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66
NonApproximability Results for Optimization Problems on Bounded Degree Instances
, 2001
"... We prove some nonapproximability results for restrictions of basic combinatorial optimization problems to instances of bounded \degree" or bounded \width." Speci cally: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to with ..."
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Cited by 78 (5 self)
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We prove some nonapproximability results for restrictions of basic combinatorial optimization problems to instances of bounded \degree" or bounded \width." Speci cally: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor 7=8+O(1= B), unless RP = NP . Hastad [18] proved that the problem is approximable to within a factor 7=8+1=64B in polynomial time, and that is hard to approximate to within a factor 7=8 + 1=(log B) 3 . Our result uses a new randomized reduction from general instances of Max 3SAT to boundedoccurrences instances. The randomized reduction applies to other Max SNP problems as well.
On Counting Independent Sets in Sparse Graphs
, 1998
"... We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if ..."
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Cited by 62 (12 self)
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We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if \Delta 25, unless RP = NP. 1 Introduction Counting independent sets in graphs is one of several combinatorial counting problems which have received recent attention. The problem is known to be #Pcomplete, even for low degree graphs [3]. On the other hand, it has been shown that, for graphs of maximum degree \Delta = 4, randomized approximate counting is possible [7, 3]. This success has been achieved using the Monte Carlo Markov chain method to construct a fully polynomial randomized approximation scheme (fpras). This has led to a natural question as to how far this success might extend. Here we consider in more detail this question of counting independent sets in graphs with constant m...
Hardness Results for the Power Range Assignment Problem in Packet Radio Networks
 in proceedings of RANDOM/APPROX
, 1999
"... Abstract. The minimum range assignment problem consists of assigning transmission ranges to the stations of a multihop packet radio network so as to minimize the total power consumption provided that the transmission range assigned to the stations ensures the strong connectivity of the network (i.e ..."
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Cited by 52 (14 self)
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Abstract. The minimum range assignment problem consists of assigning transmission ranges to the stations of a multihop packet radio network so as to minimize the total power consumption provided that the transmission range assigned to the stations ensures the strong connectivity of the network (i.e. each station can communicate with any other station by multihop transmission). The complexity of this optimization problem was studied by Kirousis, Kranakis, Krizanc, and Pelc (1997). In particular, they proved that, when the stations are located in a 3dimensional Euclidean space, the problem is NPhard and admits a 2approximation algorithm. On the other hand, they left the complexity of the 2dimensional case as an open problem. As for the 3dimensional case, we strengthen their negative result by showing that the minimum range assignment problem is APXcomplete, so, it does not admit a polynomialtime approximation scheme unless P=NP. We also solve the open problem discussed by Kirousis et al by proving that the 2dimensional case remains NPhard. 1
On the Power Assignment Problem in Radio Networks
 Electronic Colloquium on Computational Complexity (ECCC
, 2000
"... Given a finite set S of points (i.e. the stations of a radio network) on a ddimensional Euclidean space and a positive integer 1 h jSj \Gamma 1, the Min dd hRange Assignment problem consists of assigning transmission ranges to the stations so as to minimize the total power consumption, provided th ..."
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Cited by 51 (3 self)
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Given a finite set S of points (i.e. the stations of a radio network) on a ddimensional Euclidean space and a positive integer 1 h jSj \Gamma 1, the Min dd hRange Assignment problem consists of assigning transmission ranges to the stations so as to minimize the total power consumption, provided that the transmission ranges of the stations ensure the communication beween any pair of stations in at most h hops. Two main issues related to this problem are considered in this paper: the tradeoff between the power consumption and the number of hops; the computational complexity of the Min dd hRange Assignment problem. As for the first question, we provide a lower bound on the minimum power consumption of stations on the plane for constant h. The lower bound is a function of jSj, h and the minimum distance over all the pairs of stations in S. Then, we derive a constructive upper bound as a function of jSj, h and the maximum distance over all pairs of stations in S (i.e. the d...
An Empirical Comparison of Phylogenetic Methods on Chloroplast Gene Order Data in Campanulaceae
, 2000
"... The first heuristic for reconstructing phylogenetic trees from gene order data was introduced by Blanchette et al.. It sought to reconstruct the breakpoint phylogeny and was applied to a variety of datasets. We present a new heuristic for estimating the breakpoint phylogeny which, although not pol ..."
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Cited by 50 (20 self)
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The first heuristic for reconstructing phylogenetic trees from gene order data was introduced by Blanchette et al.. It sought to reconstruct the breakpoint phylogeny and was applied to a variety of datasets. We present a new heuristic for estimating the breakpoint phylogeny which, although not polynomialtime, is much faster in practice than BPAnalysis. We use this heuristic to conduct a phylogenetic analysis of chloroplast genomes in the flowering plant family Campanulaceae. We also present and discuss the results of experimentation on this real dataset with three methods: our new method, BPAnalysis, and the neighborjoining method, using breakpoint distances, inversion distances, and inversion plus transposition distances. 1
Formulations and Hardness of Multiple Sorting by Reversals
 Proc. 3rd Conf. Computational Molecular Biology RECOMB99, ACM
, 1998
"... We consider two generalizations of signed Sorting By Reversals (SBR), both aimed at formalizing the problem of reconstructing the evolutionary history of a set of species. In particular, we address Multiple SBR, calling for a signed permutation at minimum reversal distance from a given set of signed ..."
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Cited by 46 (1 self)
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We consider two generalizations of signed Sorting By Reversals (SBR), both aimed at formalizing the problem of reconstructing the evolutionary history of a set of species. In particular, we address Multiple SBR, calling for a signed permutation at minimum reversal distance from a given set of signed permutations, and Tree SBR, calling for a tree with the minimum number of edges spanning a given set of nodes in the complete graph where each node corresponds to a signed permutation and there is an edge between each pair of signed permutations one reversal away from each other. We describe a graphtheoretic relaxation of MSBR, which is the counterpart of the socalled alternatingcycle decomposition relaxation for SBR, illustrating a convenient mathematical formulation for this relaxation. Moreover, we use this relaxation to show that, even if the number of given permutations equals 3, MSBR is NPhard, and hence so is Tree SBR. In fact, we show that the two problems are APXhard, i.e. the...
Fast Convergence of the Glauber Dynamics for Sampling Independent Sets: Part II
, 1999
"... This work is a continuation of [4]. The focus is on the problem of sampling independent sets of a graph with maximum degree ffi. The weight of each independent set is expressed in terms of a fixed positive parameter 2 ffi\Gamma2 , where the weight of an indepednent set oe is joej . The Glaube ..."
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Cited by 43 (4 self)
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This work is a continuation of [4]. The focus is on the problem of sampling independent sets of a graph with maximum degree ffi. The weight of each independent set is expressed in terms of a fixed positive parameter 2 ffi\Gamma2 , where the weight of an indepednent set oe is joej . The Glauber dynamics is a simple Markov chain Monte Carlo method for sampling from this distribution. In [4], we showed fast convergence of this dynamics for trianglefree graphs. This paper proves fast convergence for arbitrary graphs. Computer Science Division, University of California at Berkeley, and International Computer Science Institute. Supported in part by National Science Foundation Fellowship. 1 Introduction For a more general introduction and a discussion of related work we refer the reader to the companion work [4]. The aim of this work is given a graph G = (V; E) to efficiently sample from the probability measure ¯G defined on the set of indepedent sets\Omega =\Omega G of G weight...
1.375Approximation Algorithm for Sorting by Reversals
, 2001
"... Analysis of genomes evolving by inversions leads to a general combinatorial problem of Sorting by Reversals, MINSBR, the problem of sorting a permutation by a minimum number of reversals. This combinatorial problem has a long history, and a number of other motivations. It was studied in a great ..."
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Cited by 37 (1 self)
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Analysis of genomes evolving by inversions leads to a general combinatorial problem of Sorting by Reversals, MINSBR, the problem of sorting a permutation by a minimum number of reversals. This combinatorial problem has a long history, and a number of other motivations. It was studied in a great detail recently in computational molecular biology. Following a series of preliminary results, Hannenhalli and Pevzner developed the first exact polynomial time algorithm for the problem of sorting signed permutations by reversals, and a polynomial time algorithm for a special case of unsigned permutations. The best known approximation algorithm for MINSBR, due to Christie, gives a performance ratio of 1.5. In this paper, by exploiting the polynomial time algorithm for sorting signed permutations and by developing a new approximation algorithm for maximum cycle decomposition of breakpoint graphs, we improve the performance ratio for MINSBR to 1.375.
On The Approximability Of The Traveling Salesman Problem
 Proceedings of the 32nd Annual ACM Symposium on Theory of Computing
, 2000
"... We show that the traveling salesman problem with triangle inequality cannot be approximated with a ratio better than 116 when the edge lengths are allowed to be asymmetric and 219 when the edge lengths are symmetric. The best previous lower bounds were 2804 and 3812 respectively. The reduction is fr ..."
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Cited by 37 (0 self)
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We show that the traveling salesman problem with triangle inequality cannot be approximated with a ratio better than 116 when the edge lengths are allowed to be asymmetric and 219 when the edge lengths are symmetric. The best previous lower bounds were 2804 and 3812 respectively. The reduction is from Håstad's maximum satisfiability of linear equations modulo 2, and is nonconstructive.
Approximation Hardness of TSP with Bounded Metrics
, 2000
"... The general asymmetric TSP with triangle inequality is known to be approximable only within an O(log n) factor, and is also known to be approximable within a constant factor as soon as the metric is bounded. In this paper we study the asymmetric and symmetric TSP problems with bounded metrics, i. ..."
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Cited by 31 (4 self)
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The general asymmetric TSP with triangle inequality is known to be approximable only within an O(log n) factor, and is also known to be approximable within a constant factor as soon as the metric is bounded. In this paper we study the asymmetric and symmetric TSP problems with bounded metrics, i.e., metrics where the distances are integers between one and some upper bound B. We first prove approximation lower bounds of 321/320 and 741/740 for the asymmetric and symmetric TSP with distances one and two, improving over the previous best lower bounds of 2805/2804 and 5381/5380. Then we consider the TSP with triangle inequality and distances that are integers between one and eight and prove approximation lower bounds of 131/130 for the asymmetric and 405/404 for the symmetric, respectively, version of that problem, improving over the previous best lower bounds of 2805/2804 and 3813/3812 by an order of magnitude.