A brief summary of greedy set-cover algorithms.

INDEX TERMS: dominating set, greedy algorithm, hitting set, set cover, minimizing a linear function subject to a submodular constraint. ]]> 01 Jan 2008 12:00:00 GMT Papers/Young08SetCover.pdf Papers/Young08SetCover.pdf

Following Mettu and Plaxton, we study incremental algorithms for the $k$-medians problem. Such an algorithm must produce a nested sequence $F_1 \subseteq F_2 \subseteq \cdots \subseteq F_n$ of sets of facilities. Mettu and Plaxton show that incremental metric medians has a (roughly) $40$-competitive deterministic polynomial-time algorithm. We give improved algorithms, including a $(24+ε)$-competitive deterministic polynomial-time algorithm and a $5.44$-competitive, randomized, non-polynomial-time algorithm.

We also consider the competitive ratio with respect to size. An algorithm is $s$-size-competitive if, for each $k$, the cost of $F_k$ is at most the minimum cost of any set of $k$ facilities, while the size of $F_k$ is at most $s k$. We present optimally competitive algorithms for this problem.

Our proofs reduce incremental medians to the following online bidding problem: faced with some unknown threshold $T>0$, an algorithm must submit ``bids'' $b>0$ until it submits a bid as large as $T$. The algorithm pays the sum of its bids. We describe optimally competitive algorithms for online bidding.

Our results on cost-competitive incremental medians extend to approximately metric distance functions, incremental fractional medians, and incremental bicriteria approximation. ]]> 01 Jan 2008 12:00:00 GMT Papers/algorithmica_2008_50_4_455.pdf Papers/algorithmica_2008_50_4_455.pdf 01 Jan 2007 12:00:00 GMT Given a rooted tree whose leaves have numeric labels, label the nodes so that each leaf label equals the sum of the node labels from the root to the leaf. Minimize the number of non-zero node labels. This problem models finding a parsimonius explanation of observed changes in hierarchical data. We give fast algorithms with applications. ]]> 01 Jun 2007 12:00:00 GMT Papers/sigkdd_2007_6.pdf Papers/sigkdd_2007_6.pdf (Corrected version of FOCS 2007 paper).

We give an approximation algorithm for packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of 1+ε of the optimal cost in time O(n+(r+c)\log(n)/ε²).

For dense problems (with r,c=O($\sqrt n$) the time is O(n+$\sqrt n$ log(n)/ε²) --- linear even as ε tends to zero. In comparison, previous Lagrangian-relaxation algorithms generally take at least Ω(n log(n)/&epsilon²) time, while (for small ε) the Simplex algorithm typically takes at least Ω(n min(r,c)) time. ]]> 01 Oct 2007 12:00:00 GMT http://arxiv.org/abs/0801.1987 http://arxiv.org/abs/0801.1987 01 Jan 2007 12:00:00 GMT We study the problem of selecting control clones in DNA array hybridization experiments. The problem arises in the OFRG method for analyzing microbial communities. The OFRG method performs classification of rRNA gene clones using binary fingerprints created from a series of hybridization experiments, where each experiment consists of hybridizing a collection of arrayed clones with a single oligonucleotide probe. This experiment produces analog signals, one for each clone, which then need to be classified, that is, converted into binary values 1 and 0 that represent hybridization and non-hybridization events. In addition to the sample rRNA gene clones, the array contains a number of control clones needed to calibrate the classification procedure of the hybridization signals. These control clones must be selected with care to optimize the classification process. We formulate this as a combinatorial optimization problem called Balanced Covering. We prove that the problem is NP-hard, and we show some results on hardness of approximation. We propose approximation algorithms based on randomized rounding, and we show that, with high probability, our algorithms approximate well the optimum solution. The experimental results confirm that the algorithms find high quality control clones. The algorithms have been implemented and are publicly available as part of the software package called CloneTools. ]]> 01 Aug 2007 12:00:00 GMT Papers/jbcb_2007_5_4_937.pdf Papers/jbcb_2007_5_4_937.pdf The Reverse Greedy algorithm (RGreedy) for the $k$-median problem works as follows. It starts by placing facilities on all nodes. At each step, it removes a facility to minimize the resulting total distance from the customers to the remaining facilities. It stops when $k$ facilities remain. We prove that, if the distance function is metric, then the approximation ratio of RGreedy is between $Ω(\log n/ \log \log n)$ and $O(\log n)$. ]]> 01 Jan 2006 12:00:00 GMT Papers/ipl_2006_97_68.pdf Papers/ipl_2006_97_68.pdf This paper is a follow-up to Topology control to simultaneously achieve near-optimal node degree and low path stretch in ad hoc networks. It gives a related heuristic with more extensive empirical evaluation. ]]> 01 Oct 2006 12:00:00 GMT Papers/MASS_2006_139.pdf Papers/MASS_2006_139.pdf 01 Jan 2006 12:00:00 GMT Papers/lncs_2006_3887_311.pdf Papers/lncs_2006_3887_311.pdf 01 Sep 2006 12:00:00 GMT Papers/SECON_2006_431.pdf Papers/SECON_2006_431.pdf 01 Jan 2006 12:00:00 GMT http://www.cs.ucsb.edu/~teo/toc.pdf http://www.cs.ucsb.edu/~teo/toc.pdf Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of minimizing c$\cdot$x subject to x$\in Z^n_+$, Ax ≥ a, Bx ≤ b, and x ≤ d. We give a bicriteria-approximation algorithm that, given ε$\in$(0, 1], finds a solution of cost O(log(m)/ε²) times optimal, meeting the covering constraints Ax≥a and multiplicity constraints (x≤d), and satisfying Bx≤(1 + ε)b + β, where β is the vector of row sums, β_i=$\sum_j$ B_ij. Here m is the number of rows of A.

This gives an O(log m)-approximation algorithm for CIP --- minimum-cost covering integer programs with multiplicity constraints, i.e., the special case when there are no packing constraints Bx ≤ b. The previous best approximation ratio has been O(log max_j $\sum_i$A_ij since 1982. CIP contains the set cover problem as a special case, so O(log m)-approximation is the best possible unless P=NP. ]]> 01 Jan 2005 12:00:00 GMT Papers/jcss_2005_71_495.pdf Papers/jcss_2005_71_495.pdf 01 Jan 2005 12:00:00 GMT http://arxiv.org/abs/cs/0504104 http://arxiv.org/abs/cs/0504104 The multiway-cut problem is, given a weighted graph and $k > 2$ terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within $1+δ$ for some small $δ > 0$.

This paper gives a 12/11-approximation algorithm for $k=3$ and a 1.3438-approximation algorithm in general. Working on this result was particularly fun because the problem of designing the algorithm was itself formulated as an optimization problem and approximately solved by computer. These computations guided the final (non-computational) solutions.

The performance guarantees are the best currently known. The geometric relaxation is due to Gruia Calinescu, Howard Karloff, and Yuval Rabani [1998], who used it to give an algorithm with (slightly worse) performance guarantee $3/2-1/k$.

The paper also gives a proof that for a general class of ``geometric embedding'' relaxations, there are always randomized rounding schemes that match the integrality gap. (Another example of such a geometric-embedding relaxation is the semidefinite-programming relaxation of max-cut by Michel Goemans and David Williamson [1995].) ]]> 01 Jan 2004 12:00:00 GMT Papers/mathofOR_2004_29_3_436.pdf Papers/mathofOR_2004_29_3_436.pdf This paper describes a modification to the heuristic algorithm presented in ``Data collection for the Sloan Digital Sky Survey: a network-flow heuristic.'' The modification takes into account that galaxies that are too close together cannot be captured in the same snapshot. This was not considered in the original algorithm. ]]> 01 Jan 2003 12:00:00 GMT Papers/astronomical_2003_125_2276.pdf Papers/astronomical_2003_125_2276.pdf Congestion control in the current Internet is accomplished mainly by TCP/IP. To understand the macroscopic network behavior that results from TCP/IP and similar end-to-end protocols, one main analytic technique is to show that the the protocol maximizes some global objective function of the network traffic.

Here we analyze a particular end-to-end, MIMD (multiplicative-increase, multiplicative-decrease) protocol. We show that if all users of the network use the protocol, and all connections last for at least logarithmically many rounds, then the total weighted throughput (value of all packets received) is near the maximum possible. Our analysis includes round-trip-times, and (in contrast to most previous analyses) gives explicit convergence rates, allows connections to start and stop, and allows capacities to change.

An application explained in the paper is multi-path bandwidth measurement. ]]> 01 Jan 2002 12:00:00 GMT Papers/FOCS_2002_303.pdf Papers/FOCS_2002_303.pdf 01 Jan 2002 12:00:00 GMT Papers/STOC_2002_785.pdf Papers/STOC_2002_785.pdf This paper introduced a new file-caching algorithm, called Landlord or Greedy-Dual-size, for use by web proxies and browsers. The paper gave theoretical analyses of the algorithm suggesting that it would perform well in practice.

A modified version of the algorithm is incorporated into the public-domain Squid web proxy.

The algorithm was independently obtained by Pei Cao and Sandy Irani [1997]. Both they and John Dilley et al [1999] found that the algorithm worked well empirically.

The results in this paper strengthen and generalize those in [1994] and are further generalized in [2009]. ]]> 01 Jan 2002 12:00:00 GMT Papers/algorithmica_2002_33_371.pdf Papers/algorithmica_2002_33_371.pdf Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients. Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set cover, bin-packing, knapsack, scheduling problems, minimum-weight triangulation, etc. This paper gives approximation algorithms for the general class of problems. The sequential algorithm can be implemented to find an $(1\pmε)$-approximate solution in $O(ε^{-2}\log m)$ linear-time iterations.

For $ε = O(1)$, these algorithms are currently the fastest known for the general problem. The results generalize previous work on pure packing and covering (the special case when the constraints are all ``less-than'' or all ``greater-than'') by Michael Luby and Noam Nisan [1993]; and Naveen Garg and Jochen Konemann [1998]; and Lisa Fleischer [1999] ]]> 01 Jan 2001 12:00:00 GMT Papers/FOCS_2001_538.pdf Papers/FOCS_2001_538.pdf 01 Jan 2001 12:00:00 GMT Papers/FOCS_2001_522.pdf Papers/FOCS_2001_522.pdf The paper gives approximation algorithms for the $k$-medians and facility-location problems (both NP-hard). For $k$-medians, the algorithm returns a solution using at most $\ln(n+n/ε) k$ medians and having cost at most $(1+ε)$ times the cost of the best solution that uses at most $k$ medians. Here $ε>0$ is an input to the algorithm. In comparison, the best previous algorithm [Lin and Jeff Vitter, 1992] had a $(1+1/ε)\ln n$ term instead of the $\ln(n+n/ε)$ term in the performance guarantee. For facility location, the algorithm returns a solution of cost at most $d+\ln(n) k$, provided there exists a solution of cost $d+k$ where $d$ is the assignment cost and $k$ is the facility cost. In comparison, the best previous algorithm [Dorit Hochbaum, 1982] returned a solution of cost at most $\ln(n)(d+k)$. For both problems, the algorithms currently provide the best performance guarantee known for the general (non-metric) problems.

The paper also introduces a new probabilistic bound (the so-called Chernoff-Wald bound) for bounding the expectation of the maximum of a collection of sums of random variables, when each sum contains a random number of terms. The bound is used to analyze the randomized rounding scheme that underlies the algorithms. ]]> 01 Jan 2000 12:00:00 GMT Papers/SODA_2000_86.pdf Papers/SODA_2000_86.pdf Elias Koutsoupias and Christos Papadimitriou introduced the so-called diffuse-adversary model for analyzing paging strategies such as least-recently-used (LRU) [1994]. Briefly, the diffuse adversary model is as follows. Fix some $ε >0$ Run the algorithm, and, for each request, allow the adversary to assign a probability distribution to the items such that no item gets probability more than $ε$, then request an item randomly from the distribution. (When $ε$ is near 1, one has the standard competitive-analysis model; when $ε$ is near 0, the adversary is severely handicapped.) The algorithm is $c$-competitive if the expected cost of the algorithm on the adversarial sequence is at most $c$ times the expected cost of the optimal algorithm.

Koutsoupias and Papadimitriou proved that LRU was optimally competitive (among deterministic on-line algorithms) according to the model, but left open the question of what the performance guarantee for LRU actually was. This paper answers the question, giving an explicit formula for the performance guarantee within a factor of two. The paper also shows that the performance guarantee of first-in-first-out (FIFO) is much worse than that of LRU, and analyzes the performance guarantees of randomized on-line paging strategies. ]]> 01 Jan 2000 12:00:00 GMT Papers/algorithms_2000_37_218.pdf Papers/algorithms_2000_37_218.pdf The data broadcast problem is to find a schedule for broadcasting a given set of messages over multiple channels. The goal is to minimize the cost of the broadcast plus the expected response time to clients who periodically and probabilistically tune in to wait for particular messages.

The problem models disseminating data to clients in asymmetric communication environments, where there is a much larger capacity from the information source to the clients than in the reverse direction. Examples include satellites, cable TV, Internet broadcast, and mobile phones. Such environments favor the ``push-based'' model where the server broadcasts (pushes) its information on the communication medium and multiple clients simultaneously retrieve the specific information of individual interest.

This paper presents the first polynomial-time approximation scheme (PTAS) for data broadcast with $O(1)$ channels and when each message has arbitrary probability, unit length and bounded cost. The best previous polynomial-time approximation algorithm for this case has a performance ratio of 9/8 [Bar-Noy et al, 2000]. ]]> 01 Jan 2000 12:00:00 GMT Papers/STOC_2000_659.pdf Papers/STOC_2000_659.pdf This paper gives a lower bound on the complexity of (1$\pm$ε)-approximately solving a packing or covering problem using a certain class of Lagrangian-relaxation algorithms: any such algorithm, given a problem formed by a random {0,1}-matrix, requires (with high probability) a number of iterations proportional to ($\rho$ε)² log m. (Here $\rho$ is a technical parameter, the``width'' of the problem instance.)

The class of algorithms in question includes Dantzig-Wolfe decomposition, Benders' decomposition, the Lagrangian-relaxation method developed by Held and Karp [1971] for lower-bounding TSP, and algorithms recently studied by many authors (including Serge Plotkin, David Shmoys, and Eva Tardos [1988]; Mike Grigoriadis and L.G. Khachiyan [1996]; Michael Luby and Noam Nisan [1993]; Naveen Garg and Jochen Konemann [1998]; and Lisa Fleischer [1999]). The lower bound matches the known upper bounds within a constant factor. The lower bound is useful because, in practice, the dependence on $ε$ limits the applicability of these algorithms. The lower bound provides some insight into what is necessary to surmount this dependence. ]]> 01 Jan 1999 12:00:00 GMT Papers/lncs_1610_320.pdf Papers/lncs_1610_320.pdf A 40-page sketch of basic techniques in the design and analysis of combinatorial approximation algorithms. Sections: Introduction, Underlying principles, Approximation algorithms with small additive error, Performance guarantees, Randomized rounding and linear programming, Performance ratios and c-approximation, Polynomial approximation schemes, Constant-factor performance guarantees, Logarithmic performance guarantees, Multi-criteria problems, Hard-to-approximate problems, Research Issues and Summary, Defining Terms, Further Information. ]]> 01 Jan 1999 12:00:00 GMT Papers/CRC_1999_ch34.pdf Papers/CRC_1999_ch34.pdf Two common objectives for evaluating a schedule are the makespan, or schedule length, and the average completion time. This short note gives improved bounds on the existence of schedules that simultaneously optimize both criteria. In particular, for any ρ > 0, there exists a schedule of makespan at most 1+ρ times the minimum, with average completion time at most 1/(1-e^-ρ) times the minimum. The proof uses an infinite-dimensional linear program to generalize and strengthen a previous analysis by Cliff Stein and Joel Wein [1997]. ]]> 01 Jan 1999 12:00:00 GMT Papers/SODA_1999_846.pdf Papers/SODA_1999_846.pdf 01 Jan 1999 12:00:00 GMT The goal of the Sloan Digital Sky Survey is ``to map in detail one-quarter of the entire sky, determining the positions and absolute brightnesses of more than 100 million celestial objects'' [1]. The survey will be performed by taking ``snapshots'' through a large telescope. Each snapshot can capture up to 600 objects from a small circle of the sky (as illustrated here). This paper describes the design and implementation of the algorithm that is being used to determine the snapshots so as to minimize their number. The problem is NP-hard in general; the algorithm described is a heuristic, based on Lagrangian-relaxation and min-cost network flow. It gets within 5-15% of a naive lower bound, whereas using a ``uniform'' cover only gets within 25-35%. ]]> 01 Jan 1998 12:00:00 GMT Papers/algorithms_1998_27_339.pdf Papers/algorithms_1998_27_339.pdf 01 Jan 1998 12:00:00 GMT Papers/SODA_1998_82.pdf Papers/SODA_1998_82.pdf 01 Jan 1998 12:00:00 GMT Papers/SODA_1998_82.pdf Papers/SODA_1998_82.pdf The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs $(x,y)$ such that there is a directed path from $x$ to $y$, and (ii) how to orient the edges so as to minimize the number of such pairs. The paper describes a quadratic-time algorithm for the first problem, and a proof that the second problem is NP-hard to approximate within some constant $1+ε > 1$. The latter proof also shows that the second problem is equivalent to ``comparability graph completion''; neither problem was previously known to be NP-hard. ]]> 01 Jan 1997 12:00:00 GMT Papers/ipl_1997_63_229.pdf Papers/ipl_1997_63_229.pdf The problem considered is the following. Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, compute a low-weight spanning tree such that the degree of each vertex is at most its specified bound. The problem is NP-hard (it generalizes Traveling Salesman (TSP)). The paper describes a network-flow heuristic for modifying a given tree T to meet the constraints. Choosing T to be a minimum spanning tree (MST) yields approximation algorithms with performance guarantee less than 2 for the problem on geometric graphs with L_p-norms. The paper also describes a Euclidean graph whose minimum TSP costs twice the MST, disproving a conjecture made in [Low-degree spanning trees of small weight]. ]]> 01 Jan 1997 12:00:00 GMT Papers/algorithms_1997_24_310.pdf Papers/algorithms_1997_24_310.pdf Pattern-matching-based document-compression systems (e.g. for faxing) rely on finding a small set of patterns that can be used to represent all of the ink in the document. Finding an optimal set of patterns is NP-hard; previous compression schemes have resorted to heuristics. This paper describes an extension of the cross-entropy approach, used previously for measuring pattern similarity, to this problem. This approach reduces the problem to a k-medians problem, for which the paper gives a new algorithm with a provably good performance guarantee. In comparison to previous heuristics (First Fit, with and without generalized Lloyd's/k-means post-processing steps), the new algorithm generates a better codebook, resulting in an overall improvement in compression performance of almost 17%. ]]> 01 Jan 1997 12:00:00 GMT Papers/DCC_1997_300.pdf Papers/DCC_1997_300.pdf 01 Jan 1996 12:00:00 GMT Papers/SODA_1996_296.pdf Papers/SODA_1996_296.pdf 1994]. To summarize: ``Our results suggest that the data structure is practical and can be faster than traditional priority queues when holding a large number of keys, and that tolerance for approximate answers can lead to significant increases in speed.'' ]]> 01 Jan 1996 12:00:00 GMT Papers/approx-VEB-impl.pdf Papers/approx-VEB-impl.pdf N.J.A. Sloane maintains an on-line encyclopedia of interesting integer sequences. This paper presents an operation for modifying such a sequence, and analyzes the operation using generating functions. If you are interested, search for the word ``boustrophedon'' at [1] or go here. ]]> 01 Jan 1996 12:00:00 GMT Papers/combthyA_76_1_44.pdf Papers/combthyA_76_1_44.pdf The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most $d$. When $d=2$ the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation algorithm (assuming the edge weights satisfy the triangle inequality).

Christos Papadimitriou and Umesh Vazirani posed the challenge of finding an algorithm with performance guarantee less than 2 for Euclidean graphs (points in $R^n$) and $d > 2$ [1984]. This paper gives the first answer to that challenge, presenting an algorithm to compute a degree-3 spanning tree of cost at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2 and the algorithm can also find a degree-4 spanning tree of cost at most 5/4 times the MST. ]]> 01 Jan 1996 12:00:00 GMT Papers/sicomp_25_2_355.pdf Papers/sicomp_25_2_355.pdf Describes a near-linear-time algorithm for a variant of Huffman coding, in which the letters may have non-uniform lengths (as in Morse code), but with the restriction that each word to be encoded has equal probability.

See also [Huffman coding with unequal letter costs]. ]]> 01 Jan 1996 12:00:00 GMT Papers/sicomp_25_6_1281.pdf Papers/sicomp_25_6_1281.pdf The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard; This paper gives a proof that, for graphs where each directed cycle has at most three edges, the MEG problem is equivalent to maximum bipartite matching, and therefore solvable in polynomial time. This leads to an improvement in the performance guarantee of the previously best approximation algorithm for the general problem, from ``Approximating the minimum equivalent digraph'' [1995].

(This result was improved by Berman et al, WADS, 2009.) ]]> 01 Jan 1996 12:00:00 GMT Papers/dam_1996_69_281.pdf Papers/dam_1996_69_281.pdf The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives an approximation algorithm with performance guarantee of pi²/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of contracting long cycles.

(This result was strengthened slightly in ``On strongly connected digraphs with bounded cycle length'' [1996].) The analysis applies directly to 2-Exchange, a simple local improvement algorithm, showing that its performance guarantee is 1.75. It was further improved by Berman et al, WADS 2009.) ]]> 01 Jan 1995 12:00:00 GMT Papers/sicomp_24_4_859.pdf Papers/sicomp_24_4_859.pdf This paper has a simple linear-time algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and $ε > 0$, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most $1+ε$ times the shortest-path distance, and yet the total weight of the tree is at most $1+2/ε$ times the weight of a minimum spanning tree. This is the best trade-off possible. ]]> 01 Jan 1995 12:00:00 GMT Papers/algorithmica_1995_14_4_305.pdf Papers/algorithmica_1995_14_4_305.pdf Randomized rounding is a standard method, based on the probabilistic method, for designing combinatorial approximation algorithms. This paper introduces a variant of randomized rounding that can be used to derive Lagrangian-relaxation algorithms and greedy algorithms. This was surprising - the approximation algorithms derived previously by the method require first solving a linear program (or some other optimization problem). Indeed, in Prabhakar Raghavan's seminal paper introducing randomized rounding [1988], he says: "The time taken to solve the linear program relaxations of the integer programs dominates the net running time theoretically (and, most likely, in practice as well)."

The variant of randomized rounding described here avoids this bottleneck, at least for packing/covering-type problems. The resulting algorithms are Lagrangian-relaxation algorithms and greedy algorithms (in fact the greedy set-cover algorithm is an example), and are faster and simpler to implement than standard randomized-rounding algorithms. This approach gives a systematic, coherent, and comprehensive understanding of Lagrangian-relaxation algorithms for packing and covering linear programs.

The ideas introduced here are also used in ``Sequential and parallel algorithms for mixed packing and covering'' [2001], in ``K-medians, facility location, and the Chernoff-Wald bound'' [2000]), and in ``On the number of iterations for Dantzig-Wolfe optimization and packing-covering approximation algorithms'' [1999]. ]]> 01 Jan 1995 12:00:00 GMT Papers/SODA_1995_170.pdf Papers/SODA_1995_170.pdf 01 Jan 1994 12:00:00 GMT Papers/SODA_1994_177.pdf Papers/SODA_1994_177.pdf This paper explores the notion of approximate data structures, which return approximately correct answers to queries, but run faster than their exact counterparts. The paper describes approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure (min, max, successor, predecessor, and existence queries, as well as insertion and deletion), except that answers to queries are approximate. The variants support all operations in constant time provided the performance guarantee is 1+1/polylog n, and in O(log log n) time provided the performance guarantee is 1+1/poly(n), for n elements in the data structure.

Applications described include Prim's minimum-spanning-tree algorithm, Dijkstra's single-source shortest paths algorithm, and an on-line variant of Graham's convex hull algorithm. To obtain output which approximates the desired output with the performance guarantee tending to 1, Prim's algorithm requires only linear time, Dijkstra's algorithm requires O(m log log n) time, and the on-line variant of Graham's algorithm requires constant amortized time per operation. ]]> 01 Jan 1994 12:00:00 GMT Papers/SODA_1994_187.pdf Papers/SODA_1994_187.pdf 01 Jan 1994 12:00:00 GMT This paper has two results. The first is based on the surprising observation that the well-known ``least-recently-used'' paging algorithm and the ``balance'' algorithm for weighted caching are linear-programming primal-dual algorithms. This observation leads to a strategy (called ``Greedy-Dual'') that generalizes them both and has an optimal performance guarantee for weighted caching.

For the second set of results, the paper presents empirical studies of paging algorithms, documenting that in practice, on ``typical'' cache sizes and sequences, the performance of paging strategies are much better than their worst-case analyses in the standard model suggest. The paper then presents theoretical results that support and explain this. For example: on any input sequence, with almost all cache sizes, either the performance guarantee of least-recently-used is $O(\log k)$ or the fault rate (in an absolute sense) is insignificant.

These results are strengthened and/or generalized in [1998] and [ 2009]. ]]> 01 Jan 1994 12:00:00 GMT Papers/algorithmica_1994_11_6_525.pdf Papers/algorithmica_1994_11_6_525.pdf We consider the problem of choosing Euclidean points to maximize the sum of their weighted pairwise distances, when each point is constrained to a ball centered at the origin. We derive a dual minimization problem and show strong duality holds (i.e., the resulting upper bound is tight) when some locally optimal configuration of points is affinely independent. We sketch a polynomial time algorithm for finding a near-optimal set of points. ]]> 01 Jan 1994 12:00:00 GMT http://arxiv.org/abs/1007.0217 http://arxiv.org/abs/1007.0217 The traditional multi-commodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multi-commodity flow network-design problem: given a set of multi-commodity flow demands, find a network subject to certain constraints such that the commodities can be maximally routed.

This paper focuses on the case when the network is required to be a tree. The main result is an approximation algorithm for the case when the tree is required to be of constant degree. The algorithm reduces the problem to the minimum-weight balanced-separator problem; the performance guarantee of the algorithm is within a factor of 4 of the performance guarantee of the balanced-separator procedure. If Tom Leighton and Satish Rao's balanced-separator procedure [1988] is used, the performance guarantee is $O(\log n)$. This improves the $O(\log^2 n)$ approximation factor that is trivial to obtain by a direct application of the balanced-separator method. ]]> 01 Jan 1994 12:00:00 GMT Papers/ipl_1994_50_49.pdf Papers/ipl_1994_50_49.pdf Von Neumann's Min-Max Theorem guarantees that each player of a zero-sum matrix game has an optimal mixed strategy. This paper gives an elementary proof that each player has a near-optimal mixed strategy that chooses uniformly at random from a multi-set of pure strategies of size logarithmic in the number of pure strategies available to the opponent.

For exponentially large games, for which even representing an optimal mixed strategy can require exponential space, it follows that there are near-optimal, linear-size strategies. These strategies are easy to play and serve as small witnesses to the approximate value of the game.

As a corollary, it follows that every language has small ``hard'' multi-sets of inputs certifying that small circuits can't decide the language. For example, if SAT does not have polynomial-size circuits, then, for each $n$ and $c$, there is a set of $n^{O(c)}$ Boolean formulae of size $n$ such that no circuit of size $n^c$ (or algorithm running in time $n^c$) classifies more than two-thirds of the formulae successfully.

The "simple strategies" observation was generalized to Nash equilibrium by Lipton et al. in their paper Playing large games using simple strategies. ]]> 01 Jan 1994 12:00:00 GMT Papers/STOC_1994_734.pdf Papers/STOC_1994_734.pdf The paper describes a simple deterministic parallel/distributed $(2+ε)$-approximation algorithm for the minimum-weight vertex-cover problem and its dual (edge/element packing). This paper was one of the first to use the primal-dual method for approximation in the distributed setting. This result was strengthened in " Distributed and parallel algorithms for weighted vertex cover and other covering problems.'' ]]> 01 Jan 1994 12:00:00 GMT Papers/algorithms_1994_17_280.pdf Papers/algorithms_1994_17_280.pdf 01 Jan 1994 12:00:00 GMT Papers/ICALP_1994_605.pdf Papers/ICALP_1994_605.pdf 01 Jan 1993 12:00:00 GMT Papers/WADS_1993_433.pdf Papers/WADS_1993_433.pdf 01 Jan 1993 12:00:00 GMT 01 Jan 1993 12:00:00 GMT Papers/SODA_1993_243.pdf Papers/SODA_1993_243.pdf

The paging problem is that of deciding which pages to keep in a memory of $k$ pages in order to minimize the number of page faults. This paper introduced the marking algorithm , a randomized on-line algorithm for the paging problem, and a proof that its performance guarantee is $O(\log k)$. This was one of the first results in the area showing that a randomized algorithm can have a strictly better performance guarantee than any deterministic on-line algorithm (no deterministic on-line algorithm can have a performance guarantee better than k). ]]> 01 Jan 1991 12:00:00 GMT Papers/algorithms_1991_12_685.pdf Papers/algorithms_1991_12_685.pdf This technical report presented notes from the first eight lectures of the class Many Models of Complexity taught by László Lovász at Princeton University in the fall of 1990. The topic is evasiveness of graph properties: given a graph property, how many edges of the graph an algorithm must check in the worst case before it knows whether the property holds. ]]> 01 Jan 1991 12:00:00 GMT http://arXiv.org/abs/cs/0205031 http://arXiv.org/abs/cs/0205031 01 Jan 1991 12:00:00 GMT Papers/SODA_1991_241.pdf Papers/SODA_1991_241.pdf The parametric shortest path problem is to find the shortest paths in graph where the edge costs are of the form $w_{ij}+\lambda$, where each $w_{ij}$ is constant and $\lambda$ is a parameter that varies. The problem is to find shortest path trees for every possible value of $\lambda$.

The minimum-balance problem is to find a ``weighting'' of the vertices so that adjusting the edge costs by the vertex weights yields a graph in which, for every cut, the minimum weight of any edge crossing the cut in one direction equals the minimum weight of any edge crossing the cut in the other direction.

The paper presents the fastest known algorithms for both problems. The algorithms run in $O(nm+n^2\log n)$ time. The paper also describes empirical studies of the algorithms on random graphs, suggesting that the expected time for finding minimum-mean cycles (an important special case of both problems) is $O(m + n \log n)$. ]]> 01 Jan 1991 12:00:00 GMT http://arXiv.org/abs/cs/0205041 http://arXiv.org/abs/cs/0205041 The thesis was about on-line algorithms. The main results were published in papers described above. The unpublished results were mainly exploring the role of linear-programming duality in on-line algorithms, including a method for deriving a potential-function analysis from a linear-programming duality analysis, and studies of some on-line matching algorithms. There were also some minor results analyzing paging strategies with lookahead. ]]> 01 Jan 1991 12:00:00 GMT http://www.cs.princeton.edu/research/techreps/TR-348-91 http://www.cs.princeton.edu/research/techreps/TR-348-91