Approximation algorithms for the arc orienteering problem

In this article we present approximation algorithms for the Arc Orienteering Problem (AOP). We propose a polylogarithmic approximation algorithm in directed graphs, while in undirected graphs we give a (6+?+o(1))$(6+?+o(1))$ and a (4+?)$(4+?)$-approximation algorithm for arbitrary instances and instances of unit profit, respectively. Also, an inapproximability result for the AOP is obtained as well as approximation algorithms for the Mixed Orienteering Problem.     

First steps to the runtime complexity analysis of ant colony optimization

The paper presents results on the runtime complexity of two ant colony optimization (ACO) algorithms: ant system, the oldest ACO variant, and GBAS, the first ACO variant for which theoretical convergence results have been established. In both cases, as the class of test problems under consideration, a slight generalization of the well-known OneMax test function has been chosen. The techniques used for the runtime analysis of the two algorithms differ: in the case of GBAS, the expected runtime until the optimal solution is reached is studied by a direct bound estimation approach inspired by comparable results for the (1+1)$(1+1)$ evolutionary algorithm (EA). A runtime bound of order O(mlogm)$O(m\log m)$, where m   is the problem instance size, is obtained. In the case of ant system, the original discrete stochastic process is approximated by a suitable continuous deterministic process. The validity of the approximation is shown by means of a rigid convergence theorem exploiting a classical result from mathematical learning theory. Using this approximation, it is demonstrated that for the considered OneMax-type problems, a runtime of order O(mlog(1/ε))$O(m\log (1/\epsilon ))$ until reaching an expected relative   solution quality of 1-ε$1-\epsilon$, and a runtime of O(mlogm)$O(m\log m)$ until reaching the optimal   solution with high probability can be predicted. Our results are the first to show competitiveness in runtime complexity with (1+1$1+1$) EA on OneMax for a proper ACO algorithm.     

Notes on a conjecture of Manoussakis concerning Hamilton cycles in digraphs

In 1992, Manoussakis conjectured that a strongly 2-connected digraph D on n vertices is hamiltonian if for every two distinct pairs of independent vertices x, y and w, z   we have d(x)+d(y)+d(w)+d(z)≥4n−3$d(x)+d(y)+d(w)+d(z)\ge 4n-3$. In this note we show that D has a Hamilton path, which gives an affirmative evidence supporting this conjecture.     

The complexity of the zero-sum 3-flows

A zero-sum k-flow for a graph G   is a vector in the null-space of the 0,1$0,1$-incidence matrix of G   such that its entries belong to {±1,?,±(k−1)}$\{\pm 1,?,\pm (k-1)\}$. Akbari et al. (2009) [5] conjectured that if G is a graph with a zero-sum flow, then G   admits a zero-sum 6-flow. (2,3)$(2,3)$-semiregular graphs are an important family in studying zero-sum flows. Akbari et al. (2009) [5] proved that if Zero-Sum Conjecture is true for any (2,3)$(2,3)$-semiregular graph, then it is true for any graph. In this paper, we show that there is a polynomial time algorithm to determine whether a given (2,3)$(2,3)$-graph G   has a zero-sum 3-flow. In fact, we show that, there is a polynomial time algorithm to determine whether a given (2,4)$(2,4)$-graph G with n   vertices has a zero-sum 3-flow, where the number of vertices of degree four is O(log?n)$O(\log ?n)$. Furthermore, we show that it is NP-complete to determine whether a given (3,4)$(3,4)$-semiregular graph has a zero-sum 3-flow.     

Finding the nucleolus of any n-person cooperative game by a single linear program

In this paper we show a new method for calculating the nucleolus by solving a unique minimization linear program with O(4ⁿ)$O(4^n)$ constraints whose coefficients belong to {−1,0,1}$\{-1,0,1\}$. We discuss the need of having all these constraints and empirically prove that they can be reduced to O(_kmax2ⁿ)$O(k_{\mathit{max}}{2}^n)$, where k_max is a positive integer comparable with the number of players. A computational experience shows the applicability of our method over (pseudo)random transferable utility cooperative games with up to 18 players.     

Approximation algorithms for facility location problems with a special class of subadditive cost functions

In this article we focus on approximation algorithms for facility location problems with subadditive costs. As examples of such problems, we present three facility location problems with stochastic demand and exponential servers, respectively inventory. We present a (1+ε,1)$(1+\epsilon ,1)$-reduction of the facility location problem with subadditive costs to the soft capacitated facility location problem, which implies the existence of a 2(1+ε)$2(1+\epsilon )$-approximation algorithm. For a special subclass of subadditive functions, we obtain a 2-approximation algorithm by reduction to the linear cost facility location problem.     

On hypercube packings,blocking sets and a covering problem

Consider sets S of hypercubes of side 2 in the discrete n-dimensional torus of side 4 with the property that every possible hypercube of side 2 has a nonempty intersection with some hypercube in S. The problem of minimizing the size of S is studied in two settings, depending on whether intersections between hypercubes in S are allowed or not. If intersections are not allowed, then one is asking for the smallest size of a non-extensible packing S  ; this size is denoted by f(n)$f(n)$. If intersections are allowed, then the structure S is called a blocking set. The smallest size of a blocking set S   is denoted by h(n)$h(n)$. By computer-aided techniques, it is shown that f(5)=12$f(5)=12$, f(6)=16$f(6)=16$, h(6)=15$h(6)=15$ and h(7)≤23$h(7)\le 23$. Also, non-extensible packings as well as blocking sets of certain small sizes are classified for n≤6$n\le 6$. There is a direct connection between these problems and a covering problem originating from the football pools.     

Flip-pushdown automata with k pushdown reversals and E0L systems are incomparable

We prove that any propagating E0L system cannot generate the language {w#w|w∈{0,1}^?}$\{w\mathrm{\#}w|w\in {\{0,1\}}^{?}\}$. This result, together with some known ones, enables us to conclude that the flip-pushdown automata with k pushdown reversals, i.e., the pushdown automata with the ability to flip the pushdown, and E0L systems are incomparable. This result solves an open problem stated by Holzer and Kutrib in 2003.     

A sufficient condition involving implicit degree and neighborhood intersection for long cycles

Let id(v)$\mathit{id}(v)$ denote the implicit degree of a vertex v in a graph G. In this paper, we prove that: Let G   be a 2-connected graph. If max?{id(u),id(v)}≥c/2$\max ?\{\mathit{id}(u),\mathit{id}(v)\}\ge c/2$ for each pair of nonadjacent vertices u and v   in an induced claw, and |N(x)∩N(y)|≥2$|N(x)\cap N(y)|\ge 2$ for each pair of nonadjacent vertices x and y in an induced modified claw, then G contains either a hamiltonian cycle or a cycle of length at least c.     

On miniaturized problems in parameterized complexity theory

We introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far. We develop parts of the basic theory of miniaturizations. Using the appropriate logical formalism, we show that the miniaturization of a definable problem in W[t]$\mathrm{W}[t]$ lies in W[t]$\mathrm{W}[t]$, too. In particular, the miniaturization of the dominating set problem is in W[2]$\mathrm{W}[2]$. Furthermore, we investigate the relation between f(k)·n^o(k)$f(k)·n^{\mathrm{o}(k)}$ time and subexponential time algorithms for the dominating set problem and for the clique problem.     

On the uniform convergence of the Chebyshev interpolants for solitons

We discuss polynomial interpolation and derive sufficient conditions for the uniform convergence of Chebyshev interpolants for different classes of functions. Rigorous results are illustrated with a number of examples which include solitons on an infinite line with algebraic, exponential and Gaussian decay rates. Suitable mappings of the real line to the interval [−1,1]$[-1\text{,}1]$ are considered for each class of solutions.     

The life-span prediction of a system connected in series

This article considers the prediction problem of the life-span of a system whose components connected in series and the lifetime of the components follows the exponential distribution with probability density f(x;θ)=θ⁻¹exp?(−x/θ)I(x>0)$f(x\text{;}\theta )={\theta }^{-1}\exp ?(-x/\theta )I(x>0)$. Employing the Bayes method, a prior distribution G(θ)$G(\theta )$ is used to describe the variability of θ$\theta$ but the form of G(θ)$G(\theta )$ is not specified and only one moment condition is assumed. Suppose the observed lifetimes of components are rightly censored, we define a prediction statistic to predict the life-span of the series-wound system which consists of some untested components, firstly, under the condition that the censoring distribution is known and secondly, that it is unknown. For several different priors, we investigate the coverage frequencies of the proposed prediction intervals as the sample size and the censorship proportion change. The simulation study shows that our predictions are efficient and applicable.     

A simple optimal representation for balanced parentheses

We consider succinct, or highly space-efficient, representations of a (static) string consisting of n   pairs of balanced parentheses, which support natural operations such as finding the matching parenthesis for a given parenthesis, or finding the pair of parentheses that most tightly enclose a given pair. This problem was considered by Jacobson [Space-efficient static trees and graphs, in: Proc. of the 30th FOCS, 1989, pp. 549–554] and Munro and Raman [Succinct representation of balanced parentheses and static trees, SIAM J. Comput. 31 (2001) 762–776] who gave O(n)$\mathrm{O}(n)$-bit and 2n+o(n)$2n+\mathrm{o}(n)$-bit representations, respectively, that supported the above operations in O(1)$\mathrm{O}(1)$ time on the RAM model of computation. This data structure is a fundamental tool in succinct representations, and has applications in representing suffix trees, ordinal trees, planar graphs and permutations.     

Topological additive numbering of directed acyclic graphs

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let D be a digraph and f   a labeling of its vertices with positive integers; denote by S(v)$S(v)$ the sum of labels over all neighbors of each vertex v. The labeling f is called topological additive numbering   if S(u)<S(v)$S(u)<S(v)$ for each arc (u,v)$(u,v)$ of the digraph. The problem asks to find the minimum number k for which D   has a topological additive numbering with labels belonging to {1,…,k}$\{1,\dots ,k\}$, denoted by _ηt(D)${\eta }_t(D)$.     

A representation theorem for (q-)holonomic sequences

Chomsky and Schützenberger showed in 1963 that the sequence _dL(n)$d_L(n)$, which counts the number of words of a given length n in a regular language L, satisfies a linear recurrence relation with constant coefficients for n  , i.e., it is C-finite. It follows that every sequence s(n)$s(n)$ which satisfies a linear recurrence relation with constant coefficients can be represented as _dL1(n)−_dL2(n)$d_{L_1}(n)-d_{L_2}(n)$ for two regular languages. We view this as a representation theorem for C-finite sequences. Holonomic or P-recursive sequences are sequences which satisfy a linear recurrence relation with polynomial coefficients. q-Holonomic sequences are the q-analog of holonomic sequences. In this paper we prove representation theorems of holonomic and q-holonomic sequences based on position specific weights on words, and for holonomic sequences, without using weights, based on sparse regular languages.