A Genetic Algorithm for the Weight Setting Problem in OSPF Routing

With the growth of the Internet, Internet Service Providers (ISPs) try to meet the increasing traffic demand with new technology and improved utilization of existing resources. Routing of data packets can affect network utilization. Packets are sent along network paths from source to destination following a protocol. Open Shortest Path First (OSPF) is the most commonly used intra-domain Internet routing protocol (IRP). Traffic flow is routed along shortest paths, splitting flow at nodes with several outgoing links on a shortest path to the destination IP address. Link weights are assigned by the network operator. A path length is the sum of the weights of the links in the path. The OSPF weight setting (OSPFWS) problem seeks a set of weights that optimizes network performance. We study the problem of optimizing OSPF weights, given a set of projected demands, with the objective of minimizing network congestion. The weight assignment problem is NP-hard. We present a genetic algorithm (GA) to solve the OSPFWS problem. We compare our results with the best known and commonly used heuristics for OSPF weight setting, as well as with a lower bound of the optimal multi-commodity flow routing, which is a linear programming relaxation of the OSPFWS problem. Computational experiments are made on the AT&T Worldnet backbone with projected demands, and on twelve instances of synthetic networks.     

Complexity and inapproximability results for the Power Edge Set problem

We consider the single channel PMU placement problem called the Power Edge Set problem. In this variant of the PMU placement problem, (single channel) PMUs are placed on the edges of an electrical network. Such a PMU measures the current along the edge on which it is placed and the voltage at its two endpoints. The objective is to find the minimum placement of PMUs in the network that ensures its full observability, namely measurement of all the voltages and currents. We prove that PES is NP-hard to approximate within a factor (1.12)-\(\epsilon \), for any \(\epsilon > 0\). On the positive side we prove that PES problem is solvable in polynomial time for trees and grids.     

Flows with unit path capacities and related packing and covering problems

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log?m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an $O(\sqrt{m})Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an O(?m)O(\sqrt{m}) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.     

Computing Optimal Beams in Two and Three Dimensions

The problem of computing an optimal beam among weighted regions (called the optimal beam problem) arises in several applied areas such as radiation therapy, stereotactic brain surgery, medical surgery, geological exploration, manufacturing, and environmental engineering. In this paper, we present computational geometry techniques that enable us to develop efficient algorithms for solving various optimal beam problems among weighted regions in two and three dimensional spaces. In particular, we consider two types of problems: the covering problems (seeking an optimal beam to contain a specified target region), and the piercing problems (seeking an optimal beam of a fixed shape to pierce the target region). We investigate several versions of these problems, with a variety of beam shapes and target region shapes in 2-D and 3-D. Our algorithms are based on interesting combinations of computational geometry techniques and optimization methods, and transform the optimal beam problems to solving a collection of instances of certain special non-linear optimization problems. Our approach makes use of interesting geometric observations, such as utilizing some new features of Minkowski sums.     

An approximation algorithm for maximum internal spanning tree

Given a graph G, the maximum internal spanning tree problem (MIST for short) asks for computing a spanning tree T of G such that the number of internal vertices in T is maximized. MIST has possible applications in the design of cost-efficient communication networks and water supply networks and hence has been extensively studied in the literature. MIST is NP-hard and hence a number of polynomial-time approximation algorithms have been designed for MIST in the literature. The previously best polynomial-time approximation algorithm for MIST achieves a ratio of \(\frac{3}{4}\). In this paper, we first design a simpler algorithm that achieves the same ratio and the same time complexity as the previous best. We then refine the algorithm into a new approximation algorithm that achieves a better ratio (namely, \(\frac{13}{17}\)) with the same time complexity. Our new algorithm explores much deeper structure of the problem than the previous best. The discovered structure may be used to design even better approximation or parameterized algorithms for the problem in the future.     

Funds versus stocks: a note on constraint classification

Even if the industrial production process forms a common frame of reference, the integration of the economist's concepts of production with those of the industrial production engineer may well present a vexed problem. Yet a basic requirement to engineers and economists alike ought to be a proper understanding of the interplay between engineering and economics in the area of industrial production. One solution to this problem may be to consider the production process in the unifying sense of a constraint satisfaction problem. This approach, which is not without merit in the history of science, has its foundation in an interdisciplinary constraint classification, developed originally in analytical mechanics. To illustrate this approach, we demonstrate in this note how the two economic notions of a fund of services and a stock of goods, are recast into two distinct constraint formulations of fundamental importance in the mathematical modelling of industrial production processes.     

On threshold BDDs and the optimal variable ordering problem

Many combinatorial optimization problems can be formulated as 0/1 integer programs (0/1 IPs). The investigation of the structure of these problems raises the following tasks: count or enumerate the feasible solutions and find an optimal solution according to a given linear objective function. All these tasks can be accomplished using binary decision diagrams (BDDs), a very popular and effective datastructure in computational logics and hardware verification. We present a novel approach for these tasks which consists of an output-sensitive algorithm for building a BDD for a linear constraint (a so-called threshold BDD) and a parallel AND operation on threshold BDDs. In particular our algorithm is capable of solving knapsack problems, subset sum problems and multidimensional knapsack problems. BDDs are represented as a directed acyclic graph. The size of a BDD is the number of nodes of its graph. It heavily depends on the chosen variable ordering. Finding the optimal variable ordering is an NP-hard problem. We derive a 0/1 IP for finding an optimal variable ordering of a threshold BDD. This 0/1 IP formulation provides the basis for the computation of the variable ordering spectrum of a threshold function. We introduce our new tool azove 2.0 as an enhancement to azove 1.1 which is a tool for counting and enumerating 0/1 points. Computational results on benchmarks from the literature show the strength of our new method.     

Parameterized lower bound and inapproximability of polylogarithmic string barcoding

String barcoding is a method that can identify microorganisms by analyzing their genome sequences. In this paper, we study the polylogarithmic string barcoding problem, where the lengths of the substrings in the testing set are polylogarithmically bounded. In particular, we show that the polylogarithmic string barcoding problem remains NP-hard and moreover, for a problem instance with n sequences, it is NP-hard to achieve an approximate ratio within dln n in polynomial time, where d is some constant. We then consider the parameterized polylogarithmic string barcoding problem, where the number of substrings in the test set is considered to be a fixed parameter k. We show that, unless W[2]=FPT, there does not exist a 2 ^O(k) n ^c algorithm that can decide whether a test set of size k exists or not, where c is a constant independent of n and k.     

Anonymizing binary and small tables is hard to approximate

The problem of publishing personal data without giving up privacy is becoming increasingly important. An interesting formalization recently proposed is the k-anonymity. This approach requires that the rows in a table are clustered in sets of size at least k and that all the rows in a cluster become the same tuple, after the suppression of some records. The natural optimization problem, where the goal is to minimize the number of suppressed entries, is known to be NP-hard when the values are over a ternary alphabet, k=3 and the rows length is unbounded. In this paper we give a lower bound on the approximation factor that any polynomial-time algorithm can achieve on two restrictions of the problem, namely (i) when the records values are over a binary alphabet and k=3, and (ii) when the records have length at most 8 and k=4, showing that these restrictions of the problem are APX-hard.     

Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree

Given a simple, undirected graph G=(V,E) and a weight function w:E→ℤ⁺, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1,k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1+1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2−1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1,k} within a ratio of 3/2 for k=2 (note that this matches the inapproximability bound above), and (2−2/(k+1)) for any k≥3, respectively, in polynomial time.     

An Algorithm for the Maximum Likelihood Problem on Evolutionary Trees

We investigate the problem of reconstructing evolutionary trees with maximum likelihood (MLET). In the MLET problem, a set of genetic sequences is given and a feasible solution is sought, consisting of an evolutionary tree (where general nodes correspond to sequences and input sequences occur as leaves) along with assignments for the interior nodes. Due to the difficulty of solving the MLET directly, we consider two restricted versions of the problem: the ancestral maximum likelihood (AML) and the maximum parsimony (MP) problems. If we let d_e denote the number of different characters occurring in two nodes linked by edge e, then the objective function of the AML problem is min ∑_{eσ E(T)} H(d_e/k), where H is the entropy function and k is the length of each sequence. In the MP we consider the objective function min σ_{e∊ E(T)} d_e/k. Both the AML and the MP are NP-hard. We propose a new approach for computing solutions for these problems, based on genetic algorithms.     

Polynomial Time Approximation Scheme for the Rectilinear Steiner Arborescence Problem

Given a set N of n terminals in the first quadrant of the Euclidean plane E ², find a minimum length directed tree rooted at the origin o, connecting to all terminals in N, and consisting of only horizontal and vertical arcs oriented from left to right or from bottom to top. This problem is called rectilinear Steiner arborescence problem, which has been proved to be NP-complete recently (Shi and Su, 11th ACM-SIAM Symposium on Discrete Algorithms (SODA), January 2000, to appear). In this paper, we present a polynomial time approximation scheme for this problem.     

A closest vector problem arising in radiation therapy planning

In this paper we consider the following closest vector problem. We are given a set of 0–1 vectors, the generators, an integer vector, the target vector, and a nonnegative integer C. Among all vectors that can be written as nonnegative integer linear combinations of the generators, we seek a vector whose ℓ _∞-distance to the target vector does not exceed C, and whose ℓ ₁-distance to the target vector is minimum.     

Random Expected Utility

We develop and analyze a model of random choice and random expected utility. A decision problem is a finite set of lotteries that describe the feasible choices. A random choice rule associates with each decision problem a probability measure over choices. A random utility function is a probability measure over von Neumann–Morgenstern utility functions. We show that a random choice rule maximizes some random utility function if and only if it is mixture continuous, monotone (the probability that a lottery is chosen does not increase when other lotteries are added to the decision problem), extreme (lotteries that are not extreme points of the decision problem are chosen with probability 0), and linear (satisfies the independence axiom).     

Simplicial Pivoting Algorithms for a Tractable Class of Integer Programs

We present a tractable class of integer feasibility problems. This class of max-closed IP problems was studied in somewhat restricted form by Glover, Pnueli, Hochbaum and Chandrasekaran and has a logic counterpart known as the class of Horn formulas. First we modify the existing algorithms in order to avoid the related recognition problem. Then we show that in order to solve these max-closed IP problems, simplicial path following methods can be used. This is important because these methods are flexible with respect to starting conditions, which make them more suitable than the top-down truncation algorithms that have been suggested.     

Exact combinatorial algorithms and experiments for?finding maximum k-plexes

We propose new practical algorithms to find maximum-cardinality k-plexes in graphs. A k-plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most k vertices in the k-plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximum-cardinality k-plexes is NP-hard. Complementing previous work, we develop exact combinatorial algorithms, which are strongly based on methods from parameterized algorithmics. The experiments with our freely available implementation indicate the competitiveness of our approach, for many real-world graphs outperforming the previously used methods.     

Node-weighted Steiner tree approximation in unit disk graphs

Given a graph G=(V,E) with node weight w:V→R ⁺ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(n ^O(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+ε)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+ε)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.     

A dynamic programming approach of finding an optimal broadcast schedule in minimizing total flow time

We study the problem of (off-line) broadcast scheduling in minimizing total flow time and propose a dynamic programming approach to compute an optimal broadcast schedule. Suppose the broadcast server has k pages and the last page request arrives at time n. The optimal schedule can be computed in O(k³(n+k)^k−1) time for the case that the server has a single broadcast channel. For m channels case, i.e., the server can broadcast m different pages at a time where m < k, the optimal schedule can be computed in O(n^k−m) time when k and m are constants. Note that this broadcast scheduling problem is NP-hard when k is a variable and will take O(n^k−m+1) time when k is fixed and m ≥ 1 with the straightforward implementation of the dynamic programming approach. The preliminary version of this paper appeared in Proceedings of the 11th Annual International Computing and Combinatorics Conference as “Off-line Algorithms for Minimizing the Total Flow Time in Broadcast Scheduling”.     

On the readability of monotone Boolean formulae

Golumbic et al. (Discrete Appl. Math. 154:1465–1477, 2006) defined the readability of a monotone Boolean function f to be the minimum integer k such that there exists an ∧−∨-formula equivalent to f in which each variable appears at most k times. They asked whether there exists a polynomial-time algorithm, which given a monotone Boolean function f, in CNF or DNF form, checks whether f is a read-k function, for a fixed k. In this paper, we partially answer this question already for k=2 by showing that it is NP-hard to decide if a given monotone formula represents a read-twice function. It follows also from our reduction that it is NP-hard to approximate the readability of a given monotone Boolean function f:{0,1} ⁿ →{0,1} within a factor of O(n)\mathcal{O}(n) . We also give tight sublinear upper bounds on the readability of a monotone Boolean function given in CNF (or DNF) form, parameterized by the number of terms in the CNF and the maximum size in each term, or more generally the maximum number of variables in the intersection of any constant number of terms. When the variables of the DNF can be ordered so that each term consists of a set of consecutive variables, we give much tighter logarithmic bounds on the readability.     

Maximum <Emphasis Type="Italic">k</Emphasis>-regular induced subgraphs

Independent sets, induced matchings and cliques are examples of regular induced subgraphs in a graph. In this paper, we prove that finding a maximum cardinality k-regular induced subgraph is an NP-hard problem for any fixed value of k. We propose a convex quadratic upper bound on the size of a k-regular induced subgraph and characterize those graphs for which this bound is attained. Finally, we extend the Hoffman bound on the size of a maximum 0-regular subgraph (the independence number) from k=0 to larger values of k.