TSP Heuristics: Domination Analysis and Complexity

   Abstract. We show that the 2-Opt and 3-Opt heuristics for the traveling salesman problem (TSP) on the complete graph K _n produce a solution no worse than the average cost of a tour in K _n in a polynomial number of iterations. As a consequence, we get that the domination numbers of the 2- Opt , 3- Opt , Carlier—Villon, Shortest Path Ejection Chain, and Lin—Kernighan heuristics are all at least (n-2)! / 2 . The domination number of the Christofides heuristic is shown to be no more than

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 $p\in\mathbb{N}$ , 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 $\tilde{O}(n^{10})$ 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 ? of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of $\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$ . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to $\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})$ . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by $\tilde{O}(n^{4-1/d}\cdot\phi)$ . In addition, we prove an upper bound of $O(\sqrt[d]{\phi})$ 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 ?=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ?～1/σ ^d .     

Theory Of A B-Spline Basis Function

This paper extends the work of a previous paper of the author. A theoretical argument is provided to justify the heuristic algorithm used in the former paper. On the basis of the theory one derives, the previous algorithm can be further simplified. In the simplified basis function algorithm, the regular basis function (where $N_i^1(t)$ is 1 for $t_i \le t \lt t_{i + 1}$ and zero elsewhere) can be used for all cases except the case of the last point of a clamped B-spline where the basis function is modified to $N_{i,1} (t)$ where is 1 for $t_i \le t \le t_{i + 1}$ and zero elsewhere. Under this simplified algorithm, the knots ( i.e. , $t_{0}$ , $t_{1}, \ldots, t_{n+k}$ ) are a k -extended partition in the interior of the knot vector with a possibility that two ends of the knot vector could be a $(k + 1)$ -extended partition (case of clamped B-spline). It is shown that given a set of $(n + 1)$ control points, $V_{0}$ , $V_{1}, \ldots, V_{n}$ , the order of k , and the knots $(t_{0}, t_{1}, \ldots, t_{n+k})$ , the B-spline $P(t) = \sum_{i = 0}^{n} N_{i}^{k}(t)V_{i}$ is a continuous function for $t \in [t_{k - 1}, t_{n + 1}]$ and maintains the partition of unity. This algorithm circumvents the problem of generating a spike at the last point of a clamped B-spline. The constraint of having k -extended partition interior knots overcomes the problem of disconnecting the B-spline at the k repeated knot.     

Fast Algorithms for Computing the Smallest k-Enclosing Circle

For a set $P$ of $n$ points in the plane and an integer $k \leq n$, consider the problem of finding the smallest circle enclosing at least $k$ points of $P$. We present a randomized algorithm that computes in $O( n k )$ expected time such a circle, improving over previously known algorithms. Further, we present a linear time $\delta$-approximation algorithm that outputs a circle that contains at least $k$ points of $P$ and has radius less than $(1+\delta)r_{opt}(P,k)$, where $r_{opt}(P,k)$ is the radius of the minimum circle containing at least $k$ points of $P$. The expected running time of this approximation algorithm is $O(n + n \cdot\min((1/k\delta^3) \log^2 (1/\delta), k))$.     

Behavior of confined fluids in nanoslit pores: the normal pressure tensor

The aim of our research is to develop a theory, which can predict the behavior of confined fluids in nanoslit pores. The nanoslit pores studied in this work consist of two structureless and parallel walls in the xy plane located at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a given uniform bulk density. We have derived the following general equation for prediction of the normal pressure tensor P _zz of confined inhomogeneous fluids in nanoslit pores:

Efficient Algorithms for k Maximum Sums

We study the problem of computing the k maximum sum subsequences. Given a sequence of real numbers and an integer parameter k, the problem involves finding the k largest values of for The problem for fixed k = 1, also known as the maximum sum subsequence problem, has received much attention in the literature and is linear-time solvable. Recently, Bae and Takaoka presented a -time algorithm for the k maximum sum subsequences problem. In this paper we design an efficient algorithm that solves the above problem in time in the worst case. Our algorithm is optimal for and improves over the previously best known result for any value of the user-defined parameter k < 1. Moreover, our results are also extended to the multi-dimensional versions of the k maximum sum subsequences problem; resulting in fast algorithms as well.     

Attractive energy contribution to nanoconfined fluids behavior: the normal pressure tensor

The aim of our research is to demonstrate the role of attractive intermolecular potential energy on normal pressure tensor of confined molecular fluids inside nanoslit pores of two structureless purely repulsive parallel walls in xy plane at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a uniform density. To achieve this we have derived the perturbation theory version of the normal pressure tensor of confined inhomogeneous fluids in nanoslit pores:

Lower and Upper Bounds on FIFO Buffer Management in QoS Switches

We consider the management of FIFO buffers for network switches providing differentiated services. In each time step, an arbitrary number of packets arrive and only one packet can be sent. The buffer can store a limited number of packets and, due to the FIFO property, the sequence of sent packets has to be a subsequence of the arriving packets. The differentiated service model is abstracted by attributing each packet with a value according to its service level. A buffer management strategy can drop packets, and the goal is to maximize the sum of the values of sent packets. For only two different packet values, we introduce the account strategy and prove that this strategy achieves an optimal competitive ratio of if the buffer size tends to infinity and an optimal competitive ratio of for arbitrary buffer sizes. For general packet values, the simple preemptive greedy strategy (PG) is studied. We show that PG achieves a competitive ratio of which is the best known upper bound on the competitive ratio of this problem. In addition, we give a lower bound of on the competitive ratio of PG which improves the previously known lower bound. As a consequence, the competitive ratio of PG cannot be further improved significantly. Supported by the DFG grant WE 2842/1. A preliminary version of this paper appeared in Proceedings of the 14th Annual European Symposium on Algorithms (ESA), 2006.     

Finding Pathway Structures in Protein Interaction Networks

The increased availability of data describing biological interactions provides important clues on how complex chains of genes and proteins interact with each other. Most previous approaches either restrict their attention to analyzing simple substructures such as paths or trees in these graphs, or use heuristics that do not provide performance guarantees when general substructures are analyzed. We investigate a formulation to model pathway structures directly and give a probabilistic algorithm to find an optimal path structure in time and space, where n and m are respectively the number of vertices and the number of edges in the given network, k is the number of vertices in the path structure, and t is the maximum number of vertices (i.e., "width") at each level of the structure. Even for the case t = 1 which corresponds to finding simple paths of length k, our time complexity is a significant improvement over previous probabilistic approaches. To allow for the analysis of multiple pathway structures, we further consider a variant of the algorithm that provides probabilistic guarantees for the top suboptimal path structures with a slight increase in time and space. We show that our algorithm can identify pathway structures with high sensitivity by applying it to protein interaction networks in the DIP database.     

Zur Charakterisierung und Berechnung von symmetrischen Kubaturformeln

The paper is concerned with the characterization and calculation of symmetric cubature formulae of degree 2k?1 for two-dimensional product-functionals. The number of knots of the cubature formulae satisfies the following relation: $$\frac{{k(k + 1)}}{2} + \left[ {\frac{k}{2}} \right] \leqslant r \leqslant \frac{{k(k + 1)}}{2} + \left[ {\frac{k}{2}} \right] + 1.$$ The systems of non-linear equations involved are either solved exactly or all solutions are computed with any precision using a program-package for symbolic and algebraic calculations.     

Stability of polynomials under coefficient perturbation

Let the real polynomiala_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n-1}x + a_{n}be stable and let the real numbersb_{k}, c_{k} geq 0, 0 leq k leq n, be given. We present a simple determinant criterion for finding the largestt_{0} geq 0such that the polynomialalpha_{0}x^{n} + alpha_{1}x^{n-1}+ ... +alpha_{n-1}x + alpha_{n}is stable for allalpha_{k} in (a_{k} - b_{k}t_{0}, a_{k} + C_{k}t_{0}) cup {a_{k}}, 0 leq k leq n. Several further observations allow us to reduce the computational cost considerably.     

On the Exact Complexity of Evaluating Quantified k -CNF

We relate the exponential complexities 2 ^s(k)n of $\textsc {$k$-sat}$ and the exponential complexity $2^{s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))n}$ of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ (the problem of evaluating quantified formulas of the form $\forall\vec{x} \exists\vec{y} \textsc {F}(\vec {x},\vec{y})$ where F is a 3-cnf in $\vec{x}$ variables and $\vec{y}$ variables) and show that s(∞) (the limit of s(k) as k→∞) is at most $s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))$ . Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ running in time 2 ^cn with c<1. On the other hand, a nontrivial exponential-time algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ would provide a $\textsc {$k$-sat}$ solver with better exponent than all current algorithms for sufficiently large k. We also show several syntactic restrictions of the evaluation problem $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ have nontrivial algorithms, and provide strong evidence that the hardest cases of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ must have a mixture of clauses of two types: one universally quantified literal and two existentially quantified literals, or only existentially quantified literals. Moreover, the hardest cases must have at least n?o(n) universally quantified variables, and hence only o(n) existentially quantified variables. Our proofs involve the construction of efficient minimally unsatisfiable $\textsc {$k$-cnf}$ s and the application of the Sparsification lemma.     

The method of interlineation of vector functions \vec{w} (x, y, z, t) on a system of vertical straight lines and its application in crosshole seismic tomography

The authors propose a method to construct interlineation operators for vector functions $ \vec{w} $ (x, y, z, t) on a system of arbitrarily located vertical straight lines. The method allows calculating the vector $ \vec{w} $ at each point (x, y, z) between straight lines Γ _k for any instant of time t ≥ 0. They are proposed to be used to construct a crosshole accelerometer to model Earth’s crust on the basis of seismic sounding data $ {{\vec{w}}_k}\left( {z,t} \right),\,k=\overline{1,M} $ , about the vector of acceleration $ \vec{w} $ (x, y, z, t) received by accelerometers at each chink Γ _k .     

Convex Drawings of 3-Connected Plane Graphs

We use Schnyder woods of 3-connected planar graphs to produce convex straight-line drawings on a grid of size The parameter depends on the Schnyder wood used for the drawing. This parameter is in the range The algorithm is a refinement of the face-counting algorithm; thus, in particular, the size of the grid is at most The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder's and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3-connected planar graphs. The algorithm takes linear time. The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to For a random 3-connected plane graph, tests show that the expected size of the drawing is      

An extension theorem for arcs and linear codes

We prove the following generalization to the extension theorem of Hill and Lizak: For every nonextendable linear [n, k, d] _q code, q = p ^s , (d,q) = 1, we have $\sum\limits_{i\not \equiv 0,d(\bmod q)} {A_i > q^{k - 3} r(q),} $ where q + r(q) + 1 is the smallest size of a nontrivial blocking set in PG(2, q). This result is applied further to rule out the existence of some linear codes over $\mathbb{F}_4 $ meeting the Griesmer bound.     

Approximation Algorithms for Multi-Criteria Traveling Salesman Problems

We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of and a randomized approximation algorithm that achieves a ratio of . In particular, we obtain a 2+ε approximation for multi-criteria metric STSP. Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio ), STSP with weights one and two (ratio 4/3) and ATSP with weights one and two (ratio 3/2). A preliminary version of this work has been presented at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006) (Lecture Notes in Computer Science, vol. 4368, pp. 302–315, 2007). B. Manthey is supported by the Postdoc-Program of the German Academic Exchange Service (DAAD). He is on leave from Saarland University and has done part of the work at the Institute for Theoretical Computer Science of the University of Lübeck supported by DFG research grant RE 672/3 and at the Department of Computer Science at Saarland University.     

On the Complexity of 2-Monotone Restarting Automata

The -automaton is the weakest form of the nondeterministic version of the restarting automaton that was introduced by Jančar et al. to model the so-called analysis by reduction. Here it is shown that the class ℒ(R) of languages that are accepted by -automata is incomparable under set inclusion to the class of Church-Rosser languages and to the class of growing context-sensitive languages. In fact this already holds for the class of languages that are accepted by 2-monotone -automata. In addition, we prove that already the latter class contains -complete languages, showing that already the 2-monotone -automaton has a surprisingly large expressive power. The results of this paper have been announced at DLT 2004 in Auckland, New Zealand. This work was mainly carried out while T. Jurdziński was visiting the University of Kassel, supported by a grant from the Deutsche Forschungsgemeinschaft (DFG). F. Mráz and M. Plátek were partially supported by the Grant Agency of the Czech Republic under Grant-No. 201/04/2102 and by the program ‘Information Society’ under project 1ET100300517. F. Mráz was also supported by the Grant Agency of Charles University in Prague under Grant-No. 358/2006/A-INF/MFF.     

The stability of an nth-order nonlinear time-varying differential system

The stability of a system described by annth order differential equationy^{(n)} + a_{n-1}y^{(n-1)} + . . . + a_{1}y + a_{0} = 0wherea_{i}=a_{i}(t, y, dot{y}, . . . , y^{(n-1)}), i=0, 1, . . . , n - 1, is considered. It is shown that if the roots of the characteristic equation of the system are always contained in a circle on the complex plane with center(-z, 0), z > 0, and radius Ω such thatfrac{z}{Omega} > 1 + nC_{[n/2]}where[n/2]= nearest integergeq n/2andnC_{m} = n!/m!(n-m)!, wherenandmare integers, then the system is uniformly asymptotically stable in the sense of Liapunov.     

Some Relations between Approximation Problems and PCPs over the Real Numbers

In [10] it was recently shown that that is the existence of transparent long proofs for was established. The latter denotes the class of real number decision problems verifiable in polynomial time as introduced by Blum et al. [6]. The present paper is devoted to the question what impact a potential full real number theorem would have on approximation issues in the BSS model of computation. We study two natural optimization problems in the BSS model. The first, denoted by MAX-QPS, is related to polynomial systems; the other, MAX-q-CAP, deals with algebraic circuits. Our main results combine the PCP framework over with approximation issues for these two problems. We also give a negative approximation result for a variant of the MAX-QPS problem.     

Cardinal interpolation with periodic polysplines on strips

Abstract  We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in , polyharmonic of order p on each strip , , and periodic in its last n variables, whose restriction to the parallel hyperplanes , , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal -splines. Keywords: cardinal -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63