Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

On the equivalence, containment, and covering problems for the regular and context-free languages

We consider the complexity of the equivalence and containment problems for regular expressions and context-free grammars, concentrating on the relationship between complexity and various language properties. Finiteness and boundedness of languages are shown to play important roles in the complexity of these problems. An encoding into grammars of Turing machine computations exponential in the size of the grammar is used to prove several exponential lower bounds. These lower bounds include exponential time for testing equivalence of grammars generating finite sets, and exponential space for testing equivalence of non-self-embedding grammars. Several problems which might be complex because of this encoding are shown to simplify for linear grammars. Other problems considered include grammatical covering and structural equivalence for right-linear, linear, and arbitrary grammars

Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every p∈N , a family of L p instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]2, where it was shown that the expected number of steps is bounded by O~(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1] d , for an arbitrary d≥2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density ϕ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of O~(n4+1/3⋅ϕ8/3) . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to O~(n4+1/3−1/d⋅ϕ8/3) . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by O~(n4−1/d⋅ϕ) . In addition, we prove an upper bound of O(ϕ√d) on the expected approximation factor with respect to all L p metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with ϕ=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ϕ∼1/σ d

Portal protein diversity and phage ecology

© 2008 The Authors. This article is distributed under the terms of the Creative Commons License, Attribution 2.5. The definitive version was published in Environmental Microbiology 10 (2008): 2810-2823, doi:10.1111/j.1462-2920.2008.01702.x.Oceanic phages are critical components of the global ecosystem, where they play a role in microbial mortality and evolution. Our understanding of phage diversity is greatly limited by the lack of useful genetic diversity measures. Previous studies, focusing on myophages that infect the marine cyanobacterium Synechococcus, have used the coliphage T4 portal-protein-encoding homologue, gene 20 (g20), as a diversity marker. These studies revealed 10 sequence clusters, 9 oceanic and 1 freshwater, where only 3 contained cultured representatives. We sequenced g20 from 38 marine myophages isolated using a diversity of Synechococcus and Prochlorococcus hosts to see if any would fall into the clusters that lacked cultured representatives. On the contrary, all fell into the three clusters that already contained sequences from cultured phages. Further, there was no obvious relationship between host of isolation, or host range, and g20 sequence similarity. We next expanded our analyses to all available g20 sequences (769 sequences), which include PCR amplicons from wild uncultured phages, non-PCR amplified sequences identified in the Global Ocean Survey (GOS) metagenomic database, as well as sequences from cultured phages, to evaluate the relationship between g20 sequence clusters and habitat features from which the phage sequences were isolated. Even in this meta-data set, very few sequences fell into the sequence clusters without cultured representatives, suggesting that the latter are very rare, or sequencing artefacts. In contrast, sequences most similar to the culture-containing clusters, the freshwater cluster and two novel clusters, were more highly represented, with one particular culture-containing cluster representing the dominant g20 genotype in the unamplified GOS sequence data. Finally, while some g20 sequences were non-randomly distributed with respect to habitat, there were always numerous exceptions to general patterns, indicating that phage portal proteins are not good predictors of a phage's host or the habitat in which a particular phage may thrive.This research was supported in part by funding from NSF (CMORE contribution #87), DOE, The Seaver Foundation and the Gordon and Betty Moore Foundation Marine Microbiology Program to S.W.C.; an NIH Bioinformatics Training Grant supported M.B.S.; MIT Undergraduate Research Opportunities Program supported V.Q., J.A.L., G.T., R.F. and J.E.R.; Howard Hughes Medical Institute funded MIT Biology Department Undergraduate Research Opportunities Program supported A.S.D.; NSERC (Canada) Discovery Grant (DG 298394) and a Grant from the Canadian Foundation for Innovation (NOF10394) to J.P.B.; NSF Graduate Fellowship funding supported M.L.C

Corrigendum "Portal protein diversity and phage ecology"

Author Posting. © The Author(s), 2011. This is the author's version of the work. It is posted here by permission of John Wiley & Sons for personal use, not for redistribution. The definitive version was published in Environmental Microbiology 13 (2011): 2832, doi:10.1111/j.1462-2920.2011.02616.x

Approximation algorithms for NEXTtime-hard periodically specified problems and domino problems

We study the efficient approximability of two general class of problems: (1) optimization versions of the domino problems studies in [Ha85, Ha86, vEB83, SB84] and (2) graph and satisfiability problems when specified using various kinds of periodic specifications. Both easiness and hardness results are obtained. Our efficient approximation algorithms and schemes are based on extensions of the ideas. Two of properties of our results obtained here are: (1) For the first time, efficient approximation algorithms and schemes have been developed for natural NEXPTIME-complete problems. (2) Our results are the first polynomial time approximation algorithms with good performance guarantees for `hard` problems specified using various kinds of periodic specifications considered in this paper. Our results significantly extend the results in [HW94, Wa93, MH+94]