Meaningfulness of conclusions from combinatorial optimization

Preface. 1 Introduction. 1.1 Two examples of combinatorial optimization. 1.2 Why study combinatorial optimization using statistical physics? 1.3 Textbooks. Bibliography. 2 Algorithms. 2.1 Pidgin Algol. 2.2 Iteration and recursion. 2.3 Divide-and-conquer. 2.4 Dynamic programming. 2.5 Backtracking. Bibliography. 3 Introduction to graphs. 3.1 Basic concepts and graph problems. 3.2 Basic graph algorithms. 3.3 Random graphs. Bibliography. 4 Introduction to complexity theory. 4.1 Turing machines. 4.2 Church's thesis. 4.3 Languages. 4.4 The halting problem. 4.5 Class P. 4.6 Class NP. 4.7 Definition of NP-completeness. 4.8 NP-complete problems. 4.9 Worst-case vs. typical-case complexity. Bibliography. 5 Statistical mechanics of the Ising model. 5.1 Phase transitions. 5.2 Some general notes on statistical mechanics. 5.3 The Curie-Weiss model of a ferromagnet. 5.4 The Ising model on a random graph. Bibliography. 6 Algorithms and numerical results for vertex covers. 6.1 Definitions. 6.2 Heuristic algorithms. 6.3 Branch-and-bound algorithm. 6.4 Results: Covering random graphs. 6.5 The leaf-removal algorithm. 6.6 Monte Carlo simulations. 6.7 Backbone. 6.8 Clustering of minimum vertex covers. Bibliography. 7 Statistical mechanics of vertex covers on a random graph. 7.1 Introduction. 7.2 The first-moment bound. 7.3 The hard-core lattice gas. 7.4 Replica approach. Bibliography. 8 The dynamics of vertex-cover algorithms. 8.1 The typical-case solution time of a complete algorithm. 8.2 The dynamics of generalized leaf-removal algorithms. 8.3 Random restart algorithms. Bibliography. 9 Towards new, statistical-mechanics motivated algorithms. 9.1 The cavity graph. 9.2 Warning propagation. 9.3 Belief propagation. 9.4 Survey propagation. 9.5 Numerical experiments on random graphs. Bibliography. 10 The satisfiability problem. 10.1 SAT algorithms. 10.2 Phase transitions in random K-SAT. 10.3 Typical-case dynamics of RandomWalkSAT. 10.4 Message-passing algorithms for SAT. Bibliography. 11 Optimization problems in physics. 11.1 Monte Carlo optimization. 11.2 Hysteric optimization. 11.3 Genetic algorithms. 11.4 Shortest paths and polymers in random media. 11.5 Maximum flows and random-field systems. 11.6 Submodular functions and free energy of Potts model. 11.7 Matchings and spin glasses. Bibliography. Index.

An agent-based model of elephant crop consumption walks using combinatorial optimization

Analysis of speedups in parallel evolutionary algorithms and [formula omitted] EAs for combinatorial optimization

Evolutionary algorithms are popular heuristics for solving various combinatorial problems as they are easy to apply and often produce good results. Island models parallelize evolution by using different populations, called islands, which are connected by a graph structure as communication topology. Each island periodically communicates copies of good solutions to neighboring islands in a process called migration. We consider the speedup gained by island models in terms of the parallel running time for problems from combinatorial optimization: sorting (as maximization of sortedness), shortest paths, and Eulerian cycles. Different search operators are considered. The results show in which settings and up to what degree evolutionary algorithms can be parallelized efficiently. Along the way, we also investigate how island models deal with plateaus. In particular, we show that natural settings lead to exponential vs. logarithmic speedups, depending on the frequency of migration.KeywordsEvolutionary AlgorithmMutation OperatorLinear SpeedupShort Path ProblemSearch PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Optimization of a large-scale Traveling Salesman Problem (TSP) especially in telecommunication networks, which is a well-known NP-hard problem in combinatorial optimization, is a time-consuming problem. In this paper, the proposed heuristic algorithm is designed for fast computing. The result will be compared with two keys parameter, accuracy and computation time. Proposed algorithm has been compared with brute force and Ant colony optimization (ACO) which known as an algorithm that is used to determine the shortest path and best cost at minimum iterations possible for a random data set on the basis of Euclidean distance formula. Proposed algorithm takes only 0.0074 seconds to provide shortest path solution with 50 nodes combination. The proposed algorithm has 5% less accuracy from brute force and provide 6.69 % better solution from ACO for 33 nodes through 50 nodes.

The notion of Shared Risk Link Groups (SRLG) captures survivability issues when a set of links of a network may fail simultaneously. The theory of survivable network design relies on basic combinatorial objects that are rather easy to compute in the classical graph models: shortest paths, minimum cuts, or pairs of disjoint paths. In the SRLG context, the optimization criterion for these objects is no longer the number of edges they use, but the number of SRLGs involved. Unfortunately, computing these combinatorial objects is NP-hard and hard to approximate with this objective in general. Nevertheless some objects can be computed in polynomial time when the SRLGs satisfy certain structural properties of locality which correspond to practical ones, namely the star property (all links affected by a given SRLG are incident to a unique node) and the span 1 property (the links affected by a given SRLG form a connected component of the network). The star property is defined in a multi-colored model where a link can be affected by several SRLGs while the span property is defined only in a mono-colored model where a link can be affected by at most one SRLG. In this paper, we extend these notions to characterize new cases in which these optimization problems can be solved in polynomial time. We also investigate the computational impact of the transformation from the multi-colored model to the mono-colored one. Experimental results are presented to validate the proposed algorithms and principles.

Proactive route guidance to avoid congestion

Multicast transmission corresponds to send data to several destinations often involving requirements of Quality of Service (QoS) and Traffic Engineering (TE). This work investigates new evolutionary models to tackle Multicast Flow Routing in a Pareto multiobjective perspective. Two multiobjective evolutionary algorithms (SPEA and SPEA2) were applied as the underlying search of such models. QoS and TE requirements are also considered in the route calculus by optimizing four objectives - maximum link utilization, total cost, maximum end-to-end delay and hops count - attending a link capacity constraint. Besides, three heuristics for subtrees reconnection to be used on crossover and mutation operators are investigated. The first heuristic uses a shortest path algorithm to improve the convergence; the second one employs a random search to reconnect subtrees into a valid route and the third one mixes the other two combining the good skills of each one. The resultant evolutionary environments were evaluated using four multiobjective metrics: two for convergence and two for diversity. SPEA2 reached better results than SPEA on the vast majority cases. The design of crossover and mutation operators that provide more diversity lead to very good improvements on both multiobjective goals - convergence and diversity - being that the heuristic which combines random and shortest path reconnections and returned the best results.

Two quasi-static minimax open shortest path first (OSPF) routing algorithms in networks supporting the Switched Multi-megabit Data Service (SMDS) are presented and compared. In OSPF routing, the network is modeled as a graph and each link is associated with a nonnegative arc weight. A shortest path spanning tree is calculated for each origin to carry both the individually addressed and the group addressed (multicast) traffic. The OSPF routing protocol is adopted as a major part of the default inter-switching system interface (ISSI) routing algorithm for SMDS networks where arc weights are inversely proportional to the aggregate link set capacities. The problem of choosing a set of link set metrics is considered so that the maximum link utilization factor is minimized in an SMDS network. The problem is formulated as a nonlinear mixed integer programming problem.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Network robustness and throughput can be improved by routing each source-to-terminal (s, t) demand via two link-disjoint paths (TLDP). However, the use of TLDP incurs higher energy cost. Henceforth, we address the problem of minimizing the energy usage of networks that use TLDP. Specifically, our problem is to maximally switch off redundant network links while maintaining at least 0≤T≤100% of (s, t) TLDP in the network, for a given T, and limiting the maximum link utilization (MLU) to no greater than a configured threshold. To address this problem, we present a fast heuristic, called TLDP by Shortest Path First (TLDP-SPF), and extensively evaluate its performance on both real and/or synthetic topologies and traffic demands. Our simulation results show that TLDP-SPF can reduce network energy usage, on average, by more than 20%, even for MLU below 50%. As compared to using Shortest Path routing, while reducing energy by about 20%, TLDP-SPF does not significantly affect (s, t) path length, even for MLU<50%.

The finding of shortest paths in networks is one of the most basic combinatorial optimization problems, and the algorithms for solving these problems are among the most widely used. The reader is probably aware that a variety of optimization problems, some of which appear to have little to do with paths in networks, can be formulated as shortest path problems.

Ant colony optimization (ACO) is a stochastic approach for solving combinatorial optimization problems like routing in computer networks. The idea of this optimization is based on the food accumulation methodology of the ant community. Zone based routing algorithms is build on the concept of individual node's position for routing of packets in mobile ad-hoc networks. Here the nodes' position can be further utilized to discover routes by the Ants in optimized way. Position based routing algorithms (POSANT) had some significant loopholes to find route (source to destination) like it never guarantees the route would be the shortest one, in cases while it is able to find it. On the contrary, routing algorithms which are based on ant colony optimization find routing paths that are close in length to the shortest paths. The drawback of these algorithms is the large number of control messages that needs to be sent or the long delay before the routes are established from a source to a destination. Here In this paper we have used Zone based ANT colony using Clustering which assures to find shortest route using the DIR principle (In this principle, the source or intermediate node transmits message to several neighbors and the node whose direction is closest to the direction of destination gets selected as the next hop forwarding node.) together with minimum overhead for route discovery and mobility management. Unlike other Zone based approach, in clustering it is not required to consider zone related information of each node while finding shortest path. Here, it is being proposed a new routing algorithm for mobile ad hoc network by combining the concept of Ant Colony approach and Zone based routing approach using clustering to get shortest path with small number of control messages to minimize the overhead. Simulations show that Zone Based ant colony routing algorithm has relatively short route establishment overhead than other zone based ant colony algorithms in highly mobile scenarios.

We study the stochastic versions of a broad class of combinatorial problems where the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings over probabilistic graphs, and other combinatorial problems like knapsack. We observe that the expected value is inadequate in capturing different types of risk averse or risk-prone behaviors, and instead we consider a more general objective which is to maximize the expected utility of the solution for some given utility function, rather than the expected weight (expected weight becomes a special case). We show that we can obtain a polynomial time approximation algorithm with additive error ϵ for any ϵ >; 0, if there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above) and the maximum value of the utility function is bounded by a constant. Our result generalizes several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. Our algorithm for utility maximization makes use of the separability of exponential utility and a technique to decompose a general utility function into exponential utility functions, which may be useful in other stochastic optimization problems.

K-adaptability in stochastic combinatorial optimization under objective uncertainty

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(logm)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an \(O(\sqrt{m})\)-approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.