Approximation and Exact Algorithms for Constructing Minimum Ultrametric Trees from Distance Matrices

An edge-weighted tree is called ultrametric if the distances from the root to all the leaves in the tree are equal. For an n by n distance matrix M, the minimum ultrametric tree for M is an ultrametric tree T = (V, E, w) with leaf set {1,..., n} such that d_T(i, j) M[i, j] for all i, j and is minimum, where d_T(i, j) is the distance between i and j on T. Constructing minimum ultrametric trees from distance matrices is an important problem in computational biology. In this paper, we examine its computational complexity and approximability. When the distances satisfy the triangle inequality, we show that the minimum ultrametric tree problem can be approximated in polynomial time with error ratio 1.5(1 + log n), where n is the number of species. We also develop an efficient branch-and-bound algorithm for constructing the minimum ultrametric tree for both metric and non-metric inputs. The experimental results show that it can find an optimal solution for 25 species within reasonable time, while, to the best of our knowledge, there is no report of algorithms solving the problem even for 12 species.     

A 2-approximation for the preceding-and-crossing structured 2-interval pattern problem

The 2-interval pattern problem over its various models and restrictions was proposed by Vialette (2004) for the application of RNA secondary structure prediction. We present an O(n ³logn)-time 2-approximation algorithm for the problem of finding a largest { < ,-structured subset of 2-intervals given an input 2-interval set of size n. This greatly improves the previous best approximation ratio of 6 by Crochemore et al. (2005).     

Simple Approximation Algorithms for MAXNAESP and Hypergraph 2-colorability

Hypergraph 2-colorability, also known as set splitting, is a widely studied problem in graph theory. In this paper we study the maximization version of the same. We recast the problem as a special type of satisfiability problem and give approximation algorithms for it. Our results are valid for hypergraph 2-colorability, set splitting and MAX-CUT (which is a special case of hypergraph 2-colorability) because the reductions are approximation preserving. Here we study the MAXNAESP problem, the optimal solution to which is a truth assignment of the literals that maximizes the number of clauses satisfied. As a main result of the paper, we show that any locally optimal solution (a solution is locally optimal if its value cannot be increased by complementing assignments to literals and pairs of literals) is guaranteed a performance ratio of . This is an improvement over the ratio of attributed to another local improvement heuristic for MAX-CUT (C. Papadimitriou, Computational Complexity, Addison Wesley, 1994). In fact we provide a bound of for this problem, where k 3 is the minimum number of literals in a clause. Such locally optimal algorithms appear to subsume typical greedy algorithms that have been suggested for problems in the general domain of satisfiability. It should be noted that the NAESP problem where each clause has exactly two literals, is equivalent to MAX-CUT. However, obtaining good approximation ratios using semi-definite programming techniques (M. Goemans and D.P. Williamson, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, 1994a, pp. 422–431) appears difficult. Also, the randomized rounding algorithm as well as the simple randomized algorithm both (M. Goemans and D.P. Williamson, SIAM J. Disc. Math, vol. 7, pp. 656–666, 1994b) yield a bound of for the MAXNAESP problem. In contrast to this, the algorithm proposed in this paper obtains a bound of for this problem.     

Stochastic Algorithms,Symmetric Markov Perfect Equilibrium,and the ‘curse’ of Dimensionality

This paper introduces a stochastic algorithm for computing symmetric Markov perfect equilibria. The algorithm computes equilibrium policy and value functions, and generates a transition kernel for the (stochastic) evolution of the state of the system. It has two features that together imply that it need not be subject to the curse of dimensionality. First, the integral that determines continuation values is never calculated; rather it is approximated by a simple average of returns from past outcomes of the algorithm, an approximation whose computational burden is not tied to the dimension of the state space. Second, iterations of the algorithm update value and policy functions at a single (rather than at all possible) points in the state space. Random draws from a distribution set by the updated policies determine the location of the next iteration's updates. This selection only repeatedly hits the recurrent class of points, a subset whose cardinality is not directly tied to that of the state space. Numerical results for industrial organization problems show that our algorithm can increase speed and decrease memory requirements by several orders of magnitude.     

Structure Prediction and Computation of Sparse Matrix Products

We consider1 the problem of predicting the nonzero structure of a product of two or more matrices. Prior knowledge of the nonzero structure can be applied to optimize memory allocation and to determine the optimal multiplication order for a chain product of sparse matrices. We adapt a recent algorithm by the author and show that the essence of the nonzero structure and hence, a near-optimal order of multiplications, can be determined in near-linear time in the number of nonzero entries, which is much smaller than the time required for the multiplications. An experimental evaluation of the algorithm demonstrates that it is practical for matrices of order 103 with 104 nonzeros (or larger). A relatively small pre-computation results in a large time saved in the computation-intensive multiplication.     

Iterative Converging Algorithms for Computing Bounds on Durations of Activities in Pert and Pert-Like Models

Since its invention in 1958, Program Evaluation and Review Technique (PERT) has been widely used during the planning, design, and implementation of projects. Pert models the activities of a project as a single source-single sink directed acyclic graph where nodes represent events (end or beginning of activities) and arcs activities. The maximum amount by which an activity can be delayed without delaying the overall project is called the slack. Critical tasks have zero slack whereas all noncritical tasks have positive slacks. Pert is a valuable tool in the management of large projects since it allows to compute the slack of each activity of the project. Such information may be crucial in avoiding cost overruns that would be caused by delays to critical activities and/or excessive delays to noncritical activities. What Pert fails to provide is how one should go about distributing remaining slack on noncritical activities while taking into consideration properties of the activities as well as precedence relationships among them, so as to have reasonable upper bounds on duration of all activities, critical or noncritical. In this paper we propose several algorithms for the distribution of slack on non-critical activities. We show that if one desires to distribute the remaining slack proportionally to the initially assigned activity durations then the problem is in P, and propose an algorithm of linear time complexity. However if one desires to use distribution functions other than the initial durations of activities, then the problem of slack distribution becomes NP-complete. Finding the maximal bounds corresponding to zero-slack solution at the sink requires iterative application of exponential algorithm. For that case we introduce an approximation algorithm of linear time complexity on each iteration. The algorithm iteratively increases bounds on durations of activities and converges to the zero-slack solution on all paths from the source node to the sink node in the Pert-like graph. The algorithms described in this paper were successfully applied to solving timing bounds problems in VLSI design.