MohammadHossein Bateni

Improved Approximation Algorithms for Prize-Collecting Steiner Tree and TSP

We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. Path-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem. We present (2-epsilon)-approximation algorithms for all three problems, connected by a unified technique for improving prize-collecting algorithms that allows us to circumvent the integrality gap barrier.

MaxMin allocation via degree lower-bounded arborescences

We consider the problem of MaxMin allocation of indivisible goods. There are m items to be distributed among n players. Each player

Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth

i

Assignment problem in content distribution networks: Unsplittable hard-capacitated facility location

has a nonnegative valuation pij for an item j, and the goal is to allocate items to players so as to maximize the minimum total valuation received by each player. There is a large gap in our understanding of this problem. The best known positive result is an ~O(√ n)-approximation algorithm, while there is only a factor 2 hardness known. Better algorithms are known for the restricted assignment case where each item has exactly one nonzero value for the players. We study the effect of bounded degree for items: each item has a nonzero value for at most D players. We show that essentially the case D = 3 is equivalent to the general case, and give a 4-approximation algorithm for D = 2. The current algorithmic results for MaxMin Allocation are based on a complicated LP relaxation called the configuration LP. We present a much simpler LP which is equivalent in power to the configuration LP. We focus on a special case of MaxMin Allocation-a family of instances on which this LP has a polynomially large gap. The technical core of our result for this case comes from an algorithm for an interesting new optimization problem on directed graphs, MaxMinDegree Arborescence, where the goal is to produce an arborescence of large outdegree. We develop an nε-approximation for this problem that runs in nO(1/ε) time and obtain a a polylogarithmic approximation that runs in quasipolynomial time, using a lift-and-project inspired LP formulation. In fact, we show that our results imply a rounding algorithm for the relaxations obtained by t rounds of the Sherali-Adams hierarchy applied to a natural LP relaxation of the problem. Roughly speaking, the integrality gap of the relaxation obtained from t rounds of Sherali-Adams is at most n1/t. We are able to extend the latter result to a more general class of instances. Along the way, we prove a result about the existence of a perfect matching in a probabilistically pruned graph which may be of independent interest.

The cooperative game theory foundations of network bargaining games

We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prize-collecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Each subinstance has a relatively inexpensive Steiner tree connecting all its terminals, and the subinstances can be solved (almost) separately. Another building block is a PTAS for Steiner forest on graphs of bounded treewidth. Surprisingly, Steiner forest is NP-hard even on graphs of treewidth 3. Therefore, our PTAS for bounded-treewidth graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved in polynomial time for series-parallel graphs (graphs of treewidth at most two) by a novel combination of dynamic programming and minimum cut computations, completing our thorough complexity study of Steiner forest in the range of bounded-treewidth graphs, planar graphs, and bounded-genus graphs.

Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP

In a Content Distribution Network (CDN), there are m servers storing the data; each of them has a specific bandwidth. All the requests from a particular client should be assigned to one server because of the routing protocol used. The goal is to minimize the total cost of these assignments—cost of each is proportional to the distance between the client and the server as well as the request size—while the load on each server is kept below its bandwidth limit. When each server also has a setup cost, this is an unsplittable hard-capacitated facility location problem. As much attention as facility location problems have received, there has been no nontrivial approximation algorithm when we have hard capacities (i.e., there can only be one copy of each facility whose capacity cannot be violated) and demands are unsplittable (i.e., all the demand from a client has to be assigned to a single facility). We observe it is NP-hard to approximate the cost to within any bounded factor in this case. Thus, for an arbitrary constant ε>0, we relax the capacities to a 1+ε factor. For the case where capacities are almost uniform, we give a bicriteria O(log n, 1+ε)-approximation algorithm for general metrics and a (1+ε, 1+ε)-approximation algorithm for tree metrics. A bicriteria (α,β)-approximation algorithm produces a solution of cost at most α times the optimum, while violating the capacities by no more than a β factor. We can get the same guarantees for nonuniform capacities if we allow quasipolynomial running time. In our algorithm, some clients guess the facility they are assigned to, and facilities decide the size of the clients they serve. A straightforward approach results in exponential running time. When costs do not satisfy metricity, we show that a 1.5 violation of capacities is necessary to obtain any approximation. It is worth noting that our results generalize bin packing (zero connection costs and facility costs equal to one), knapsack (single facility with all costs being zero), minimum makespan scheduling for related machines (all connection costs being zero), and some facility location problems.

Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth

We study bargaining games between suppliers and manufacturers in a network context. Agents wish to enter into contracts in order to generate surplus which then must be divided among the participants. Potential contracts and their surplus are represented by weighted edges in our bipartite network. Each agent in the market is additionally limited by a capacity representing the number of contracts which he or she may undertake. When all agents are limited to just one contract each, prior research applied natural generalizations of the Nash bargaining solution to the networked setting, defined the new solution concepts of stable and balanced, and characterized the resulting bargaining outcomes. We simplify and generalize these results to a setting in which participants in only one side of the market are limited to one contract each. The core of our results uses a linear-programming formulation to establish a novel connection between well-studied cooperative game theory concepts and the solution concepts of core and prekernel defined for the bargaining games. This immediately implies one can take advantage of the results and algorithms in cooperative game theory to reproduce results such as those of Azar et al. [1] and Kleinberg and Tardos [28] and generalize them to our setting. The cooperative-game-theoretic connection also inspires us to refine our solution space using standard solution concepts from that literature such as nucleolus and lexicographic kernel. The nucleolus is particularly attractive as it is unique, always exists, and is supported by experimental data in the network bargaining literature. Guided by algorithms from cooperative game theory, we show how to compute the nucleolus by pruning and iteratively solving a natural linear-programming formulation.

Improved approximation algorithms for (budgeted) node-weighted steiner problems

We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. Path-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem. We present (2-epsilon)-approximation algorithms for all three problems, connected by a unified technique for improving prize-collecting algorithms that allows us to circumvent the integrality gap barrier.

Revenue Maximization for Selling Multiple Correlated Items

We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prize-collecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Each subinstance has a relatively inexpensive Steiner tree connecting all its terminals, and the subinstances can be solved (almost) separately. Another building block is a PTAS for Steiner forest on graphs of bounded treewidth. Surprisingly, Steiner forest is NP-hard even on graphs of treewidth 3. Therefore, our PTAS for bounded treewidth graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved in polynomial time for series-parallel graphs (graphs of treewidth at most two) by a novel combination of dynamic programming and minimum cut computations, completing our thorough complexity study of Steiner forest in the range of bounded treewidth graphs, planar graphs, and bounded genus graphs.

Approximation algorithms for the directed k-tour and k-stroll problems

Moss and Rabani [13] study constrained node-weighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an O(logn)-approximation algorithm for the prize-collecting node-weighted Steiner tree problem (PCST)--where the goal is to minimize the cost of a tree plus the penalty of vertices not covered by the tree. They use the algorithm for PCST to obtain a bicriteria (2, O(logn))-approximation algorithm for the Budgeted node-weighted Steiner tree problem--where the goal is to maximize the prize of a tree with a given budget for its cost. Their solution may cost up to twice the budget, but collects a factor