fixed) labeling of the l accepting interactions that are defined by the pairs of answers from q v and q c. For every level y ( y < Y ) and every i l, for the i th accepting interaction that exists between q v and q c (say corresponding to answer a v from q v and b c from q c ) we have a random permutation π i,y : [X] [X], where [X] denotes the set {,...,X}. Note that π i,y is independent of π i,y for y y. Suppose that y is odd, then the answer edge a x,y v y th level of super-node q v ) is connected to answer edge b π i,y(x),y+ c l(b π i,y(x),y+ c ). For even values of y, we add a regular edge between r(b x,y c (copy of a v in the x th copy of the using a regular edge between r(a x,y v ) and ) and l(a π i,y(x),y+ v ) (See Figure ). So random permutation π i,y, for odd values of y defines how the regular edges are added between answer edges for v-query super-nodes in level y and the answer edges for c-query super-nodes in level y +, whereas for even values of y, π i,y defines how the regular edges go from answer edges of c-query super-nodes in level y to answer edges of v-query super-nodes in level y +. Now we explain the regular edges connected to source and sin nodes and also how the demand pairs are defined. For each query q v we have X source nodes s v,...,s X v and for each query q c we have X sin nodes t c,...,t x c. The (regular) edges between source nodes and answer edges in level and between the sin nodes and answer edges in level Y are defined based on canonical paths, described below. Consider a random string r which corresponds to query pair q v,q c. Suppose that answer a v from q v and b c from q c are consistent (form an accepting interaction) and this is the i th accepting interaction (for some i l ) between this query-pair. In the basic construction, answer edges a v and b c together with three regular edges define a canonical path P v,c (for demand d v,c ) of length five: P v,c = s v,l(a v ),r(a v ),l(b c ),r(b c ),t c. Here we have X corresponding 0

12 Page of Transactions on Algorithms s v y = y = y = y = y = y = a, v b, c a, v Figure : Main construction: with X = and Y = ; only three canonical paths between s v and t c are shown, corresponding to the st, nd, and the th accepting interactions between q v and q c in the basic instance. canonical paths. Each of these X canonical paths starts from a unique copy of the X copies of answer edge a v at level and travels by a regular edge (defined by the random permutation π i, ) to a copy of b c in level, then to a copy of a v (based on random permutation π i, ) in level, and so on. For example, the canonical path that starts from l(a x, v ) uses the regular edge between answer edges a x, v and b π i,(x), c to go to l(b π i,(x), c ); assuming that π i, (x) = x, then exiting from the right end of answer edge b x, c it goes to answer edge a π i,(x ), v ; assuming that π i, (x ) = x, the path continues by using the regular to the left node of answer edge, and so on, until it reaches the right node of answer edge b π i,y (x Y ),Y. All these X canonical b π i,(x ), c paths are disjoint and each has length Y, with Y answer edges and Y regular edges in between them. We have a random permutation π i,0 : [X] [X] which defines which of the X copies s v,...,s X v of source node s v will have a regular edge to which of the X copies a, v,...,a X, v of answer edge a v at level. If a,...,a z are the answers of q v, then s x v will have a regular edge to the left-nodes of the following answer edges: a π,0(x),a π,0(x),...a π z,0(x) z. Then we choose a random copy t x c, and after following the A c canonical paths that start from each of a π,0(x),a π,0(x),... a π z,0(x) z, the right node of the last answer edges of these will have a regular edge to t x c. This source-sin pair s x v and t x c will be a demand pair d v,c,x. We have a total of XR demand pairs, X corresponding to each demand pair in the instance I 0. Note that the degree of each source node s x v will be A v and the degree of each sin node t x c will be A c. Each answer edge has stage cost of L a and each regular edge has stage price of L r. The inflation factor for answer edges is λ a and for regular edges is λ r. Note that corresponding to each demand pair d v,c in I 0, we have X demand pairs d v,c,x, x X. This completes the construction of our graph G. Our final instance I of robust Steiner forest consist of this graph G together with the XR demand source-sin pairs defined above. We prove our hardness result based on this instance I. b, c c a, v b, c t c

13 Transactions on Algorithms Page of The parameters We will choose our parameters as follows. Here n is the number of variables in the formula φ, l is the parallel repetition, δ is the constant in Raz theorem, g is a parameter (related to the girth of the construction), and h is a parameter used in the analysis of the construction: Q c = (n/) l, Q v = n l, A c = l, A v = l, R = (n) l, D c = R/Q c = l, D v = R/Q v = l, g = log σ n, Y = g = log σ/ n, X = ( 0 Y A c Q c ) g, l = σ log log n/δ, and h = δl/ 0.. Analysis of the yes-instances Lemma. If φ is a yes-instance of Max-SAT() then the solution to robust Steiner forest on instance I has cost at most XY (Q v + Q c )L a + λ r (Y + )L r. Proof: Assume that φ is a yes-instance of Max-SAT(). So for every query q v (similarly q c ) there is an answer a v (b c ), such that for every random string r which corresponds to a query pair q v,q c, the answers of q v and q c are consistent (i.e. form an accepting interaction). We refer to these Q c + Q v answers as chosen answers of queries. From this we build a robust Steiner forest solution for I as follows. In stage, we purchase all the XY copies of each chosen answer, for a total price of XY (Q c + Q v )L a. Suppose that in stage, the demand pair revealed is s x v,t x c. Consider the query-pair q v,q c and the corresponding chosen answers, call them a v and b c, respectively. So in the basic instance I 0, path s v,l(a v ),r(a v ),l(b c ),r(b c ),t c is a canonical path. Corresponding to this we have a canonical path in I between s x v and t x c from which all the Y answer edges (Y copies of a v and Y copies of b c ) are bought in stage. So it is enough to buy the remaining Y (regular) edges of this canonical path plus the two regular edges incident with source and sin nodes in stage which will cost λ r (Y + )L r. So there is a solution of cost at most XY (Q v + Q c )L a + λ r (Y + )L r.. Analysis of the no-instances From now on, we assume that φ is a no instance and we shall prove that in this case the cost of the solution to instance I is at least Ω(Y L r ), and by choosing the parameters carefully, this will give a gap of Ω(Y ) w.r.t. the yes instance. The main steps of the proof are as follows. Assume that E is the set of answer edges bought in stage of the optimal solution, and assume that E is not too large, e.g. E XY (Q v + Q c ) δl/. We show that the number of demands that can be routed in stage on short non-canonical paths is much smaller than XR. In other words, we show that w.h.p. for Ω(XR) demands, the length of non-canonical paths is at least Ω(Y ). Using this we show that if there is a demand pair s x v,tx c such that we cannot form a canonical path for it by using the edges in E and perhaps buying only regular edges in stage then the cost of the solution for that demand is at least Ω(Y L r ). Thus, for all demand pairs, we must be able to extend E in stage to form a canonical path. This however is not doable unless E is much larger (which incur a large cost in stage ) or we can extract a correct answer for formula φ. For a demand pair, a path of length at most g is called a short path, otherwise it is a long path. It can be shown that, due to the randomness in the construction of G, not many demand pairs can have short noncanonical paths. Specifically, we will show the following lemma: Lemma. With probability at least /, the number of demand pairs that have a short non-canonical path is at most XR/.

14 Page of Transactions on Algorithms Proof: To prove this lemma we first define graph G which is obtained from G by first removing all the source and sin nodes and then contracting each answer edge into a single node. It is easy to see that every path p in G between a demand pair, say s x v and tx c, corresponds to a path in G between a vertex in the first level (i.e. the first vertex of p after s x v ) and a vertex in the last level (i.e. the last vertex of p before tx c ). In particular, the notions of canonical and non-canonical paths carry over to graph G. For example for every canonical path of G, by removing the the source and sin and contracting the answer edges we find a canonical path in G, also each such path in G corresponds to a canonical path in G. Thus, to bound the number of demand pairs with short non-canonical paths in G it is sufficient to bound the number of demand pairs with a short non-canonical path in G. One property of G that will be used is, since it consists of only regular edges, it is a random graph. In particular, for any pair of nodes u,v G, the probability of having an edge e = uv in G even given the existence of other edges in G, is at most X. Consider an arbitrary demand pair s x v,tx c in G and the set of all canonical paths between them. Let P be the corresponding set of canonical paths in G and let V be the set of all the nodes that belong to a path in P. Note that since P = A c : V Y A c () We first show that with high probability the canonical paths in P are (internally) vertex disjoint. Furthermore, the probability of having a short path between two vertices u,v V without using the edges of paths in P is small. Roughly speaing, the reason is that the number of canonical paths for a fixed demand pair and also the length of each such path is much smaller compared to the size of the graph (and in particular to the total number of demand pairs). Given that the graph is (almost) a random graph, the probability of these small paths being close to each other is very small. We formalize this below. Consider two canonical paths P i,p j P. We say P i and P j meet at level < y < Y if they have a common vertex (corresponding to an answer edge in G that is now contracted into a vertex in G ) in level y. Since each of P i,p j has length at most Y and by the random selection of edges in each level, given the existence of up to Y other edges, the probability that they meet at level y is at most /(X Y ). Thus, using a union bound, the probability that P i and P j meet (at some internal level) is at most Y/(X Y ). Given that there are A c paths in P, the probability that two of them intersect (internally) is at most A cy/(x Y ) which for the given values for X,Y,A c is arbitrary small, say at most /. Therefore: Claim. With probability at least /, the canonical paths in P are (internally) vertex disjoint. Note that every short non-canonical path between s x v and tx c in G corresponds to a short non-canonical path in G that starts from the start point of one of the paths in P and ends at the end-point of a (possibly different) path in P. Let s assume that the canonical paths in P are (internally) vertex disjoint (which happens with high probability by the above claim). Therefore, each short non-canonical path between s x v and t x c in G must contain a sub-path P between two vertices of V such that P does not have any edges in common with any path in P. So, if G is the graph obtained from G by deleting the edges of paths in P, in order to bound the number of short non-canonical paths between s x v and t x c in G it is sufficient to bound the number of short paths between a pair of nodes of V in G. Claim. The probability of having a path between two nodes of V in G is at most /. Proof: If we denote the number of nodes in G (which is the same as the number of nodes in G ) by N then N XY (A v Q v + A c Q c ) XY A c Q c. Consider a fixed pair of nodes u,v V and a fixed integer g. We first bound the expected number of paths of length between u,v in G. There are N choices for the

15 Transactions on Algorithms Page of middle points (with order). For each pair the probability of having an edge between them is at most /(X ) (even if other edges are given). Since << X, the expected number of such paths (from u to v) is at most N (X ) ( ) XY Ac Q c X X ( Y A c Q c ). X Summing this over all values of g and considering all the pairs of nodes in V, the expected number of paths of length at most g between a pair of nodes in V is at most V g( Y A c Q c ) g X g A c ( Y A c Q c ) g X ( Y A c Q c ) g X <. Therefore, using Claims. and., the probability of having a short non-canonical path between s x v and t x c is at most / + /. Summing over all the XR demand pairs, the expected number of demand pairs that have a short non-canonical path in G is at most XR/. Using Marov s inequality, the probability that the number of demand pairs having a short non-canonical paths in G is larger than X/ is at most /, and this completes the proof of Lemma. Let us define the bad event B to be the event that more than XR/ demand pairs have short non-canonical paths. By lemma. we can assume that Pr[B ]. Using this, we can focus only on canonical paths and long non-canonical paths. Our next goal is to show that if φ is a no-instance, then the answer edges bought in stage cannot form many canonical paths (or else from that we can form a solution to φ). For each y Y, let E y be the set of answer edges of level y bought in stage. Without loss of generality, assume that y is odd (the same analysis carries over for even values of y). For every answer a v to a v-query q v, let f y (a v ) be the fraction of X copies of a v that are bought in level y, i.e. /X times the number of a x,y v s that are bought (for x X). We define f y+ (b c ) for an answer b c to a c-query q c similarly. We call a v-query q v (or c-query q c ) heavy at level y if a v q v f y (a v ) h for a parameter h. A demand pair d v,c,x is called heavy (at level y), if either q v is heavy at level y or q c is heavy at level y + (or both). We say a demand d v,c,x is routable at level y if on at least one of its canonical paths, the two answer edges at levels y and y + are bought. The following two lemmas are similar to Lemmas and in []. Lemma. If E y E y+ h(q v +Q c )X then the number of heavy (at levels y) demands is at most XR/. Proof: For every heavy v-query (similarly c-query), the number of answer edges bought at level y (or y + if it is a c-query) is at least hx. So the total number of heavy queries in each prover is at most h(qv+qc)x hx = (Q v + Q c )/. For each c-query in I 0, the number of demands going through it is D c and for each v-query it is D v. So the total number of heavy demands is at most XD c (Q v + Q c )/ XR/. Lemma. If E y E y+ h(q v + Q c )X then the probability that more than XR/ non-heavy demands are routed at level y is at most e XR/. Proof: Consider fixed sets E y and E y+ with E y E y+ h(q v + Q c )X and an arbitrary accepting interactions defined by random string r and answers a v and b c from queries q v and q c, respectively. This