Disjoint bases in a polymatroid

Let f : 2^N → ⁺ be a polymatroid (an integer‐valued non‐decreasing submodular set function with f(??) = 0). We call S ? N a base if f(S) = f(N). We consider the problem of finding a maximum number of disjoint bases; we denote by m* be this base packing number. A simple upper bound on m* is given by k* = max{k : Σ_iεNf_A(i) ≥ kf_A(N),?A ? N} where f_A(S) = f(A ∪ S) ‐ f(A). This upper bound is a natural generalization of the bound for matroids where it is known that m* = k*. For polymatroids, we prove that m* ≥ (1 ? o(1))k*/lnf(N) and give a randomized polynomial time algorithm to find (1 ? o(1))k*/lnf(N) disjoint bases, assuming an oracle for f. We also derandomize the algorithm using minwise independent permutations and give a deterministic algorithm that finds (1 ? ε)k*/lnf(N) disjoint bases. The bound we obtain is almost tight because it is known there are polymatroids for which m* < (1 + o(1))k*/lnf(N). Moreover it is known that unless NP ? DTIME(n^{log log n}), for any ε > 0, there is no polynomial time algorithm to obtain a (1 + ε)/lnf(N)‐approximation to m*. Our result generalizes and unifies two results in the literature. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

An Efficient Approximation Algorithm for Minimizing Makespan on Uniformly Related Machines

We give a new and efficient approximation algorithm for scheduling precedence-constrained jobs on machines with different speeds. The problem is as follows. We are given n jobs to be scheduled on a set of m machines. Jobs have processing times and machines have speeds. It takes p_j/s_i units of time for machine i with speed s_i to process job j with processing requirement p_j. Precedence constraints between jobs are given in the form of a partial order. If j k, processing of job k cannot start until job j's execution is completed. The objective is to find a non-preemptive schedule to minimize the makespan of the schedule. Chudak and Shmoys (1997, “Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA),” pp. 581–590) gave an algorithm with an approximation ratio of O(log m), significantly improving the earlier ratio of due to Jaffe (1980, Theoret. Comput. Sci.26, 1–17). Their algorithm is based on solving a linear programming relaxation. Building on some of their ideas, we present a combinatorial algorithm that achieves a similar approximation ratio but runs in O(n³) time. Our algorithm is based on a new and simple lower bound which we believe is of independent interest.     

On average throughput and alphabet size in network coding

We examine the throughput benefits that network coding offers with respect to the average throughput achievable by routing, where the average throughput refers to the average of the rates that the individual receivers experience. We relate these benefits to the integrality gap of a standard linear programming formulation for the directed Steiner tree problem. We describe families of configurations over which network coding at most doubles the average throughput, and analyze a class of directed graph configurations with N receivers where network coding offers benefits proportional to /spl radic/N. We also discuss other throughput measures in networks, and show how in certain classes of networks, average throughput bounds can be translated into minimum throughput bounds, by employing vector routing and channel coding. Finally, we show configurations where use of randomized coding may require an alphabet size exponentially larger than the minimum alphabet size required.     

Non-Cooperative Multicast and Facility Location Games

We consider a multicast game with selfish non- cooperative players. There is a special source node and each player is interested in connecting to the source by making a routing decision that minimizes its payment. The mutual influence of the players is determined by a cost sharing mechanism, which in our case evenly splits the cost of an edge among the players using it. We consider two different models: an integral model, where each player connects to the source by choosing a single path, and a fractional model, where a player is allowed to split the flow it receives from the source between several paths. In both models we explore the overhead incurred in network cost due to the selfish behavior of the users, as well as the computational complexity of finding a Nash equilibrium. The existence of a Nash equilibrium for the integral model was previously established by the means of a potential function. We prove that finding a Nash equilibrium that minimizes the potential function is NP-hard. We focus on the price of anarchy of a Nash equilibrium resulting from the best-response dynamics of a game course, where the players join the game sequentially. For a game with in players, we establish an upper bound of O(radicnlog² n) on the price of anarchy, and a lower bound of Omega(log n/log log n). For the fractional model, we prove the existence of a Nash equilibrium via a potential function and give a polynomial time algorithm for computing an equilibrium that minimizes the potential function. Finally, we consider a weighted extension of the multicast game, and prove that in the fractional model, the game always has a Nash equilibrium.     

Routing bandwidth guaranteed paths with local restoration in label switched networks

The emerging multiprotocol label switching (MPLS) networks enable network service providers to route bandwidth guaranteed paths between customer sites. This basic label switched path (LSP) routing is often enhanced using restoration routing which sets up alternate LSPs to guarantee uninterrupted connectivity in case network links or nodes along primary path fail. We address the problem of distributed routing of restoration paths, which can be defined as follows: given a request for a bandwidth guaranteed LSP between two nodes, find a primary LSP, and a set of backup LSPs that protect the links along the primary LSP. A routing algorithm that computes these paths must optimize the restoration latency and the amount of bandwidth used. We introduce the concept of "backtracking" to bound the restoration latency. We consider three different cases characterized by a parameter called backtracking distance D: 1) no backtracking (D=0); 2) limited backtracking (D=k); and 3) unlimited backtracking (D=/spl infin/). We use a link cost model that captures bandwidth sharing among links using various types of aggregate link-state information. We first show that joint optimization of primary and backup paths is NP-hard in all cases. We then consider algorithms that compute primary and backup paths in two separate steps. Using link cost metrics that capture bandwidth sharing, we devise heuristics for each case. Our simulation study shows that these algorithms offer a way to tradeoff bandwidth to meet a range of restoration latency requirements.     

A greedy approximation algorithm for the group Steiner problem

In the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices . Each subset g_i is called a group and the vertices in ?_ig_i are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265-285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73-91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59-63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε>0, our algorithm gives an approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(max_i|g_i|)·logm) provided by the LP based approaches.