Kamesh Munagala

Local search heuristic for k-median and facility location problems

In this paper, we analyze local search heuristics for the k-median and facility location problems. We define the {\em locality gap\/} of a local search procedure as the maximum ratio of a locally optimum solution (obtained using this procedure) to the global optimum. For k-median, we show that local search with swaps has a locality gap of exactly 5. When we permit p facilities to be swapped simultaneously then the locality gap of the local search procedure is exactly 3+2/p. This is the first analysis of local search for k-median that provides a bounded performance guarantee with only k medians. This also improves the previous known 4 approximation for this problem. For Uncapacitated facility location, we show that local search, which permits adding, dropping and swapping a facility, has a locality gap of exactly 3. This improves the 5 bound of Korupolu et al. We also consider a capacitated facility location problem where each facilitym has a capacity and we are allowed to open multiple copies of a facility. For this problem we introduce a new operation which opens one or more copies of a facility and drops zero or more facilities. We prove that local search which permits this new operation has a locality gap between 3 and 4. instances where it is not necessary to satisfy every demand. Our algorithms provide the optimum total profit, while stretching the definition of locality by a constant and violating the required demands by a constant. We prove that without this stretch, the problem becomes NP-Hard to approximate. facility location, we show that local search, which permits adding, dropping and swapping a facility, has a locality gap of exactly 3. This improves the 5 bound of Korupolu et al. We also consider a capacitated facility location problem where each facilitym has a capacity and we are allowed to open multiple copies of a facility. For this problem we introduce a new operation which opens one or more copies of a facility and drops zero or more facilities. We prove that local search which permits this new operation has a locality gap between 3 and 4.

Local Search Heuristics for k -Median and Facility Location Problems

We analyze local search heuristics for the metric k-median and facility location problems. We define the locality gap of a local search procedure for a minimization problem as the maximum ratio of a locally optimum solution (obtained using this procedure) to the global optimum. For k-median, we show that local search with swaps has a locality gap of 5. Furthermore, if we permit up to p facilities to be swapped simultaneously, then the locality gap is 3+2/p. This is the first analysis of a local search for k-median that provides a bounded performance guarantee with only k medians. This also improves the previous known 4 approximation for this problem. For uncapacitated facility location, we show that local search, which permits adding, dropping, and swapping a facility, has a locality gap of 3. This improves the bound of 5 given by M. Korupolu, C. Plaxton, and R. Rajaraman [Analysis of a Local Search Heuristic for Facility Location Problems, Technical Report 98-30, DIMACS, 1998]. We also consider a capacitated facility location problem where each facility has a capacity and we are allowed to open multiple copies of a facility. For this problem we introduce a new local search operation which opens one or more copies of a facility and drops zero or more facilities. We prove that this local search has a locality gap between 3 and 4.

Adaptive ordering of pipelined stream filters

We consider the problem of pipelined filters, where a continuous stream of tuples is processed by a set of commutative filters. Pipelined filters are common in stream applications and capture a large class of multiway stream joins. We focus on the problem of ordering the filters adaptively to minimize processing cost in an environment where stream and filter characteristics vary unpredictably over time. Our core algorithm, A-Greedy (for Adaptive Greedy), has strong theoretical guarantees: If stream and filter characteristics were to stabilize, A-Greedy would converge to an ordering within a small constant factor of optimal. (In experiments A-Greedy usually converges to the optimal ordering.) One very important feature of A-Greedy is that it monitors and responds to selectivities that are correlated across filters (i.e., that are nonindependent), which provides the strong quality guarantee but incurs run-time overhead. We identify a three-way tradeoff among provable convergence to good orderings, run-time overhead, and speed of adaptivity. We develop a suite of variants of A-Greedy that lie at different points on this tradeoff spectrum. We have implemented all our algorithms in the STREAM prototype Data Stream Management System and a thorough performance evaluation is presented.

A Sampling-Based Approach to Optimizing Top-k Queries in Sensor Networks

Wireless sensor networks generate a vast amount of data. This data, however, must be sparingly extracted to conserve energy, usually the most precious resource in battery-powered sensors. When approximation is acceptable, a model-driven approach to query processing is effective in saving energy by avoiding contacting nodes whose values can be predicted or are unlikely to be in the result set. To optimize queries such as top-k, however, reasoning directly with models of joint probability distributions can be prohibitively expensive. Instead of using models explicitly, we propose to use samples of past sensor readings. Not only are such samples simple to maintain, but they are also computationally efficient to use in query optimization. With these samples, we can formulate the problem of optimizing approximate top-k queries under an energy constraint as a linear program. We demonstrate the power and flexibility of our sampling-based approach by developing a series of topk query planning algorithms with linear programming, which are capable of efficiently producing plans with better performance and novel features. We show that our approach is both theoretically sound and practically effective on simulated and real-world datasets.

Cost-distance: two metric network design

Presents the cost-distance problem, which consists of finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of source-sink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for the cost-distance problem, where k is the number of sources. We reduce several common network design problems to cost-distance problems, obtaining (in some cases) the first known logarithmic approximation for them. These problems include a single-sink buy-at-bulk problem with variable pipe types between different sets of nodes, facility location with buy-at-bulk-type costs on edges, constructing single-source multicast trees with good cost and delay properties, and multi-level facility location. Our algorithm is also easier to implement and significantly faster than previously known algorithms for buy-at-bulk design problems.

Optimization of continuous queries with shared expensive filters

We consider the problem of optimizing and executing multiple continuous queries, where each query is a conjunction of filters and each filter may occur in multiple queries. When filters are expensive, significant performance gains are achieved by sharing filter evaluations across queries. A shared execution strategy in our scenario can either be fixed, in which filters are evaluated in the same predetermined order for all input, or adaptive, in which the next filter to be evaluated is chosen at runtime based on the results of the filters evaluated so far. We show that as filter costs increase, the best adaptive strategy is superior to any fixed strategy, despite the overhead of adaptivity. We show that itis NP-hard to find the optimal adaptive strategy, even if we are willing to approximate within any factor smaller than m where m is the number of queries. We then present a greedy adaptive execution strategy and show that it approximates the best adaptive strategy to within a factor O(log2m log n) where n is the number of distinct filters. We also give a precomputation technique that can reduce the execution overhead of adaptive strategies.

A constant factor approximation for the single sink edge installation problems

We present the first constant approximation to the single sink buy-at-bulk network design problem, where we have to design a network by buying pipes of different costs and capacities per unit length to route demands at a set of sources to a single sink. The distances in the underlying network form a metric. This result improves the previous bound of O(\log |R|), where R is the set of sources. Our algorithms are combinatorial and can be derandomized easily at the cost of a constant factor loss in the approximation ratio.

Energy-efficient monitoring of extreme values in sensor networks

Monitoring extreme values (MAX or MIN) is a fundamental problem in wireless sensor networks (and in general, complex dynamic systems). This problem presents very different algorithmic challenges from aggregate and selection queries, in the sense that an individual node cannot by itself determine its inclusion in the query result. We present novel query processing algorithms for this problem, with the goal of minimizing message traffic in the network. These algorithms employ a hierarchy of local constraints, or thresholds, to leverage network topology such that message-passing is localized. We evaluate all algorithms using simulated and real-world data to study various trade-offs.

The pipelined set cover problem

A classical problem in query optimization is to find the optimal ordering of a set of possibly correlated selections. We provide an ion of this problem as a generalization of set cover called pipelined set cover, where the sets are applied sequentially to the elements to be covered and the elements covered at each stage are discarded. We show that several natural heuristics for this NP-hard problem, such as the greedy set-cover heuristic and a local-search heuristic, can be analyzed using a linear-programming framework. These heuristics lead to efficient algorithms for pipelined set cover that can be applied to order possibly correlated selections in conventional database systems as well as datastream processing systems. We use our linear-programming framework to show that the greedy and local-search algorithms are 4-approximations for pipelined set cover. We extend our analysis to minimize the l P -norm of the costs paid by the sets, where p > 2 is an integer, to examine the improvement in performance when the total cost has increasing contribution from initial sets in the pipeline. Finally, we consider the online version of pipelined set cover and present a competitive algorithm with a logarithmic performance guarantee. Our analysis framework may be applicable to other problems in query optimization where it is important to account for correlations.

Approximation algorithms for budgeted learning problems

We present the first approximation algorithms for a large class of budgeted learning problems. One classicexample of the above is the budgeted multi-armed bandit problem. In this problem each arm of the bandithas an unknown reward distribution on which a prior isspecified as input. The knowledge about the underlying distribution can be refined in the exploration phase by playing the arm and observing the rewards. However, there is a budget on the total number of plays allowed during exploration. After this exploration phase,the arm with the highest (posterior) expected reward is hosen for exploitation. The goal is to design the adaptive exploration phase subject to a budget constraint on the number of plays, in order to maximize the expected reward of the arm chosen for exploitation. While this problem is reasonably well understood in the infinite horizon discounted reward setting, the budgeted version of the problem is NP-Hard. For this problem and several generalizations, we provide approximate policies that achieve a reward within constant factor of the reward optimal policy. Our algorithms use a novel linear program rounding technique based on stochastic packing.