A linear time algorithm for full steiner trees

Full Steiner trees are building blocks for the construction of Steiner minimal trees. Melzak gave an elegant construction for full Steiner tree. However no polynomial time implementation of Melzak's construction was previously known. In this paper we show how it can be implemented to run in linear time.     

A faster strongly polynomial time algorithm for submodular function minimization

We consider the problem of minimizing a submodular function f defined on a set V with n elements. We give a combinatorial algorithm that runs in O(n ⁵EO  +  n ⁶) time, where EO is the time to evaluate f(S) for some . This improves the previous best strongly polynomial running time by more than a factor of n. We also extend our result to ring families.     

A polynomial time algorithm for a hemispherical minimax location problem

A polynomial time algorithm to obtain an exact solution for the equiweighted minimax location problem when the demand points are spread over a hemisphere is presented. It is shown that the solution of the minimax problem when the norm under consideration is geodesic is equivalent to solving a maximization problem using the Euclidean norm.     

A polynomial time approximation scheme for the two-source minimum routing cost spanning trees

Let G be an undirected graph with nonnegative edge lengths. Given two vertices as sources and all vertices as destinations, we investigated the problem how to construct a spanning tree of G such that the sum of distances from sources to destinations is minimum. In the paper, we show the NP-hardness of the problem and present a polynomial time approximation scheme. For any >0, the approximation scheme finds a (1+)-approximation solution in O(n^1/+1) time. We also generalize the approximation algorithm to the weighted case for distances that form a metric space.     

A polynomial time primal network simplex algorithm for minimum cost flows

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n ²m lognC, n ²m² logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm ² logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).     

A polynomial time algorithm for a chance-constrained single machine scheduling problem

This paper gives an O(n√nlog³n) time algorithm for the chance-constrained sequencing problem on a single machine, where n is the number of jobs and the objective is to minimize the number of jobs which are early with probability not smaller than α (a given constant) against the common due time d.     

A linear time algorithm for edge coloring of binomial trees

We consider the problem of efficient coloring of the edges of a so-called binomial tree T, i.e. acyclic graph containing two kinds of edges: those which must have a single color and those which are to be colored with L consecutive colors, where L is an arbitrary integer greater than 1. We give an O(n) time algorithm for optimal coloring of such a tree, where n is the number of vertices of T. Also, we give simple bounds on the chromatic index of T and a division of all binomial trees into two classes depending on their chromaticity.     

A polynomial time approximation algorithm for the two-commodity splittable flow problem

We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity ${i \in \{1, 2\}}$ can be split into a bounded number k _i of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k _i and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k ₁, k ₂)-splittable flow without chunk size restrictions for fixed demand ratios.     

Finding minimum height elimination trees for interval graphs in polynomial time

The elimination tree plays an important role in many aspects of sparse matrix factorization. The height of the elimination tree presents a rough, but usually effective, measure of the time needed to perform parallel elimination. Finding orderings that produce low elimination is therefore important. As the problem of finding minimum height elimination tree orderings is NP-hard, it is interesting to concentrate on limited classes of graphs and find minimum height elimination trees for these efficiently. In this paper, we use clique trees to find an efficient algorithm for interval graphs which make an important subclass of chordal graphs. We first illustrate this method through an algorithm that finds minimum height elimination for chordal graphs. This algorithm, although of exponential time complexity, is conceptionally simple and leads to a polynomial-time algorithm for finding minimum height elimination trees for interval graphs.This work was supported by grants from the Norwegian Research Council.     

A polynomial time algorithm for finding the prime factors of cartesian-product graphs

We consider the computational complexity of recognizinf concerned cartesian product graphs. Sabidussi gives a non-algorithmic proof that the cartesian factorization is unique. He uses a tower of successively coarser equivalence relations on the edge set in which each prime factor of the graph is identified with an equivalence class in the coarsest of the relations. We first explore the structure and size of the relation at the base of the tower. Then we give a polynomial-time algorithm to compute the relations and to construct the prime factors of any connected graph. The bounds on the size of the relations are crucial to the runtime analysis of our algorithm.     

A polynomial time algorithm for solving a quality control station configuration problem

We study unreliable serial production lines with known failure probabilities for each operation. Such a production line consists of a series of stations; existing machines and optional quality control stations (QCS). Our aim is to simultaneously decide where and if to install the QCSs along the line and to determine the production rate, so as to maximize the steady state expected net profit per time unit from the system.We use dynamic programming to solve the cost minimization auxiliary problem where the aim is to minimize the time unit production cost for a given production rate. Using the above developed O(N²) dynamic programming algorithm as a subroutine, where N stands for the number of machines in the line, we present an O(N⁴) algorithm to solve the Profit Maximization QCS Configuration Problem.     

A polynomial time algorithm to compute the Abelian kernel of a finite monoid

The time complexity of the best previously known algorithm to compute the Abelian kernel of a finite monoid with a fixed number of generators is exponential. In this paper we use results on subgroups of the free Abelian group and constructions on labeled graphs to develop a polynomial time algorithm for this problem.     

A polynomial time algorithm for solving the fermat-weber location problem with mixed norms

This paper discusses the Fermat-Weber location problem, manages to apply the ellipsoid method to this problem and proves the ellipsoid method can be terminated at an approximately optimal location in polynomial time, verifies the ellipsoid method is robust for the lower dimensional location problem     

An exact and polynomial distance-based algorithm to reconstruct single copy tandem duplication trees

The problem of reconstructing the duplication tree of a set of tandemly repeated sequences which are supposed to have arisen by unequal recombination, was first introduced by Fitch (1977), and has recently received a lot of attention. In this paper, we place ourselves in a distance framework and deal with the restricted problem of reconstructing single copy duplication trees. We describe an exact and polynomial distance based algorithm for solving this problem, the parsimony version of which has previously been shown to be NP-hard (like most evolutionary tree reconstruction problems). This algorithm is based on the minimum evolution principle, and thus involves selecting the shortest tree as being the correct duplication tree. After presenting the underlying mathematical concepts behind the minimum evolution principle, and some of its benefits (such as statistical consistency), we provide a new recurrence formula to estimate the tree length using ordinary least-squares, given a matrix of pairwise distances between the copies. We then show how this formula naturally forms the dynamic programming framework on which our algorithm is based, and provide an implementation in O(n³) time and O(n²) space, where n is the number of copies.     

A fully polynomial time projective method

This paper proposes a modification of the algorithm of de Ghellinck and Vial, which keeps the size of the numbers occurring in the calculation by a fixed bound, independently of the number of iterations. This algorithm is fully polynomial in time.     

On trees with real-rooted independence polynomial

The independence polynomial of a graph $G$ is

$I(G,x)=\sum _{k\ge 0}{i}_k\left(G\right)x^k,$

where $i_k\left(G\right)$ denotes the number of independent sets of $G$ of size $k$ (note that $i_0\left(G\right)=1$). In this paper we show a new method to prove real-rootedness of the independence polynomials of certain families of trees.In particular we will give a new proof of the real-rootedness of the independence polynomials of centipedes (Zhu’s theorem), caterpillars (Wang and Zhu’s theorem), and we will prove a conjecture of Galvin and Hilyard about the real-rootedness of the independence polynomial of the so-called Fibonacci trees.     

Choiceless polynomial time

Turing machines define polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? This question can be recast as follows: Does there exist a logic that captures polynomial time (without presuming the presence of a linear order)? Earlier, one of us conjectured a negative answer. The problem motivated a quest for stronger and stronger PTime logics. All these logics avoid arbitrary choice. Here we attempt to capture the choiceless fragment of PTime. Our computation model is a version of abstract state machines (formerly called evolving algebras). The idea is to replace arbitrary choice with parallel execution. The resulting logic expresses all properties expressible in any other PTime logic in the literature. A more difficult theorem shows that the logic does not capture all of PTime.