Vinayaka Pandit

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.

Robust fingerprint authentication using local structural similarity

Fingerprint matching is challenging as the matcher has to minimize two competing error rates: the False Accept Rate and the False Reject Rate. We propose a novel, efficient, accurate and distortion-tolerant fingerprint authentication technique based on graph representation. Using the fingerprint minutiae features, a labeled, and weighted graph of minutiae is constructed for both the query fingerprint and the reference fingerprint. In the first phase, we obtain a minimum set of matched node pairs by matching their neighborhood structures. In the second phase, we include more pairs in the match by comparing distances with respect to matched pairs obtained in first phase. An optional third phase, extending the neighborhood around each feature, is entered if we cannot arrive at a decision based on the analysis in first two phases. The proposed algorithm has been tested with excellent results on a large private livescan database obtained with optical scanners.

Decision trees for entity identification: approximation algorithms and hardness results

We consider the problem of constructing decision trees for entity identification from a given relational table. The input is a table containing information about a set of entities over a fixed set of attributes and a probability distribution over the set of entities that specifies the likelihood of the occurrence of each entity. The goal is to construct a decision tree that identifies each entity unambiguously by testing the attribute values such that the average number of tests is minimized. This classical problem finds such diverse applications as efficient fault detection, species identification in biology, and efficient diagnosis in the field of medicine. Prior work mainly deals with the special case where the input table is binary and the probability distribution over the set of entities is uniform. We study the general problem involving arbitrary input tables and arbitrary probability distributions over the set of entities. We consider a natural greedy algorithm and prove an approximation guarantee of O(rK • log N), where N is the number of entities and K is the maximum number of distinct values of an attribute. The value rK is a suitably defined Ramsey number, which is at most log K. We show that it is NP-hard to approximate the problem within a factor of Ω(log N), even for binary tables (i.e. K=2). Thus, for the case of binary tables, our approximation algorithm is optimal up to constant factors (since r2=2). In addition, our analysis indicates a possible way of resolving a Ramsey-theoretic conjecture by Erdos.

Analysis of sampling techniques for association rule mining

In this paper, we present a comprehensive theoretical analysis of the sampling technique for the association rule mining problem. Most of the previous works have concentrated only on the empirical evaluation of the effectiveness of sampling for the step of finding frequent itemsets. To the best of our knowledge, a theoretical framework to analyze the quality of the solutions obtained by sampling has not been studied. Our contributions are two-fold. First, we present the notions of ε-close frequent itemset mining and ε-close association rule mining that help assess the quality of the solutions obtained by sampling. Secondly, we show that both the frequent items mining and association rule mining problems can be solved satisfactorily with a sample size that is independent of both the number of transactions size and the number of items. Let θ be the required support, ε the closeness parameter, and 1/h the desired bound on the probability of failure. We show that the sampling based analysis succeeds in solving both ε-close frequent itemset mining and ε-close association rule mining with a probability of at least (1 - 1/h) with a sample of size S = O(1/ε2θ [Δ + log h/(1 - ε)θ]), where Δ is the maximum number of items present in any transaction. Thus, we establish that it is possible to speed up the entire process of association rule mining for massive databases by working with a small sample while retaining any desired degree of accuracy. Our work gives a comprehensive explanation for the well known empirical successes of sampling for association rule mining.

Order scheduling models: hardness and algorithms

We consider scheduling problems in which a job consists of components of different types to be processed on m machines. Each machine is capable of processing components of a single type. Different components of a job are independent and can be processed in parallel on different machines. A job is considered as completed only when all its components have been completed. We study both completion time and flowtime aspects of such problems. We show both lowerbounds and upperbounds for the completion time problem. We first show that even the unweighted completion time with single release date is MAX-SNP hard. We give an approximation algorithm based on linear programming which has an approximation ratio of 3 for weighted completion time with multiple release dates. We give online algorithms for the weighted completion time which are constant factor competitive. For the flowtime, we give only lowerbounds in both the offline and online settings. We show that it is NP-hard to approximate flowtime within Ω(log m) in the offline setting. We show that no online algorithm for the flowtime can have a competitive ratio better than Ω(√m).

Online sorting buffers on line

We consider the online scheduling problem for sorting buffers on a line metric, motivated by an application to disc scheduling. Input is an online sequence of requests. Each request is a block of data to be written on a specified track of the disc. To write a block on a particular track, the scheduler has to bring the disc head to that track. The cost of moving the disc head from a track to another is the distance between those tracks. A sorting buffer that can store at most k requests at a time is available to the scheduler. This buffer can be used to rearrange the input sequence. The objective is to minimize the total cost of head movement while serving the requests. On a disc with n uniformly-spaced tracks, we give a randomized online algorithm with a competitive ratio of O(log2n) in expectation against an oblivious adversary. We show that any deterministic strategy which makes scheduling decisions based only on the contents of the buffer has a competitive ratio of Ω(k) or Ω(log n/loglog n).

Bandwidth Maximization in Multicasting

We formulate bandwidth maximization problems in multicasting streaming data. Multicasting is used to stream data to many terminals simultaneously. The goal here is to maximize the bandwidth at which the data can be transmitted satisfying the capacity constraints on the links. A typical network consists of the end-hosts which are capable of duplicating data instantaneously, and the routers which can only forward the data. We show that if one insists that all the data to a terminal should travel along a single path, then it is NP-hard to approximate the maximum bandwidth to a factor better than 2. We also present a fast 2-approximation algorithm. If different parts of the data to a terminal can travel along different paths, the problem can be approximated to the same factor as the minimum Steiner tree problem on undirected graphs. We also prove that in case of a tree network, both versions of the bandwidth maximization problem can be solved optimally in polynomial time. Of independent interest is our result that the minimum Steiner tree problem on tree-metrics can be solved in polynomial time.

Approximation Algorithms for Budget-Constrained Auctions

Recently there has been a surge of interest in auctions research triggered on the one hand by auctions of bandwidth and other public assets and on the other by the popularity of Internet auctions and the possibility of new auction formats enabled by e-commerce. Simultaneous auction of items is a popular auction format. We consider the problem of maximizing total revenue in the simultaneous auction of a set of items where the bidders have individual budget constraints. Each bidder is permitted to bid on all the items of his choice and specifies his budget constraint to the auctioneer, who must select bids to maximize the revenue while ensuring that no budget constraints are violated. We show that the problem of maximizing revenue is such a setting is NP-hard, and present a factor-1.62 approximation algorithm for it. We formulate the problem as an integer program and solve a linear relaxation to obtain a fractional optimal solution, which is then deterministically rounded to obtain an integer solution. We argue that the loss in revenue incurred by the rounding procedure is bounded by a factor of 1.62.

Price of anarchy, locality gap, and a network service provider game

In this paper, we define a network service provider game. We show that the price of anarchy of the defined game can be bounded by analyzing a local search heuristic for a related facility location problem called the k-facility location problem. As a result, we show that the k-facility location problem has a locality gap of 5. This result is of interest on its own. Our result gives evidence to the belief that the price of anarchy of certain games are related to analysis of local search heuristics.