Chaitanya Swamy

Facility location with client latencies: linear programming based techniques for minimum latency problems

We introduce a problem that is a common generalization of the uncapacitated facility location and minimum latency (ML) problems, where facilities need to be opened to serve clients and also need to be sequentially activated before they can provide service. Formally, we are given a set F of n facilities with facility-opening costs fi, a set D of m clients, connection costs cij specifying the cost of assigning a client j to a facility i, a root node r denoting the depot, and a time metric d on F∪{r}. Our goal is to open a subset F of facilities, find a path P starting at r and spanning F to activate the open facilities, and connect each client j to a facility φ(j) e F, so as to minimize Σi∈Ffi + Σj∈D(Cφ(j),j +tj), where tj is the time taken to reach φ(j) along path P. We call this the minimum latency uncapacitated facility location (MLUFL) problem. n nOur main result is an O(log n ċ max(log n, logm))-approximation for MLUFL. We also show that any improvement in this approximation guarantee, implies an improvement in the (current-best) approximation factor for group Steiner tree. We obtain constant approximations for two natural special cases of the problem: (a) related MLUFL (metric connection costs that are a scalar multiple of the time metric); (b) metric uniform MLUFL (metric connection costs, uniform time-metric). Our LP-based methods are versatile and easily adapted to yield approximation guarantees for MLUFL in various more general settings, such as (i) when the latency-cost of a client is a function of the delay faced by the facility to which it is connected; and (ii) the k-route version, where k vehicles are routed in parallel to activate the open facilities. Our LP-based understanding of MLUFL also offers some LP-based insights into ML, which we believe is a promising direction for obtaining improvements for ML.

Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications

We consider the matroid median problem [Krishnaswamy et al. 2011], wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation in Krishnaswamy et al. [2011]. We illustrate the power and versatility of our techniques by deriving (a) an 8-approximation for the two-matroid median problem, a generalization of matroid median that we introduce involving two matroids; and (b) a 24-approximation algorithm for matroid median with penalties, which is a vast improvement over the 360-approximation obtained in Krishnaswamy et al. [2011]. We show that a variety of seemingly disparate facility-location problems considered in the literature—data placement problem, mobile facility location, k-median forest, metric uniform minimum-latency Uncapacitated Facility Location (UFL)—in fact reduce to the matroid median or two-matroid median problems, and thus obtain improved approximation guarantees for all these problems. Our techniques also yield an improvement for the knapsack median problem.

Hardness Results for Signaling in Bayesian Zero-Sum and Network Routing Games

We study the optimization problem faced by a perfectly informed principal in a Bayesian game, who reveals information to the players about the state of nature to obtain a desirable equilibrium. This signaling problem is the natural design question motivated by uncertainty in games and has attracted much recent attention. We present new hardness results for signaling problems in (a) Bayesian two-player zero-sum games, and (b) Bayesian network routing games. For Bayesian zero-sum games, when the principal seeks to maximize the equilibrium utility of a player, we show that it is NP-hard to obtain an additive FPTAS. Our hardness proof exploits duality and the equivalence of separation and optimization in a novel way. Further, we rule out an additive PTAS assuming planted clique hardness, which states that no polynomial time algorithm can recover a planted clique from an ErdH{o}s-Renyi random graph. Complementing these, we obtain a PTAS for a structured class of zero-sum games (where obtaining an FPTAS is still NP-hard) when the payoff matrices obey a Lipschitz condition. Previous results ruled out an FPTAS assuming planted-clique hardness, and a PTAS only for implicit games with quasi-polynomial-size strategy sets. For Bayesian network routing games, wherein the principal seeks to minimize the average latency of the Nash flow, we show that it is NP-hard to obtain a (multiplicative) (4/3 - epsilon)-approximation, even for linear latency functions. This is the optimal inapproximability result for linear latencies, since we show that full revelation achieves a (4/3)-approximation for linear latencies.

Approximating Min-Cost Chain-Constrained Spanning Trees: A Reduction from Weighted to Unweighted Problems

We study the min-cost chain-constrained spanning-tree abbreviated MCCST problem: find a min-cost spanning tree in a graph subject to degree constraints on a nested family of node sets. We devise the first polytime algorithm that finds a spanning tree that i violates the degree constraints by at most a constant factor and ii whose cost is within a constant factor of the optimum. Previously, only an algorithm for unweighted CCST was knowni¾ź[13], which satisfied i but did not yield any cost bounds. This also yields the first result that obtains an O1-factor for both the cost approximation and violation of degree constraints for any spanning-tree problem with general degree bounds on node sets, where an edge participates in multiple degree constraints. n nA notable feature of our algorithm is that we reduce MCCST to unweighted CCST and then utilizei¾ź[13] via a novel application of Lagrangian duality to simplify the cost structure of the underlying problem and obtain a decomposition into certain uniform-cost subproblems. n nWe show that this Lagrangian-relaxation based idea is in fact applicable more generally and, for any cost-minimization problem with packing side-constraints, yields a reduction from the weighted to the unweighted problem. We believe that this reduction is of independent interest. As another application of our technique, we consider the k-budgeted matroid basis problem, where we build upon a recent rounding algorithm ofi¾ź[4] to obtain an improved

Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers

n^{Ok^{1.5}/epsilon }

Near-Optimal and Robust Mechanism Design for Covering Problems with Correlated Players

-time algorithm that returns a solution that satisfies any one of the budget constraints exactly and incurs a

On the Integrality Gap of the Prize-Collecting Steiner Forest LP

1+epsilon