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397
Which Problems Have Strongly Exponential Complexity?
- Journal of Computer and System Sciences
, 1998
"... For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of sub-exponential algorithms for these problems. We introduce a generalized reduction which we call Sub-Exponential Reduction Family (SERF) t ..."
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Cited by 242 (11 self)
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For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of sub-exponential algorithms for these problems. We introduce a generalized reduction which we call Sub-Exponential Reduction Family (SERF) that preserves sub-exponential complexity. We show that CircuitSAT is SERF-complete for all NP-search problems, and that for any fixed k, k-SAT, k-Colorability, k-Set Cover, Independent Set, Clique, Vertex Cover, are SERF--complete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, sub-exponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds for AC 0 ; that is, bounds of the form 2 \Omega\Gamma n) . This problem is even open for depth-3 circuits. In fact, such a bound for depth-3 circuits with even l...
Improved Steiner Tree Approximation in Graphs
, 2000
"... The Steiner tree problem in weighted graphs seeks a minimum weight connected subgraph containing a given subset of the vertices (terminals). We present a new polynomial-time heuristic with an approximation ratio approaching 1 + 2 1:55, which improves upon the previously best-known approximation ..."
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Cited by 225 (6 self)
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The Steiner tree problem in weighted graphs seeks a minimum weight connected subgraph containing a given subset of the vertices (terminals). We present a new polynomial-time heuristic with an approximation ratio approaching 1 + 2 1:55, which improves upon the previously best-known approximation algorithm of [10] with performance ratio 1:59.
Efficient Algorithms for Online Decision Problems
- J. Comput. Syst. Sci
, 2003
"... In an online decision problem, one makes a sequence of decisions without knowledge of the future. Tools from learning such as Weighted Majority and its many variants [13, 18, 4] demonstrate that online algorithms can perform nearly as well as the best single decision chosen in hindsight, even when t ..."
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Cited by 192 (3 self)
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In an online decision problem, one makes a sequence of decisions without knowledge of the future. Tools from learning such as Weighted Majority and its many variants [13, 18, 4] demonstrate that online algorithms can perform nearly as well as the best single decision chosen in hindsight, even when there are exponentially many possible decisions. However, the naive application of these algorithms is inefficient for such large problems. For some problems with nice structure, specialized efficient solutions have been developed [10, 16, 17, 6, 3].
Geometric Shortest Paths and Network Optimization
- Handbook of Computational Geometry
, 1998
"... Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 187 (15 self)
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Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Guillotine subdivisions approximate polygonal subdivisions: Part II - A simple polynomial-time approximation scheme for geometric k-MST, TSP, and related problems
, 1996
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A polylogarithmic approximation algorithm for the group Steiner tree problem
- Journal of Algorithms
, 2000
"... The group Steiner tree problem is a generalization of the Steiner tree problem where we ae given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimum-weight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich a ..."
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Cited by 149 (9 self)
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The group Steiner tree problem is a generalization of the Steiner tree problem where we ae given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimum-weight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich and Widmayer and finds applications in VLSI design. The group Steiner tree problem generalizes the set covering problem, and is therefore at least as had. We give a randomized O(log 3 n log k)-approximation algorithm for the group Steiner tree problem on an n-node graph, where k is the number of groups. The best previous ink)v/ (Bateman, Helvig, performance guarantee was (1 + - Robins and Zelikovsky).
Efficient algorithms for geometric optimization
- ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear progra ..."
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Cited by 114 (10 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.
Networked Slepian-Wolf: Theory, Algorithms and Scaling Laws
- IEEE Transactions on Information Theory
, 2003
"... Consider a set of correlated sources located at the nodes of a network, and a set of sinks that are the destinations for some of the sources. We consider the minimization of cost functions which are the product of a function of the rate and a function of the path weight. We consider both the data ..."
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Cited by 93 (8 self)
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Consider a set of correlated sources located at the nodes of a network, and a set of sinks that are the destinations for some of the sources. We consider the minimization of cost functions which are the product of a function of the rate and a function of the path weight. We consider both the data gathering scenario, which is relevant in sensor networks, and general tra#c matrices, relevant for general networks. The minimization is achieved by jointly optimizing (a) the transmission structure, which we show consists in general of a superposition of trees from each of the source nodes to its corresponding sink nodes, and (b) the rate allocation across the source nodes, which is done by Slepian-Wolf coding. We show that the overall minimization can be achieved in two concatenated steps.
Algorithms for Facility Location Problems with Outliers (Extended Abstract)
- In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms
, 2000
"... ) Moses Charikar Samir Khuller y David M. Mount z Giri Narasimhan x Abstract Facility location problems are traditionally investigated with the assumption that all the clients are to be provided service. A significant shortcoming of this formulation is that a few very distant clients, called outlier ..."
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Cited by 92 (9 self)
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) Moses Charikar Samir Khuller y David M. Mount z Giri Narasimhan x Abstract Facility location problems are traditionally investigated with the assumption that all the clients are to be provided service. A significant shortcoming of this formulation is that a few very distant clients, called outliers, can exert a disproportionately strong influence over the final solution. In this paper we explore a generalization of various facility location problems (K-center, K-median, uncapacitated facility location etc) to the case when only a specified fraction of the customers are to be served. What makes the problems harder is that we have to also select the subset that should get service. We provide generalizations of various approximation algorithms to deal with this added constraint. 1 Introduction The facility location problem and the related clustering problems, k-median and k-center, are widely studied in operations research and computer science [3, 7, 22, 24, 32]. Typically in...
Tighter Bounds for Graph Steiner Tree Approximation
- SIAM Journal on Discrete Mathematics
, 2005
"... Abstract. The classical Steiner tree problem in weighted graphs seeks a minimum weight connected subgraph containing a given subset of the vertices (terminals). We present a new polynomial-ln 3 time heuristic that achieves a best-known approximation ratio of 1 + ≈ 1.55 for general graphs 2 and best- ..."
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Cited by 88 (7 self)
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Abstract. The classical Steiner tree problem in weighted graphs seeks a minimum weight connected subgraph containing a given subset of the vertices (terminals). We present a new polynomial-ln 3 time heuristic that achieves a best-known approximation ratio of 1 + ≈ 1.55 for general graphs 2 and best-known approximation ratios of ≈ 1.28 for both quasi-bipartite graphs (i.e., where no two nonterminals are adjacent) and complete graphs with edge weights 1 and 2. Our method is considerably simpler and easier to implement than previous approaches. We also prove the first known nontrivial performance bound (1.5 · OPT) for the iterated 1-Steiner heuristic of Kahng and Robins in quasi-bipartite graphs.
