Consider the edge-connectivity survivable network design problem: given a graph G = (V, E) with edge-costs, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimum-cost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized O(rmax log 3 n) competitive online algorithm for this edge-connectivity network design problem, where rmax = maxij rij. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. Our results for the online problem give us approximation algorithms that admit strict cost-shares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rent-or-buy version and (b) the (twostage) stochastic version of the edge-connected network design problem with independent arrivals. For these two problems, if we are in the case when the underlying graph is complete and the edge-costs are metric (i.e., satisfy the triangle inequality), we improve our results to give O(1)-strict cost shares, which gives constant-factor rent-or-buy and stochastic algorithms for these instances.

The Steiner tree problem is one of the most fundamental-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robins,Zelikovsky-SIDMA’05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than [Vazirani,Rajagopalan-SODA’99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation a� algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the�-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together

We consider the stochastic Steiner forest problem: suppose we were given a collection of Steiner forest instances, and were guaranteed that a random one of these instances would appear tomorrow; moreover, the cost of edges tomorrow will be λ times the cost of edges today. Which edges should we buy today so that we can extend it to a solution for the instance arriving tomorrow, to minimize the expected total cost? While very general results have been developed for many problems in stochastic discrete optimization over the past years, the approximation status of the stochastic Steiner Forest problem has remained open, with previous works yielding constant-factor approximations only for special cases. We resolve the status of this problem by giving a constant-factor primal-dual based approximation algorithm.

We consider optimization problems for which the best known approximation algorithms are randomized algorithms: these algorithms make random choices during their execution, and it has been shown that in expectation the cost of the algorithm’s solution is at most a known constant factor more than optimal. We show how to give deterministic variants of these algorithms that have similar performance guarantees. In particular, we give conditions under which the Sample-Augment algorithms proposed by Gupta et al. [42] can be derandomized, thus obtaining the best known deterministic algorithms for a number of network design problems such as the connected facility location, virtual private network design and single sink buy-at-bulk problems. We also give deterministic variants of the “pivoting ” algorithms proposed by Ailon et al. [4] for several ranking and clustering problems. In addition to obtaining the same performance guarantees, the analysis of our algorithms is actually simpler than that of their randomized counterparts. Finally, we take a more practical approach to one of the ranking problems considered: the rank aggregation problem. We perform an extensive evaluation of several known and new algorithms for rank aggregation on web search data. We argue that there are two important classes of algorithms for rank aggregation: positional methods and comparison sort methods. We find that hybrid algorithms, that combine a positional and comparison sort approach, work especially well on our data sets.

Abstract. We consider a general class of non-cooperative buy-at-bulk cost sharing games, in which k players must contribute to purchase a number of resources. The resources have costs and must be paid for to be available to players. Each player can specify payments and has a certain constraint on the number and types of resources that she needs to have available. She strives to fulfill this constraint with the smallest investment possible. Our model includes a natural economy of scale: for a subset of players, capacity must be installed at the resources. The cost increase for larger sets of players is composed of a fixed price c(r) for each resource r and a global concave capacity function g. This cost can be shared arbitrarily between players. We consider the quality and existence of pure-strategy exact and approximate Nash equilibria. In general, prices of anarchy and stability depend heavily on the economy of scale and are Θ(k/g(k)). For non-linear functions g pure Nash equilibria might not exist and deciding their existence is NPhard. For subclasses of games corresponding to covering problems, primal-dual methods can be applied to derive cheap and stable approximate Nash equilibria in polynomial time. In addition, for singleton games optimal Nash equilibria exist. In this case expensive exact as well as cheap approximate Nash equilibria can be computed in polynomial time. Some of our results can be extended to games based on facility location problems. 1

Abstract. For several NP-hard network design problems, the best known approximation algorithms are remarkably simple randomized algorithms called Sample-Augment algorithms in [11]. The algorithms draw a random sample from the input, solve a certain subproblem on the random sample, and augment the solution for the subproblem to a solution for the original problem. We give a general framework that allows us to derandomize most Sample-Augment algorithms, i.e. to specify a specific sample for which the cost of the solution created by the Sample-Augment algorithm is at most a constant factor away from optimal. Our approach allows us to give deterministic versions of the Sample-Augment algorithms for the connected facility location problem, in which the open facilities need to be connected by either a tree or a tour, the virtual private network design problem, 2-stage rooted stochastic Steiner tree problem with independent decisions, the a priori traveling salesman problem and the single sink buy-at-bulk problem. This partially answers an open question posed in Gupta et al. [11]. 1

We consider robust network design problems where the set of feasible demands may be given by an arbitrary polytope or convex body more generally. This model, introduced by Ben-Ameur and Kerivin [2], generalizes the well studied virtual private network (VPN) problem. Most research in this area has focused on finding constant factor approximations for specific polytope of demands, such as the class of hose matrices used in the definition of VPN. As pointed out in [4], however, the general problem was only known to be APX-hard (based on a reduction from the Steiner tree problem). We show that the general robust design is hard to approximate to within polylogarithmic factors. We establish this by showing a general reduction of buy-at-bulk network design to the robust network design problem. In the second part of the paper, we introduce a natural generalization of the VPN problem. In this model, the set of feasible demands is determined by a tree with edge capacities; a demand matrix is feasible if it can be routed on the tree. We give a constant factor approximation algorithm for this problem that achieves factor 8 in general, and 2 for the case where the tree has unit capacities.

Abstract—We consider the problem of constructing a single spanning tree for the single-sink buy-at-bulk network design problem for doubling-dimension graphs. We compute a spanning tree to route a set of demands along a graph G to or from a designated sink node. The demands could be aggregated at (or symmetrically distributed to) intermediate edges where the fusion-cost is specified by a non-negative concave function f. We describe a novel approach for developing an oblivious spanning tree in the sense that it is independent of the number and location of data sources (or demands) and cost function at the edges. We present a deterministic, polynomial-time algorithm for constructing a spanning tree in low doubling-dimension graphs that guarantees a log 3 D-approximation over the optimal cost, where D is the diameter of the graph G. With a constant fusion-cost function, our spanning tree gives a O(log 3 D)-approximation for every Steiner tree that includes the sink. We also provide a Ω(logn) lower-bound for any oblivious tree in low doubling-dimension graphs. To our knowledge, this is the first paper to propose a single spanning tree solution to the single-sink buy-at-bulk network design problem (as opposed to multiple overlay trees).

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This thesis presents approximation algorithms for some N P-Hard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widely-believed complexity-theoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomial-time) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N P-Hard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing fault-tolerant networks. Two vertices u, v in a network are said to be k-edge-connected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are k-vertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning