Project Summary. Approximating Network Design Problems on Directed and Undirected Graphs

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1 Project Summary Approximating Network Design Problems on Directed and Undirected Graphs Network design on directed graphs are problems that require to select a minimum cost directed subgraph of a given edge-weighted directed graph, under some constraints. One of the main goals of this proposal is to understand some of the most fundamental network design problems on directed graphs whose exact approximability status remains unclear for a very long time. Directed graphs appear frequently, when modeling the problem by an undirected graph is not enough (see, for example, [71]). Applications for which the graphs arising are naturally directed include questions related to the web graph (c.f., [68]), social networks graphs (c.f., [72]), dynamic graphs in artificial intelligence (c.f. [25]), and more. Theoretical importance and intellectual merit: The core problem we wish to deal with is the Directed Steiner Tree (DST) problem. Given a directed graph with cost over the edges and a root r and a set of terminals the goal to is to pick a subgraph of minimum cost so that the root r could reach by a path any terminal. The theoretical importance of the DST problem is two-fold. On the one hand, the DST problem generalizes many of the problems on undirected graphs we shall present and deal with later such as buyat-bulk problems (MCBB) (see [58]) the Group Steiner problem (GSP) (see [37]) and shallow-light trees (SLT) (see [53]). On the other hand the DST problem is a bottleneck whose solution is required for various generalizations of DST. For example the directed pairs connection (DPC) problem that given a directed graph a collection {s, t i } k i=1 of pairs looks for a subgraph G of minimum cost so that every s i could reach t i by a path in G. Almost all problems presented are special cases or generalizations of DST and DPC that serve as a unifying theme for the project. Dealing with these probelem and the related problem presents a taught intelectual challange. Practical importance: While DST has practical importance, for example, in web applications and more, its special cases have importance in may other fields. An example of a special case is finding a low cost tree with bounded height. The applications of such trees are, among others, designing networks for fast broadcast (see [5]), fast routing (see [49]), VLSI circuit design (see [26]) and bitmaps compression (see [11]). The GSP problem. has applications in VLSI network design (see [37]). For more applications of special cases of DST, for example, recuperating from power outage. wireless networks and more (see [23, 33, 69, 50, 57, 55, 67]). In summary to the opinion of the PI the DST problem is one of the most central, not well understood, problems left in the field of approximation algorithms. The proposal uses this problem as a focal point. Broader impact Rutgers Camden starts a Masters studies in Computer Science in The project will include the creation of a new graduate course on approximation algorithms. Students taking the course will be encourage to undertake research work in this subject. The PI has vast experience in guiding students to masters degree from his time in service in the Open University of Israel. The PI is interested in promoting the participation of women and other under-represented groups in this project. Students will also be encouraged to seek opportunities for internships and participate in educational programs for high school students. In fact, the PI belongs (together with two other faculty) to a committee that tries to find ways to enhance the interaction of the computer science departments with high schools in the area. 1

2 Project Description 1 Research Plan We consider network design problems on directed graphs that revolve around the Directed Steiner Tree (DST) problem. Consider the following fundamental scenario. Clearly, not all the nodes in the web of equal importance. There are central nodes on the web that are major servers for many clients. Consider a central server that needs to broadcast an important message to a selected set U of its clients. Usually it is impractical for the server to be directly connected to all its clients. The delivery of the message is performed along paths, that iteratively use direct connection between two given users. Using such a direct link incurs a fixed usage cost, for buying bandwidth. Because of the different locations and attributes of users, directed edges may have different usage costs. Therefore the problem resulting is buying bandwidth on some of the directed links existing in the web such that a path from r to u is obtained for every u U. The goal is to accomplish this at minimum cost. This alludes to the Directed Steiner Tree (DST) problem. Input: A directed graph G(V, E, c) with costs c(e) R + on the edges, a root r and a set of terminals U. Output: Find directed tree (arborescence) rooted at r, containing all of U, of minimum cost. While the undirected variant of the Steiner tree problem admits an approximation algorithm of ratio 1 + ln 3/2 [65], not even a polylogarithmic ratio in n = V algorithm that runs in polynomial time in n is known for the DST problem. The problem is known to be hard to approximate within log 2 ɛ n for any universal constant ɛ [45], unless N P admits a quasi-polynomial algorithm, namely admits an algorithm of time n to the power of some polynomial of log n. The best known approximation for the problem is O ( 1/ɛ 3 n ɛ) ( ) for any universal constant ɛ [15]. The running time of the [15] is O n O(1/ɛ3 ). Thus to get a polylogarithmic approximation ratio, requires quasi-polynomial in n time. More generally, the input may contain (instead of r and the set of terminals) a collection {s i, t i } k i=1 of pairs and the goal is to find a network G (V, E ) of minimum cost so that s i can reach t i in a directed path in G for every 1 i k. This is the directed version of the Steiner forest problem. We call it the directed pairs connection (DPC) problem. Input A directed graph G(V, E) with cost c : E R + on the edges and a collection of demand pairs {s i, t i } k i=1. Output: A subtree G (V, E ) of minimum cost c(e ) so that s i can reach t i by a path in G for every 1 i k. The unifying theme of the project: The project can be described in a unified way as follows: all (but one) of the problems studied are strongly connected to DST or DPC (for example may be extensions or special cases of these two problems). 1.1 The methodology: Approximation Algorithms and Inapproximability Optimization problems that are known to be NP-hard [34] are believed not to have polynomial time algorithms. It is widely believed that P NP. It is, therefore, of interest to study how well a problem can be approximated, i.e. to devise efficient algorithms that can produce solutions not too far from the optimal ones and also, to establish approximation thresholds for a specific problem. The theoretical framework for these questions has been well developed in the past two decades. Vazirani, [70] provides an excellent starting point for a tour of results in this area. Formally, let OP T (x) be the cost of an optimal solution to some instance x of an optimization problem L, where the cost is suitably defined. We 1

3 say that an algorithm for problem L achieves a ratio of approximation α > 0 if for every such instance x, the algorithm runs in time polynomial in the size of x and produces a solution with cost at most α OP T (x) if L is a minimization problem or at least OP T (x)/α if L is a maximization problem (for the sake of uniformity, all the ratios in the proposal are numbers larger than 1). An algorithm in this framework must provide a provable performance guarantee; that is, it should be supplemented with a proof that the value of the solution it finds is not too far from the optimum. Recent work on numerous classes of optimization problems has addressed both sides of the issue of approximability. From the point of view of upper bounds, the idea is to design an appropriate algorithm that achieves a certain provable approximation ratio. Equally important is to explore whether designing an approximation algorithm with a given ratio for a specific problem is as hard as solving an NP -complete problem. Such a proof is referred to as proving a hardness of approximation result. Indeed, some of the most notable and beautiful results in this area have shown that certain problems are provably inapproximable under reasonable assumptions of intractability. A large contribution to the lower bounds area is due to the so called PCP theorem [3]. This theorem implies that certain problems like M ax-3sat are N P hard to approximate within some constant larger than 1. 1 Later, the seminal paper of Johan Hastad [47] showed that unless P=NP, the Max-3SAT problem can not be approximated within 8/7 ɛ for any constant ɛ > 0. This matches the simple 8/7 upper bound resulting from a random choice. The list of inapproximability results is quite long and growing each year. For the most part in this list, the lower bound and the upper bound do not match. We merely wish to point out here that many of our proposed research problems inhabit the same realm of inapproximability. 2 Some open problems 2.1 The DST problem and related problems DST and sepcial cases: The PI has tried to find a polynomial time, polylogarithmic ratio approximation algorithm for DST ever since 1997 (thus 11 years already). One of the motivation is that the algorithm used by [15] which gives the best approximation known for the DST problem is due to Kortsarz and Peleg in the context of shallow-light trees [53]. The [53] paper was published (in a conference) a year before the conference version of [15] paper. However, fairness dictates to say that [15] found the same algorithm independently, and gave a better and very elegant analysis. We survey some of the ideas the PI tried to find a polynomial time polylogarithmic ratio approximation algorithm for DST. But before that let us define the Group Steiner (GSP) problem. Input: An edge weighted undirected graph, a root r, a collection of terminals T V and a collection of groups {g i } k i=1, g i T. Output: Find a minimum cost tree rooted at r containing at least one terminal from every group. If the g i may intersect, the problem is set-cover hard even on stars [37] hence admits no c ln n ratio approximation for some c < 1 unless P = NP [64]. The best approximation known for the GSP problem is O(log 3 n) [37]. If the input graph is a tree, [37] give an elegant O(log 2 n) approximation algorithm. Further, it was shown that the algorithm of [37] for trees is almost the best possible. In [44] Halperin, Kortsarz, Krauthgamer, Srinivasan and Wang prove an integrality gap of Ω(log 2 k) for the natural LP for GSP on trees with k the number of groups. This integrality gap is equal in many cases to Ω(log 2 n), for example if there are, say, k = n disjoint groups all of size n. This integrality gap is used as a gadget in [45] to prove a hardness of log 2 ɛ n for GSP on trees for any universal constant ɛ. This nearly matches the upper bound [37]. 1 In the Max-3Sat problem, a conjunctive normal form (CNF) formula is given with each clause having exactly 3 variables and the goal is to give truth values to the variables so as to maximize the number of satisfied clauses. 2

4 Both algorithms of [37] are LP based. The [37] paper left it as open problem whether it is possible to find a polynomial time greedy approximation for group Steiner on trees, with polylogarithmic ratio. In [18] Chekuri, Even and Kortsarz answer this open question positively and also introduced tools that might help approximating DST. The authors of [18] designed for every universal (small) constant ɛ a general combinatorial scheme that reduces any tree with possibly very large height to a log n/ log log n height tree. This reduction incurs a penalty of log ɛ n in the approximation ratio. The authors also give a tool to reduce the maximum degree of the tree into O(log n) with essentially no penalty in the ratio and essentially no increase in the height. Then the authors apply the so called the recursive greedy technique [73, 53, 15] to approximate the new instance. Let r be the root of the tree. Loosely speaking in the recursive greedy technique the algorithm guesses a child u of r (in the optimum) whose subtree has the minimum density among all children (the density of a subtree is its cost over the number of terminals it contains. The cost also includes the cost of the edge ur). By averaging there must be at least one child with density no worse than the density of the optimum tree. Thus, a search is conducted over all neighbors u of r and over all possible numbers n u of terminals in the subtree of u in the optimum solution. Then the algorithm recursively calls for finding a tree rooted at u with n u terminals for all possible u, n u pairs. The best density tree returned T out of all recursive calls over all u, n u pairs is used. The terminals belonging to T are removed, and the algorithm recurses. The analysis of the recursive greedy algorithm is extremely delicate (see [15]). The techniques of [15] would have led, if applied in a straitforward way, to super-polynomial in n running time. Thus [18] adds to these techniques extra tricks such as geometric searching for the correct number of terminals n u in the subtree of a given u in OP T (hence the n u we use may be smaller by a 1+/ log n factor than correct n u in OP T ). This trick saves a lot of time. In addition, in [18] small subtrees of the optimum containing only few terminals are ignored (by restricting n u to large enough values). It is possible to show that this has only a small affect on the ratio but it saves a lot of time, because the number of recursive calls goes down (only trees with many terminals are found, and so, the number of recursive calls becomes small). The bottom line is a polynomial in n greedy, log 2+ɛ n approximation ratio for GSP on trees. The crucial property used by the algorithm that secures its polynomiality is that for any constant c, (log n) c log n log log n, is a polynomial in n. In fact, the running time is O( ) O(h) with the largest degree in the graph. This means that it is enough to have reduce the degrees to any polylogarithmic value as long as the height is O(log n/ log log n). In the conference version of the paper, [29], Even and Kortsarz show that if the input can be transformed into a directed tree with arbitrary (polynomial in n) degrees, it is enough to reduce the height to O(log n/ log log n) and use a technique called lazy density updates (that re-checks density of trees only if many of their terminals were covered) getting a log 2+ɛ n ratio, polynomial time algorithm. Hence [29] shows how to avoid the step of degree reduction. Some of these techniques described above are valid for the DST problem as well. The techniques of height reduction holds (see [73]). Binary searching over n u, and ignoring small trees of the optimum, is possible. But it seems hard to reduce the maximum degree to polylogarithmic in n value. Even the lazy update technique [29] does not seem to help, as it needs the graph to be transformed into a directed tree, albeit, allowing the degrees to grow polynomially n, which does not seem possible. The PI tried other technique for DST like the techniques of Hajiaghayi, Kortsarz and Salavatipour [43] (which seemed promising at first, coupled with many additional ideas, but failed in the end). The PI also tried various combination of the recursive greedy technique with dynamic programming, to no avail. What about LP methods? There are two natural LPs for DST. One is the cut variant and one is the flow variant. However: 3

5 Conjecture 2.1 The integrality gap of both LPs for DST is n ɛ for some universal constant ɛ If this is true, LP techniques can not help us with approximation DST. The only integrality gap known is k (k is the number of terminals) by [74]. But the construction of [74] requires k = O(log n) to hold, and thus a polynomial in n integrality gap for DST is not known. We shall return to Conjecturegap In Section 2.3 where we describe a methodical way that may lead to a large integrality gap for the natural LP for the shallow-light trees (which would imply a large integrality gap for DST because SLT is a special case of DST). In [22], a possible explanation for these difficulties is provided. It is shown there that unless NP admits a sub-exponential time algorithm, the gap reduction of [45] that proves a log 2 ɛ n lower bound for GSP on trees has to use quasi-polynomial time in the size of the NPC instance. To put it in other way, in [22] an algorithm for solving an NP C problem is presented, that runs in O(log 2 n) 2 logδ n time, where n is the size of the tree instance in the gap reduction of [45] and δ is any universal constant δ < 1. The ( fact that this ) does not achieve a sub-exponential algorithm for NP is because in the [45] reduction n = Ω exp(log 1/δ I ) with I the NP C instance you start with. Thus, unfortunately the running time is larger than 2 I. Note that if n would have been polynomial in I, (with the degree of the polynomial independent of δ) a subexponential algorithm for N P would have been derived! This leads to the following conjecture. Conjecture 2.2 The DST problem admits no polynomial time polylogarithmic ratio algorithm, unless N P admits a subexponential time algorithm The PI believes the conjecture is not speculative and has its reasons. Indeed, [22] does show that unless N P admit a subexponential algorithm, there are things that can be done in quasi-polynomial time but can not be done in polynomial time. In addition, the PI and many other researchers have a vast experience in trying to achieve a polynomial in n polylogarithmic ratio in n algorithm for the DST problem for the last 11 years. The concrete idea we would like to try in order to prove Conjecture 2.2 is to be described now. But first we need to define the Labelcover max problem. Input: A bipartite graph B(X, Y, E) and two disjoint partitions q i=1 A i = X and q i=1 B i = Y of X and Y. Two sets A i and B j are said to form a superedge if there exist a A i, b B j so that ab E. A superedge A i B j is covered by Z X Y if there exists a i A i Z, b j B j Z so that a i b j E Output: A choice Z X Y so that for every A i and B j, Z A i 1, and Z B j 1 that covers the maximum number of superedges. It follows directly from the PCP theorem [3] that there some universal constant c > 1 and a reduction from (say) max-3-sat to Labelcover max so that a yes instance is mapped to a Labelcover max instance that admits a solution that covers all the superedges, and a no instance is mapped to a Labelcover max instance so that any solution cover at most 1/c fraction of the edges. The parallel repetition theorem [63] gives a way enhance the gap getting an instance of size I l with gap 2 dl for some universal constant d. Hence the gap grows fast. Let I be an instance Labelcover max problem with constant gap. Use parallel repetition theorem to increase the gap [63] to 2 d l so that n = I l may be very large but still subexponential in I. Use [27] to show how to reduce the new Labelcover max instance of into a DPC instance of size linear in n. Then we should show that if Conjecture 2.2 does not hold we could produce an algorithm that distinguish in time f(n), for some function f, between the resulting yes instance of DPC and the resulting no instance of DPC. Here the negation of Conjecture 2.2 should be of big help because a good approximation for DST implies a good approximation for DPC. The crux of the mater is that we should achieve the goal that f(n) = f( I l ) should be subexponential in I. If this goal is achieved it implies that N P has a subexponential algorithm and would be a strong evidence that Conjecture 2.2 holds. 4

6 Problem 1: Prove that if DST admits a polynomial time polylogarithmic ratio approximation, N P admits a sub-exponential time algorithm. We now talk on special cases of DST. While the upper and lower bound for GSP on trees are nearly equal, the following question stands up: Problem 2: Can the group Steiner problem on general graphs be approximated within O(log 2 n) in polynomial time? We now discuss some ideas that can be used to solve Problem 2. The O(log 3 n) ratio approximation of [37] uses a transformation of a graph into a tree by the Bartal technique. See [8] and Fakcharoenphol, Rao, and Talwar [30]. The transformation incurs a O(log n) loss in the ratio (see [30]). Then [37] applies an O(log 2 n) LP approximation algorithm for trees. Since approximating the group Steiner problem on trees within log 2 ɛ n was proved to be hard in [45], for any universal constant ɛ > 0, any (significant) improvement in the ratio should come from not using Bartal and working directly on the original graph. An idea that may be viable is to directly work on the LP for the original graph. One has to find a way to round paths that is highly dependent (rounding paths by independent probability will not work). Let us propose a completely different method. A sparse cover of a graph is a partition of the graph into non disjoint sets, so that no vertex appears in many sets (see [6, 10]). Some sparse covers have the property that sets (clusters) in the cover intersect with only few other subsets. The number of other sets a given set S intersects is called the super-degree of S and many of the sparse covers constructions assure small maximum super-degree (see [61]). It may be that non-disjoint covers of graphs can be used to derive better approximation for the GSP problem. The reason we believe so is that given some parameter k there are sparse covers so that for every path of length at most k, there is a cluster in the partition that contains all the edges of this path (see [61]). We may assume the optimum solution for the group Steiner problem is a path by doubling every edge and taking the Euler tour resulting and short-cutting as in the classic approximation for the TSP problem [66]. Thus we may partition the optimum into strictly smaller but intersecting clusters, so that one of those clusters fully contains the optimum. Hence we may try to recurse on all the clusters in the cover (the size of each cluster is much smaller than the original graph) until the size of the cluster become small enough to use some other technique. This idea depends on the question if the running time resulting would be polynomial in n. The sparsity properties of the graph depend on k and the question if a log 2 n polylogarithmic in n ratio that runs in time polynomial algorithm n could be derived this way, needs a thorough inspection. In fact the dependency of the algorithm of [37] on the [8] techniques turns up problematic if we change the GSP even by a little. Consider the interesting fault-tolerance GSP version of finding a subgraph that contains at least 2 edgedisjoint paths from every group to r. 2 The two paths to for the same group can start from different vertices in the group or can start from the same vertex of the group, but must be edge disjoint. Paths into r for different groups may intersect. It is possible to get a O( n) ratio approximation by the technique of iterative merging (details omitted). However, the question is: Problem 3: Can the two disjoint path group Steiner problem on general graphs be approximated within polylogarithmic ratio in polynomial time? If we change G into the Bartal tree we will not be able to deal with Problem 3 as the disjoint paths are lost. Hence we have to operate on the original graph. We may attempt to better use a combinatorial 2 As far as the PI can see this is the only problem in this proposal that is not directly related to DST or DPC 5

7 technique such as a better version of iterative merging. The problem in iterative merging (that relies on the pairing lemma, see [56]) is when it merges two subgraphs that contain almost the same set of groups. Then no significant progress is achieved. It may be possible to perform a pre-processing that will assure that every two terminals that belong to the same group will have large distance. If this is achieved in a significant way, it is unlikely that two clusters containing many mutual groups will be merged. Indeed, the pairing lemma uses short paths for merging two clusters and any two terminals in the same group have large distance. The DPC problem and generalizations We now define a more general version of the DPC problem. The p-dpc problem: Input: A directed graph G(V, E) with costs c(e) over edges, and a collection of pairs (s i, t i ) k i=1 and a number p k Output: A subgraph G (V, E ) with minimum cost c(e ) so that for at least p of the s i, t i pairs G admits a directed path from s i to t i in G. We now describe previous work on DPC and p-dpc. The first approximation for DPC and p-dpc was given by [15]. The ratio was Õ(k2/3 ) (k is the number of pairs) and Õ(p2/3 ) respectively (the Õ() notation ignores polylogarithmic factors). In [27] it is proven that DPC is Labelcover max -hard which implies that unless NP DT IME (n polylog ) n the problem admits no 2 log1 ɛ n ratio for any universal constant ɛ. Recently, in [17], Chekuri, Even, Gupta and Segev improved [15] and gave a k 1/2+ɛ ratio approximation for DPC. One of the main techniques used by [17] are junction trees. An r-junction tree is an in arborescence T I into r and and out arborescence T O from r that contains the complementary t i for every s i T I. This tool was first introduced for use in approximation algorithm by Chekuri, Hajiaghayi, Kortsarz, and Salavatipour, in the context of buy-at-bulk [19] (see next subsection). We informally present some details of the [17] algorithm. It is simple to discard s i t i of short distance (taking the whole path into the solution has good density, if the path is short enough, even though we cover a single pair). Else, all the s i to t i path are long and in such a case a simple averaging lemma over the vertices shows that there is a junction tree for some r of density (cost over number of pairs connected) at most k 1/2+ɛ times the optimum density. This does not end the trouble [17] were facing. Say that we just guess r and the number j sources in T I which equals the number j of sinks in T O. Calling the DST algorithm of [15] twice, once to cover j sources by an in-arborescence rooted at r (this requires reversing the direction of the edges) and after that [15] to cover j sinks from r by an out-arborescence rooted at r may not derive a good solution. The sources and the sinks may no not match. In fact, they may not even intersect. As a result [17] had to apply series of time and space consuming transformations to assure that the s i and t i match. Their algorithm iteratively solves many density type LPs. However, more recently, Feldam,Kortsarz and Nutov [32], provide the same ratio while largely improving the time and space complexity of the algorithm. The [32] is a purely combinatorial algorithm. In fact [32] reduces the DPC problem to the DST problem as follows. Consider the metric completion of the graph so that u, v with no directed connection are connected by new uv edge of cost equal to dist G (u, v) (namely the length of the shortest path in G from u to v). Clearly this does not decrease the optimum as if an edge uv that did not exist before is used in a solution, this edge can be replaced by the directed path from u to v with no increase (but perhaps a decrease) in the solution cost. Now [32] introduce a new concept called the star-junction tree structure. A star-junction tree is a collection s 1,..., s p of sources connected by the directed edges (s i, r), i = 1,..., p into r, combined with 6

8 an arborescence T O leaving r containing every corresponding t i, 1 i p. Thus a star-junction tree is an in-star into r from some p sources plus an out- arborescence from r leading by paths to the p matching sinks t i of the sources of the star. Lemma 2.3 [32] There exist a junction-star tree in the transitive clousare of G with density at most O( k) times the optimum DPC density subtree The proof of Lemma 2.3 is quite non-trivial. This allows to reduce DPC to DST with the following trick. Define a graph G as follows. Add a collection of new vertices {s i }. For every i add a t i, s i edge whose cost is the minimum distance from s i to r in G. The DST instance results by setting r as root and s i as the terminals. The star-junction tree is embedded as an r-out-arborescence in G with the same cost as the star-junction tree (because the cost of of the edges t i s i equals the cost of the edge s ir in the star-junction tree). Hence guessing r and the number p of pairs covered by the star-junction tree, we apply the p-dst approximation of [15] and get the required good density subgraph (after the t i s i edges are replaced by the shortest s i to r paths). We iteratively find subgraphs with density k 1/2+ɛ times the optimum. This leads by a set-cover like analysis to a Õ(k1/2+ɛ ) ratio approximation for DPC. Besides being purely combinatorial, the algorithm has a huge save in running time and space complexity compared to [17]. Turning our attention to p-dpc, the algorithm of [17] can not deal with covering exactly p pairs. This is because it uses an LP that returns a solution with good density (covers relatively many pairs compared to its cost) but the number of pairs covered by the LP can not be controlled. Hence, the cost of the solution returned by the LP can be much larger than the optimum cost for covering p pairs. Given the combinatorial nature of the [32] algorithm, it can use [15] that is able to cover p pairs with approximation ratio Õ(p1/2+ɛ ). This improves the Õ(p2/3+ɛ ) ratio of [15]. One more interesting challenge is giving a sublinear approximation for DPC and/or p-dpc. As the number of pairs k can be as large as k = Θ(n 2 ) (for example if every two vertices form a pair) non of the algorithm above gives a sublinear in n approximation for DPC. In [32] Feldman, Kortsarz and Nutov give a Õ(n 4/5 ) ratio algorithm for DPC is given, being the first sublinear algorithm for the problem. We informally present some of the details of the algorithm. If there are many long paths between s i t i we are able to prove the existence of a junction tree with density O(n 4/5 ) times the optimum density. Thus we call the algorithm of [17] getting a good density subgraph, and iterate. Hence we may assume that all distances between s i, t i are short. We say that s i, t i have many good connectors if there is a large enough set C(s i, t i ) = {x 1, x 2,...} so that short s i to p i path via each of the x j C(s i, t i ) exists. Let these pairs be called good pairs. For covering good pairs we use the probabilistic method [1]. We select a random set X by adding each v V into X by a carefully chosen probability p. We obtain a small set X, that nevertheless has at least one good connector for any good pair s i, t i. Thus we connect all good pairs via X. Because X is small enough, and the paths from s i to X and from X to t i are short enough, these stage does not add to the cost too much. At the absence of good pairs, we write a novel LP that sets capacities to each edge so that for every i, s i can send to t i a unit of flow but only along short paths. The capacity constraints should be preserved (separately) for every commodity i. Say that s i t i are not connected. By definition, the capacity of every cut between s i and t i is at least 1. We claim that every such cut must contain at least one edge of relatively large capacity. Indeed, if all capacities are small, the min-cut must have many edges to get to capacity 1. But this means that there are also many vertices that are in the middle of a short s i to t i paths. This would imply s i, t i are a good pair and would lead to a contradiction. Hence, at least one of the capacities of one of the edges is large. This 7

9 edge is added into the solution (set to 1) and we recurse. This process is similar to the iterative rounding method [48]. However, what is the best possible ratio for DPC? We conjecture: Conjecture 2.4 DPC admits an O( n) ratio algorithm. Problem 4: Give an O( n) ratio approximation for DPC The PI has very concrete ideas of how to improve the n 4/5 ratio for DPC. There are two bottlenecks that define lead to the Õ(n4/5 ) ratio. One of them is when finding a junction trees. Whenever we use a junction tree we assume its cost is equal to the cost of the entire tree (this may be far from being true). The other is when we find an s i to t i cut. We assume no edge in the cut has capacity larger than 1/n 4/5 (otherwise we choose this edge at fractional local ratio of O(n 4/5 )). Since the cut has to deliver a unit flow, there are at least n 4/5 edges in the cut. Thus many vertices touch the cut edges. In the worst case the two sides of the cut form a complete bipartite graph (then we get the required capacity 1 of the cut using the least amount of vertices) giving only O(n 2/5 ) vertices touching cut edges. However it is clear to the PI (because of some technical details) that those two bottleneck can not occur simultaneously. If the two bottlenecks can t happen together, the ratio is improved. And what about p-dpc? Unfortunately, the Õ(n4/5 ) algorithm does not apply for p-dpc as it uses LP that return a good density solution but perhaps with much more than p terminals [17] and thus with large cost. In fact as we shall now see, even the undirected version of p-dpc seems to be a hard problem. The undirected case of p-dpc: We now digress into one interesting problem on undirected graphs. The state of the art above may be compared to the p Steiner forest on undirected graph. Thus problem is given an undirected edge-weighted graph and a collection {s i, t i } k i=1 of pairs, and a number p, find a subgraph that connects at least p pairs. The best ratio for the problem on undirected graphs is min{ n, p} [41]. In terms of p, the two ratios are p 1/2 [41] versus p 1/2+ɛ [17]. These two ratios are almost identical which may be somewhat of a surprise. The weak ratio for p-steiner forest on undirected graphs may look surprising at first glance. But in [42] it is shown that a ρ ratio approximation for undirected p Steiner forest implies a ρ 2 ratio for the dense k-subgraph problem (see [31]). Feige, Kortsarz and Peleg [31] gave an n 1/3 ɛ ratio (with ɛ about 1/60) for the dense k-subgraph problem. This ratio has not been improved for 12 years now. Hence even getting better than n 1/6 ɛ/2 approximation for p Steiner forest seems hard. Still, it seems that undirected version should admits a better than p ratio approximation: Problem 5: Does undirected p-steiner forest admit better than p approximation? A first try would be to see what do the technique for directed graphs give for undirected graphs. In particular, the flow techniques. An improvement for undirected p-steiner forest leads to an improvement for a problems called dial a ride [41]. Lower bounds for DPC and p-dpc Even though Labelcover max was invented in the context of proving inapproximability results, it turns out that it is an important problem by itself as for example it is related to red-blue set-cover (see [62]) and has vital applications in data mining (see [12] for a large elaboration on this subject). If we consider the general Labelcover max problem (and do not restrict the inputs to those resulting from parallel repetition [63]), the problem changes character quite drastically. The best known upper bound for Labelcover max is O( n) by Peleg [62]. This means that unless the result of Peleg [62] is improved, [27] implies that DPC admits no better than Ω( n) ratio. Problem 6: Prove that the general version of Labekcover max admits no better than Ω( n) ratio approximation, unless P = NP 8

10 Solving both Problem 4 and Problem 6 and using the direct nature of the reduction of [27] from Labelcover max to DPC, implies that Θ( n) is the best possible approximation for DPC. It seems to the PI that Problem 6 can be solve via a reduction from set-cover packing which is the problem of finding the largest collection of vertices of the same side of a bipartite graph so that no two vertices in the collection share a neighbor. And what about p-dpc? Problem 7: Does p-dpc admits a sublinear ratio approximation? Unfortunately, the PI is not optimistic about p-dpc having o(n) approximation algorithm. More likely, an Ω(n) hardness of approximation can be proved 2.2 Buy-at-bulk network-design problems and cost-distance problems We now describe buy-at-bulk problems. It can be shown that rooted buy-at-bulk is strongly related to DST and multicommodity buy-at-bulk is strongly related to DPC. Buy-at-bulk network-design problems are common in settings where buying a large number of items reduces the per-capita price of the purchase. This is also called economies of scale. We study economies of scale in the purchase of bandwidth for establishing connectivity demands between certain chosen pairs in a network. The network is represented by a given graph G = (V, E). Every edge e represents a cable of a certain type (there may be many different types of cables). For any given edge (of some fixed cable type) we are to decide how much bandwidth should we buy for this cable. One of the main application areas is in the design of telecommunication networks [59]. Recall that a function f is subadditive if f(a + b) f(a) + f(b). The typical scenario is that capacity (or bandwidth) on a link of a given fixed type can be purchased in some discrete units u 1 < u 2 <... < u r with costs c 1 < c 2 <... < c r such that the cost per bandwidth decreases c 1 /u 1 > c 2 /u 2 >... > c r /u r. The capacity units are sometimes referred to as cables or pipes. For each cable e there is a monotone non-decreasing concave (or more generally a subadditive) function f e : R + R + where f e (x) is the cost of purchasing cables along e of total capacity x. A basic problem in this setting is: given a set of bandwidth demands among pairs s i, t i, install sufficient capacity x e on every link e E of the underlying network topology so as to be able to route the demands, and so that the total cost e E f e(x e ) is minimized. Let the set of h demand pairs be T = {s 1 t 1, s 2 t 2,..., s h t h }. Pair i has a non-negative flow demand d i. The routing of the ith demand between s i, t i is performed on a single path. Hence, given the selected path P i from s i to t i all flow from s i to t i goes via P i (we later show that routing all flow along a single path is an optimal strategy up to constants). As the demands are routed simultaneously along the paths P i, the total flow over e is: x e = i:e P i d i. The objective is to minimize the cost of the flow. The cost of the flow is given by e f(x e). If the function f e is the same for every e, we get the uniform version of the problem. An instance is called a single-source instance if all the pairs have a common source s. A typical telecommunications problem with discrete capacity units gives rise to a uniform problem. However, non-uniform cases arise often for several reasons including the following: First, not all capacity units are available at all links due to various constraints. Second, when designing networks incrementally, existing links can have different unused spare capacity available and this leads to non-uniformity. In summary, the non-uniform buy-at-bulk multicommodity problem is: Input: A graph G = (V, E), a monotone non-decreasing and subadditive function f e for every e E, and a set s 1 t 1,..., s h t h of h pairs. For every i a demand d i is also given. 9

11 Output: Select h paths {P i } h i=1 so that P i is an s i to t i path. and minimize f e (x e ), where for every e E e E x e = P i: e P i d i. For short, we denote this problem by MCBB. For algorithmic purposes, it is convenient to approximate each function f e by a collection of simple piece-wise linear functions of the form a + bx. We replace an edge e with cost function f e by a a collection of parallel edges, one for each of the linear functions in the piecewise linear representation. We can find such a reduction to preserve the value of the solutions to the network design problems to within a factor that is arbitrarily close to 2. For discrete cable types, Andrews and Zhang [2] pointed out this reduction and subsequently Meyerson et al. [58] used it for arbitrary concave functions. Given a function f : R + R +, and a fixed ɛ 0, for integer i 0, let g i : R + R + be a linear function defined by g i (x) = f(a i ) + f(a i )/a i x where a = (1 + ɛ). It can be verified that if f 1 is monotone and subadditive, then for all x 1, 2+ɛ min i g i (x) f(x) min i g i (x). This allows us to reformulate the buy-at-bulk network-design problem as a bicriteria cost-distance network-design problem. In the cost-distance network design problem, we are given two separate edge-weight functions c : E R + and l : E R +. For an edge e E, we let c e and l e denote the cost and length of e. We think of c e as the fixed cost of e and l e as the incremental or flow-cost of e. Thus for x e > 0, a flow of x e on e costs c e + l e x e. Now consider the problem of routing the demands for the given pairs. A feasible solution consists of a set of edges E E and a routing of flow in G[E ] for each given pair. We observe that it is optimal for each pair to route all its flow along a shortest path in G[E ] where the length of the path is computed using the function l. This also shows that unsplittable routing of the flow for each pair (namely sending all flow along one path) does not cost more than a factor of 2 in buy-at-bulk network-design. Hence, up to a constant in the approximation ratio MCBB is, given the above input, find E E that minimizes the following cost measure: min E E c(e ) + h d i l E (s i, t i ) i=1 In the above objective function, l E (u, v) is the shortest l-path in G[E ] and c(e ) = e E c e. Observe that every s i, t i must be in the same connected component in the graph induced by E. For the rest of the proposal, we adopt this slightly simpler cost-distance formulation above. The first term in the total cost (i.e., c(e )) is referred to as the fixed cost, and the second term (which involves lengths) is referred to as the incremental cost. Note that the reformulation of the problem in this way allows us to avoid the explicit reference to flow. The first non-trivial approximation for MCBB (namely, for the most general case) is due to Charikar and Karagiazova [16]. They gave a simple and elegant algorithm and showed that it achieves a log D exp(o( log h log log h)) approximation. 3 This result was improved by Chekuri, Hajiaghayi, Kortsarz, and Salavatipour [19]: Theorem 2.5 [19] MCBB admits a polynomial time O(log 4 h) = O(log 4 n) ratio approximation with h the number of pairs. 3 The algorithm in [16] is essentially for the case when d i = d for 1 i h. For this case the ratio obtained is exp(o( log h log log h)). For the case of arbitrary demands with total demand D, one can obtain, using simple reductions, either a ratio of exp(o( log D log log D)) or a ratio of log D exp(o( log h log log h)). 10

12 A notable feature of the [19] algorithm is that it can deal equally well with demands that are exponential in n. In [20] the same authors study the harder case of monotone subadditive functions on the vertices. Even though this case is harder, [20] gives an approximation of the same O(log 4 h) ratio for this more general variant. For general demands, the above ratio is the best known one. Kortsarz and Nutov [51] recently improved the algorithm and analysis of [19] to give an O(log 3 n) ratio for the case of demands polynomial in n (for the harder vertex costs case). The most natural open question in that regard is: Problem 9: Determine the best possible approximation ratio for MCBB. In fact, we conjecture: Conjecture 2.6 The MCBB problem (with vertex or edge costs) admits an O(log 2 n) approximation. However, for every constant ɛ > 0 the problem admits no log 2 ɛ n-approximation algorithm, unless P = NP. We now discuss several possible directions that can be taken towards proving the conjecture. Perhaps the hardness result conjectured for the problem may be proved by a reduction from the GSP on trees [45]. It may be simpler to first prove the upper bound part of the conjecture for demands polynomial in n. One reason is that besides the LP-based approximations [19, 51], the polynomial demands case also admits a greedy approximation algorithm [51] with ratio O(log 4 n) opt C + O(log 3 n) opt L with opt C and opt L, respectively, the optimal fixed and incremental cost of a given optimum. The greedy algorithm of [19] uses as a black box a solution to the buy-at-bulk k-steiner tree problem. The buy-at-bulk k-steiner tree problem is given a graph with costs and lengths on edges (or more generally, on vertices), a source s, a collection T of terminals, and a number k, find a subtree containing s and at least k terminals that minimizes the sum of costs of edges plus the sum of distances of the terminals to s. It is shown [19] that if there is a ρ-ratio for the buy-at-bulk k Steiner tree problem then there is a (greedy) O(ρ log n opt C + ρ log 2 n opt L )-ratio approximation for MCBB. Hence further studying the buy-at-bulk k-steiner tree problem may shed further light on MCBB. The buy-at-bulk k Steiner problem is related to the prize-collecting Steiner tree problem (see [9]) and its generalization into vertex costs versions (see [39]). Therefore, it is natural to discuss the vertex costs variants of these problems (the vertex variants are more general than the edge variants). In [39] the budgeted Steiner tree problem with vertex costs ( BST ) problem is defined. A graph G = (V, E) with vertex costs and a budget B are given. Every vertex v has a profit p(v). The goal is to select a subtree T of G of budget (sum of cost of vertices) at most B and to maximize the sum of profits of vertices included in T. The initial motivation for studying BST was fast recovery from a power outage [39]. In [39] Guha et al. give an O(log 2 n) pseudo-approximation algorithm is given for BST. Namely, the cost is at most O(log 2 n) times the optimum cost for budget B, but the budget used can be as large as 2B. In [60], Moss and Rabani give an O(log n) ratio approximation algorithm for BST that uses at most 2B budget. In [60], it is conjectured that BST admits a constant approximation algorithm. In [51], Kortsarz and Nutov disprove the conjecture by showing that unless N P admits a quasi-polynomial time algorithm, the BST problem admits no ln ln n/4 ratio approximation. The technique applied by [60] is the primal-dual technique. The BST problem is related to the buy-atbulk k-steiner tree problem only if we can add lengths on vertices and impose diameter constraints on the required network. The hope would be that the techniques of [60] or related techniques can handle diameter constraints equally well. Otherwise, the hope is to combine these technique with the iterative merging techniques (see [56]). Another venue for improving our understanding of the MCBB problem, including perhaps shedding light on possible ways of proving lower bounds, is studying the natural LP relaxation of MCBB. Determining 11

13 the integrality gap of the natural flow-type linear program for MCBB is an interesting open question. Integrality gaps of natural programs are a subject of research by itself (see, for example, [36, 13]) and can lead to hardness of approximation results (see [24, 44]). Let P i be all the s i to t i paths in G. The MCBB LP is: LP-BB min e E c ex(e) + h i=1 d i p inp i l(p)f(p) f(p) x(e) e E, 1 i h p P i f(p) 1 1 i h x(e), f(p) 0 e E, p i P i p P i e p The results of [19, 51] while using LP method still do not guarantee any bound on the integrality gap of this LP because it reduces MCBB to the (density variant of the) rooted case (with logarithmic loss in the ratio) in which an integrality gap of O(log n) is already known [21]. Problem 9: Determine the integrality gap of the linear program LP-BB. We note that in [40] an O(log 2 n) bound was given for the the integrality gap of a related LP which however does not imply an integrality gap for LP-BB. Finally, we consider the relation of problems and their density variants. [19] reduces MCBB with O(log n) loss in the approximation ratio to the following rooted density MCBB problem: Input: A root r and a collection of terminals t i Output: A tree tree rooted at r with minimum cost-distance density. The cost-distance density is the cost of the edges plus the sum of distances of the terminals to the root, over the number of terminals in the tree. In [19], LP techniques are used providing an O(log 2 n) ratio approximation for density rooted MCBB Thus the density version of the rooted MCBB as worst ratio (namely, O(log 2 n)) than the rooted MCBB problem itself who has an O(log n) ratio [58]. In the opinion of the PI this is an anomaly. The density version of a problem should in almost all cases have a no worse approximation ratio than the problem itself. We give some well known example when this holds. For example, the k MST problem is given a weighted complete graph to select k vertices so that the minimum spanning tree over those vertices has minimum cost. This problem is NP-hard but admits a ratio 2 approximation due to Garg [35]. The density variant is to compute a tree that minimizes the cost of the tree over the number of vertices it spans and is trivially solvable in polynomial time. Another notable example is finding dense subgraphs. The density of a subset V of the vertices is e(v )/ V with e(v ) the number of edges that belong to the graph induced by V. The problem of finding the densest subgraph has a simple flow solution (see [54]). If the number of vertices in the subgraph is fixed to some parameter k, the resulting problem is called the densest k subgraph problem which is a very hard problem (see [31]). The most recent and notable problem whose density version admits a better approximation algorithm is the minimum cost multicut problem versus its density version, which is the sparsest cut problem. The best ratio known for the min-cost multicut problem is O(log n) by Garg, Vazirani and Yanakakis [38]. In [4], Arora, Rao and Vazirani give an O( log n) ratio approximation for the sparsest cut problem (see [4] for precise definitions of these problems). If indeed the rooted density MCBB should have a better approximation ratio than rooted MCBB then a natural open problem is: Problem 10: Find an O(1) ratio approximation for the density version of rooted MCBB A solution to the above open problem would give the O(log 2 n) ratio approximation for the most general case of MCBB namely even for the super-polynomial in n demands case. 12