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We consider Newton's method for the linear fractional combinatorial optimization. First we show a strongly polynomial bound on the number of iterations for the general case. Then we consider the transshipment problem when the maximum arc cost is being minimized. This problem can be reduced to the maximum mean-weight cut problem, which is a special case of the linear fractional combinatorial optimization. We prove that Newton's method runs in O(m) iterations for the maximum mean weight cut problem. One iteration is dominated by the maximum flow computation, so the overall running time is O(m2n). The previous fastest algorithm is based on Meggido's parametric search method and runs in O(n3m) time.

We design an expected polynomial-time, truthful-in-expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are m matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions, matroid weighted-rank functions, and convex combinations thereof. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P != NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm, rather than over the input to the rounding algorithm. This approach leads to truthful-in-expectation mechanisms, and these mechanisms can be implemented efficiently when the corresponding objective function is concave. For bidders with MRS valuations, we give a novel randomized rounding algorithm that leads to both a concave objective function and a (1-1/e)-approximation of the optimal welfare.

Research supported by NASA Langley Research Center includes many applications of aerospace design optimization and is conducted by teams of applied mathematicians and aerospace engineers. This paper investigates the benefits from this combined expertise in formulating and solving integer and combinatorial optimization problems. Applications range from the design of large space antennas to interior noise control. A typical problem. for example, seeks the optimal locations for vibration-damping devices on an orbiting platform and is expressed as a mixed/integer linear programming problem with more than 1500 design variables.

Consider the following problem: given a metric space, some of whose points are "clients", open a set of at most $k$ facilities to minimize the average distance from the clients to these facilities. This is just the well-studied $k$-median problem, for which many approximation algorithms and hardness results are known. Note that the objective function encourages opening facilities in areas where there are many clients, and given a solution, it is often possible to get a good idea of where the clients are located. However, this poses the following quandary: what if the identity of the clients is sensitive information that we would like to keep private? Is it even possible to design good algorithms for this problem that preserve the privacy of the clients? In this paper, we initiate a systematic study of algorithms for discrete optimization problems in the framework of differential privacy (which formalizes the idea of protecting the privacy of individual input elements). We show that many such problems indeed have good approximation algorithms that preserve differential privacy; this is even in cases where it is impossible to preserve cryptographic definitions of privacy while computing any non-trivial approximation to even the_value_ of an optimal solution, let alone the entire solution. Apart from the $k$-median problem, we study the problems of vertex and set cover, min-cut, facility location, Steiner tree, and the recently introduced submodular maximization problem, "Combinatorial Public Projects" (CPP).

We investigate the performance of a variant of Axelrod's model for dissemination of culture - the Adaptive Culture Heuristic (ACH) - on solving an NP-Complete optimization problem, namely, the classification of binary input patterns of size $F$ by a Boolean Binary Perceptron. In this heuristic, $N$ agents, characterized by binary strings of length $F$ which represent possible solutions to the optimization problem, are fixed at the sites of a square lattice and interact with their nearest neighbors only. The interactions are such that the agents' strings (or cultures) become more similar to the low-cost strings of their neighbors resulting in the dissemination of these strings across the lattice. Eventually the dynamics freezes into a homogeneous absorbing configuration in which all agents exhibit identical solutions to the optimization problem. We find through extensive simulations that the probability of finding the optimal solution is a function of the reduced variable $F/N^{1/4}$ so that the number of agents must increase with the fourth power of the problem size, $N \propto F^ 4$, to guarantee a fixed probability of success. In this case, we find that the relaxation time to reach an absorbing configuration scales with $F^ 6$ which can be interpreted as the overall computational cost of the ACH to find an optimal set of weights for a Boolean Binary Perceptron, given a fixed probability of success.

In the classic multi-armed bandits problem, the goal is to have a policy for dynamically operating arms that each yield stochastic rewards with unknown means. The key metric of interest is regret, defined as the gap between the expected total reward accumulated by an omniscient player that knows the reward means for each arm, and the expected total reward accumulated by the given policy. The policies presented in prior work have storage, computation and regret all growing linearly with the number of arms, which is not scalable when the number of arms is large. We consider in this work a broad class of multi-armed bandits with dependent arms that yield rewards as a linear combination of a set of unknown parameters. For this general framework, we present efficient policies that are shown to achieve regret that grows logarithmically with time, and polynomially in the number of unknown parameters (even though the number of dependent arms may grow exponentially). Furthermore, these policies only require storage that grows linearly in the number of unknown parameters. We show that this generalization is broadly applicable and useful for many interesting tasks in networks that can be formulated as tractable combinatorial optimization problems with linear objective functions, such as maximum weight matching, shortest path, and minimum spanning tree computations.

In this talk we will demonstrate iterative methods as a general technique to analyze linear programming formulations of combinatorial optimization problems. We first show an application of the method to the Minimum Bounded Degree Spanning Tree problem. We present a polynomial time algorithm that returns a spanning tree of optimal cost while exceeding the degree bound of any vertex by at most an additive one. This is the best possible result for this problem and settles a 15-year-old conjecture of Goemans affirmatively. We will present a new short proof of the result. We will also discuss extensions to degree constrained versions of more general network design problems and give first additive approximation algorithms using the iterative method. These results add to a rather small list of combinatorial optimization problems which have an additive approximation algorithm. I will also discuss applications of the method to various multi-criteria problems. This talk will contain joint works with Lap Chi Lau, Seffi Naor, Mohammad Salavatipour and R. Ravi when candidates can influence their position, can lead to sub-optimal result and challenges the basic assumption that the candidates arrive in a random order. This issue gains more importance since secretary problem and its variants have been used to design online auctions. In this talk, I will describe a general framework for dealing with the issue of incentives in secretary problems. We formalize an intuitive notion of incentive compatible mechanisms in which the position of the candidate is independent of his chances of being hired. We then construct optimal incentive compatible mechanisms which select the best secretary with high probability. This is joint work with Niv Buchbinder and Kamal Jain. ©2009 Microsoft Corporation. All rights reserved.

1 Multi�ows and Disjoint Paths - Let G = (V,E) be a graph and let...

We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.

We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.

Since the introduction of mathematical programming it has been all too easy to identify real-world problems that could be formulated as math programs but could not be solved to a provable optimum within a reasonable amount of time. As computing power continues to increase, so too does the size of the mathematical programs to be solved. This situation has given rise to a multitude of heuristic solution techniques that seek to provide good approximate solutions within a reasonable amount of time. Designers and users of heuristic solution techniques would like to assess the quality of their heuristics, where heuristic quality is defined in terms of the characteristics of the solutions returned by the heuristic, often emphasizing the objective function values. Fixed bounds on worst case performance are available for some heuristics, but in many cases heuristic-quality assessment approaches must take a sampling perspective and apply statistical tools to derive their assessment. Although many authors have proposed statistical methods for assessing heuristic quality, there has not been a foundation for a single unified approach or a framework for comparison of the distinct approaches to heuristic-quality assessment. The primary contribution of this research is that it presents a unifying probability modeling framework that applies whenever randomized heuristic solution techniques are applied to instances of combinatorial optimization problems. With this probability model in hand, we can better understand the relative strengths and weaknesses of the existing statistical approaches to assessing heuristic quality in combinatorial optimization. Moreover, the probability model suggests new avenues for the development of heuristic quality assessment approaches, and we present empirical results from initial applications.

There are two principal technologies for these large-scale combinatorial optimization problems: 1) exact algorithms and 2) metaheuristic algorithms. This project will integrate concepts from these two technologies to develop generic optimization frameworks to find provably good solutions to large-scale discrete optimization problems often encountered in many real applications. The way that these two sets of methods will be used, and in particular the way in which they will be used together so that each complements the strengths of the other, will be novel and pioneering. In particular, this project will begin by exploring and capitalizing on the links between mixed integer programming decomposition approaches, such as Dantzig Wolfe decomposition and Lagrangian relaxation, and metaheuristics such as the Nested Partitions framework. While the relationship between these seemingly disparate approaches has not been exploited before, there is already evidence to suggest that capitalizing on this relationship to integrate these methods could yield solution frameworks that are more powerful than using either approach on its own.

We saw last time that orientation of a 2k-edge connected graph into a k-arc connected digraph and the Lucchesi and Younger Theorem were special cases of submodular �ows. Other familiar problems can also be phrased as submodular �ows.

It is well-known that methods of statistical physics are applicable to computational optimization problems like the traveling salesman problem. In the random link model, where all pairs of cities receive independent 'distances', several features of the optimum solution have been predicted non-rigorously based on so-called replica symmetry. I will present a rigorous approach where each of several optimization problems leads to a two-person game. The assumptions underlying the replica symmetric ansatz are essentially equivalent to the statement that the associated game can be effectively analyzed by a game-tree search. This approach has led to proofs of several conjectures originating from the physics community. A paper is available at arXiv:0908.1920. ©2009 Microsoft Corporation. All rights reserved.

It is well-known that methods of statistical physics are applicable to computational optimization problems like the traveling salesman problem. In the random link model, where all pairs of cities receive independent 'distances', several features of the optimum solution have been predicted non-rigorously based on so-called replica symmetry. I will present a rigorous approach where each of several optimization problems leads to a two-person game. The assumptions underlying the replica symmetric ansatz are essentially equivalent to the statement that the associated game can be effectively analyzed by a game-tree search. This approach has led to proofs of several conjectures originating from the physics community. A paper is available at arXiv:0908.1920. ©2009 Microsoft Corporation. All rights reserved.

It is well-known that methods of statistical physics are applicable to computational optimization problems like the traveling salesman problem. In the random link model, where all pairs of cities receive independent 'distances', several features of the optimum solution have been predicted non-rigorously based on so-called replica symmetry. I will present a rigorous approach where each of several optimization problems leads to a two-person game. The assumptions underlying the replica symmetric ansatz are essentially equivalent to the statement that the associated game can be effectively analyzed by a game-tree search. This approach has led to proofs of several conjectures originating from the physics community. A paper is available at arXiv:0908.1920. ©2009 Microsoft Corporation. All rights reserved.

A real symmetric matrix A with the property that Axx^(T) > 0 for all x � 0 is called positive de�nite. If A is positive de�nite, then there exists an invertible matrix P , such that A = PP^(T)

In this lecture, we will introduce three related topics: graph orientations, directed cuts, and submodular �ows. In fact, we will use submodular �ows to prove results from the other topics.

This is my reply to Zalka and Brun's criticism of my recent paper on quantum optimization heuristics. Essentially, this criticism is shown to be utterly irrelevant.

The aggregation principle in stochastic optimization was exploited to build an algorithmic, progressive hedging, for solving such problems. Relationships were developed between asymptotic results for statistical estimators and those for the solution of stochastic optimization problems.

The aggregation principle in stochastic optimization was exploited to build an algorithmic, progressive hedging, for solving such problems. Relationships were developed between asymptotic results for statistical estimators and those for the solution of stochastic optimization problems.

We describe an approach that uses combinatorial optimization and machine learning to share the work between the host and device of heterogeneous computing systems such that the overall application execution time is minimized. We propose to use combinatorial optimization to search for the optimal system configuration in the given parameter space (such as, the number of threads, thread affinity, work distribution for the host and device). For each system configuration that is suggested by combinatorial optimization, we use machine learning for evaluation of the system performance. We evaluate our approach experimentally using a heterogeneous platform that comprises two 12-core Intel Xeon E5 CPUs and an Intel Xeon Phi 7120P co-processor with 61 cores. Using our approach we are able to find a near-optimal system configuration by performing only about 5% of all possible experiments.

An introduction to the application of combinatorial optimization methods to ground state calculations of frustrated, disordered systems is given. We discuss the interface problem in the random bond Ising ferromagnet, the random field Ising model, the diluted antiferromagnet in an external field, the spin glass problem, the solid-on-solid model with a disordered substrate and other convex cost flow problems occurring in superconducting flux line lattices and traffic flow networks. On the algorithmic side we present a concise introduction to a number of elementary algorithms in combinatorial optimization, in particular network flows: the shortest path algorithm, the maximum-flow algorithms and minimum-cost-flow algorithms. We present a short glance at the minimum weighted matching and branch-and-cut algorithms.

There are also problems for which we have found no polynomial-time algorithms. The theory of NP-completeness uni�es these failures. Roughly speaking, an NP-complete problem is one that is as hard as any problem in a large class of problems.

We provide a numerical refutation of the developments of Fiorini et al. (2015)* for models with disjoint sets of descriptive variables. We also provide an insight into the meaning of the existence of a one-to-one linear map between solutions of such models. *: Fiorini, S., S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf (2015). Exponential Lower Bounds for Polytopes in Combinatorial Optimization. Journal of the ACM 62:2, Article No. 17.

We establish the soundness of the replica symmetric ansatz introduced by M. Mezard and G. Parisi for minimum matching and the traveling salesman problem in the pseudo-dimension d mean field model for d\geq 1. The case d=1 of minimum matching corresponds to the pi^2/6 limit for the assignment problem established by D. Aldous in 2001, and the analogous limit for the d=1 case of TSP was recently established by the author with a different method. We introduce a game-theoretical framework by which we prove the correctness of the replica-cavity prediction of the corresponding limits also for d>1.

Given a combinatorial optimization problem, we aim at characterizing the set of all instances for which every feasible solution has the same objective value. Our central result deals with multi-dimensional assignment problems. We show that for the axial and for the planar $d$-dimensional assignment problem instances with constant objective value property are characterized by sum-decomposable arrays. We provide a counterexample to show that the result does not carry over to general $d$-dimensional assignment problems. Our result for the axial $d$-dimensional assignment problem can be shown to carry over to the axial $d$-dimensional transportation problem. Moreover, we obtain characterizations when the constant objective value property holds for the minimum spanning tree problem, the shortest path problem and the minimum weight maximum cardinality matching problem.

The paper is devoted to upper bounds on run-time of Non-Elitist Genetic Algorithms until some target subset of solutions is visited for the first time. In particular, we consider the sets of optimal solutions and the sets of local optima as the target subsets. Previously known upper bounds are improved by means of drift analysis. Finally, we propose conditions ensuring that a Non-Elitist Genetic Algorithm efficiently finds approximate solutions with constant approximation ratio on the class of combinatorial optimization problems with guaranteed local optima (GLO).

Theorem 1. If G is bipartite, then P=M.

Theorem 1. If G is bipartite, then P=M.

[CITATION] Solving combinatorial optimization problems by simulated annealing, genetic algorithms, and neural networks Y Lu - 1991 - University of Minnesota Cited by 4

[CITATION] Solving combinatorial optimization problems by simulated annealing, genetic algorithms, and neural networks Y Lu - 1991 - University of Minnesota Cited by 4

[CITATION] Solving combinatorial optimization problems by simulated annealing, genetic algorithms, and neural networks Y Lu - 1991 - University of Minnesota Cited by 4

Let C be a uniform clutter and let I=I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizations. Then we show some applications to edge ideals and clutters which are related to a conjecture of Conforti and Cornu\'ejols and to max-flow min-cut problems.

Estimation of Distribution Algorithms (EDAs) require flexible probability models that can be efficiently learned and sampled. Deep Boltzmann Machines (DBMs) are generative neural networks with these desired properties. We integrate a DBM into an EDA and evaluate the performance of this system in solving combinatorial optimization problems with a single objective. We compare the results to the Bayesian Optimization Algorithm. The performance of DBM-EDA was superior to BOA for difficult additively decomposable functions, i.e., concatenated deceptive traps of higher order. For most other benchmark problems, DBM-EDA cannot clearly outperform BOA, or other neural network-based EDAs. In particular, it often yields optimal solutions for a subset of the runs (with fewer evaluations than BOA), but is unable to provide reliable convergence to the global optimum competitively. At the same time, the model building process is computationally more expensive than that of other EDAs using probabilistic models from the neural network family, such as DAE-EDA.

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.

In this paper, we present approximation algorithms for combinatorial optimization problems under probabilistic constraints. Specifically, we focus on stochastic variants of two important combinatorial optimization problems: the k-center problem and the set cover problem, with uncertainty characterized by a probability distribution over set of points or elements to be covered. We consider these problems under adaptive and non-adaptive settings, and present efficient approximation algorithms for the case when underlying distribution is a product distribution. In contrast to the expected cost model prevalent in stochastic optimization literature, our problem definitions support restrictions on the probability distributions of the total costs, via incorporating constraints that bound the probability with which the incurred costs may exceed a given threshold.

The research proposed in this document investigated multiobjective optimization approaches based upon the Genetic Algorithm (GA). Several versions of the GA have been adopted for multiobjective design, but, prior to this research, there had not been significant comparisons of the most popular strategies. The research effort first generalized the two-branch tournament genetic algorithm in to an N-branch genetic algorithm, then the N-branch GA was compared with a version of the popular Multi-Objective Genetic Algorithm (MOGA). Because the genetic algorithm is well suited to combinatorial (mixed discrete / continuous) optimization problems, the GA can be used in the conceptual phase of design to combine selection (discrete variable) and sizing (continuous variable) tasks. Using a multiobjective formulation for the design of a 50-passenger aircraft to meet the competing objectives of minimizing takeoff gross weight and minimizing trip time, the GA generated a range of tradeoff designs that illustrate which aircraft features change from a low-weight, slow trip-time aircraft design to a heavy-weight, short trip-time aircraft design. Given the objective formulation and analysis methods used, the results of this study identify where turboprop-powered aircraft and turbofan-powered aircraft become more desirable for the 50 seat passenger application. This aircraft design application also begins to suggest how a combinatorial multiobjective optimization technique could be used to assist in the design of morphing aircraft.

This survey attempts, without including many details of algorithms or of the underlying theory, to answer the questions of what is combinatorial optimization.

Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colorable subgraph. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows. We show that the minimum-degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3- colorability, we found only graphs with Nullstellensatz certificates of degree four.

This article is from Nucleic Acids Research , volume 40 . Abstract A major challenge in metabolic engineering and synthetic biology is to balance the flux of an engineered heterologous metabolic pathway to achieve high yield and productivity in a target organism. Here, we report a simple, efficient and programmable approach named ‘customized optimization of metabolic pathways by combinatorial transcriptional engineering (COMPACTER)’ for rapid tuning of gene expression in a heterologous pathway under distinct metabolic backgrounds. Specifically, a library of mutant pathways is created by de novo assembly of promoter mutants of varying strengths for each pathway gene in a target organism followed by high-throughput screening/selection. To demonstrate this approach, a single round of COMPACTER was used to generate both a xylose utilizing pathway with near-highest efficiency and a cellobiose utilizing pathway with highest efficiency that were ever reported in literature for both laboratory and industrial yeast strains. Interestingly, these engineered xylose and cellobiose utilizing pathways were all host-specific. Therefore, COMPACTER represents a powerful approach to tailor-make metabolic pathways for different strain backgrounds, which is difficult if not impossible to achieve by existing pathway engineering methods.

In this paper a local integral simplex method will be described which, starting with the initial tableau of a set partitioning problem, makes pivots using the pivot on one rule until no more such pivots are possible because a local optimum has been found. If the local optimum is also a global optimum the process stops. Otherwise, a global integral simplex method creates and solves a search tree consisting of a polynomial number of subproblems, subproblems of subproblems, etc., and the solution to at least one of which is guaranteed to be an optimal solution to the original problem. If that solution has a bounded objective then it is the optimal set partitioning solution of the original problem, but if it has an unbounded objective then the original problem has no feasible solution. It will be shown that the total number of pivots required for the global integral simplex method to solve a set partitioning problem having m rows, where m is an arbitrary but fixed positive integer, is bounded by a polynomial function of n. Preliminary computational experience is given which indicates that global method has a low order polynomial empirical performance when solving such problems.

In this paper a local integral simplex method will be described which, starting with the initial tableau of a set partitioning problem, makes pivots using the pivot on one rule until no more such pivots are possible because a local optimum has been found. If the local optimum is also a global optimum the process stops. Otherwise, a global integral simplex method creates and solves a search tree consisting of a polynomial number of subproblems, subproblems of subproblems, etc., and the solution to at least one of which is guaranteed to be an optimal solution to the original problem. If that solution has a bounded objective then it is the optimal set partitioning solution of the original problem, but if it has an unbounded objective then the original problem has no feasible solution. It will be shown that the total number of pivots required for the global integral simplex method to solve a set partitioning problem having m rows, where m is an arbitrary but fixed positive integer, is bounded by a polynomial function of n. Preliminary computational experience is given which indicates that global method has a low order polynomial empirical performance when solving such problems.

In this paper a local integral simplex method will be described which, starting with the initial tableau of a set partitioning problem, makes pivots using the pivot on one rule until no more such pivots are possible because a local optimum has been found. If the local optimum is also a global optimum the process stops. Otherwise, a global integral simplex method creates and solves a search tree consisting of a polynomial number of subproblems, subproblems of subproblems, etc., and the solution to at least one of which is guaranteed to be an optimal solution to the original problem. If that solution has a bounded objective then it is the optimal set partitioning solution of the original problem, but if it has an unbounded objective then the original problem has no feasible solution. It will be shown that the total number of pivots required for the global integral simplex method to solve a set partitioning problem having m rows, where m is an arbitrary but fixed positive integer, is bounded by a polynomial function of n. Preliminary computational experience is given which indicates that global method has a low order polynomial empirical performance when solving such problems.

We consider combinatorial optimization problems defined over random ensembles, and study how solution cost increases when the optimal solution undergoes a small perturbation delta. For the minimum spanning tree, the increase in cost scales as delta^2; for the mean-field and Euclidean minimum matching and traveling salesman problems in dimension d>=2, the increase scales as delta^3; this is observed in Monte Carlo simulations in d=2,3,4 and in theoretical analysis of a mean-field model. We speculate that the scaling exponent could serve to classify combinatorial optimization problems into a small number of distinct categories, similar to universality classes in statistical physics.

Typical-case computation complexity is a research topic at the boundary of computer science, applied mathematics, and statistical physics. In the last twenty years the replica-symmetry-breaking mean field theory of spin glasses and the associated message-passing algorithms have greatly deepened our understanding of typical-case computation complexity. In this paper we use the vertex cover problem, a basic nondeterministic-polynomial (NP)-complete combinatorial optimization problem of wide application, as an example to introduce the statistical physical methods and algorithms. We do not go into the technical details but emphasize mainly the intuitive physical meanings of the message-passing equations. A nonfamiliar reader shall be able to understand to a large extent the physics behind the mean field approaches and to adjust them in solving other optimization problems.

A procedure is presented which considerably improves the performance of local search based heuristic algorithms for combinatorial optimization problems. It increases the average `gain' of the individual local searches by merging pairs of solutions: certain parts of either solution are transcribed by the related parts of the respective other solution, corresponding to flipping clusters of a spin glass. This iterative partial transcription acts as a local search in the subspace spanned by the differing components of both solutions. Embedding it in the simple multi-start-local-search algorithm and in the thermal-cycling method, we demonstrate its effectiveness for several instances of the traveling salesman problem. The obtained results indicate that, for this task, such approaches are far superior to simulated annealing.

Contents: Introduction; Minimum Spanning Trees; Shortest Paths; Polyhedral Combinatorics; Facts of Polyhedra; Ellipsoids.

The primary activities supported by the grant during the reporting period include a new result showing the hardness of the single-source robust network design and an invitation to include this in the special issue devoted to selected papers in FOCS 2005. A summer intern was hosted, Andrew McGregor from UPenn, who worked with Shepherd on recognizing Hilbert Bases and other theoretical topics in Math Programming. A visit was also supported for Gianpaolo Oriolo (Rome), which resulted in some new joint work on robust network design. In addition, there was a week visit from Seffi Naor (Technicion). Travel supported during this period includes trips by Shepherd to UPenn to work with Sanjeev Khanna and C. Chekuri on the mutlicommodity flow problem. This work has resulted in the FOCS 2005 paper, which in addition was invited into a special issue of selected papers. Conferences attended were the 2004 APPROX/RANDOM (Chekuri) and CORC 4th Optimization Day (Shepherd).

Combinatorial optimization is widely applied in a number of areas nowadays. Unfortunately, many combinatorial optimization problems are NP-hard which usually means that they are unsolvable in practice. However, it is often unnecessary to have an exact solution. In this case one may use heuristic approach to obtain a near-optimal solution in some reasonable time. We focus on two combinatorial optimization problems, namely the Generalized Traveling Salesman Problem and the Multidimensional Assignment Problem. The first problem is an important generalization of the Traveling Salesman Problem; the second one is a generalization of the Assignment Problem for an arbitrary number of dimensions. Both problems are NP-hard and have hosts of applications. In this work, we discuss different aspects of heuristics design and evaluation. A broad spectrum of related subjects, covered in this research, includes test bed generation and analysis, implementation and performance issues, local search neighborhoods and efficient exploration algorithms, metaheuristics design and population sizing in memetic algorithm. The most important results are obtained in the areas of local search and memetic algorithms for the considered problems. In both cases we have significantly advanced the existing knowledge on the local search neighborhoods and algorithms by systematizing and improving the previous results. We have proposed a number of efficient heuristics which dominate the existing algorithms in a wide range of time/quality requirements. Several new approaches, introduced in our memetic algorithms, make them the state-of-the-art metaheuristics for the corresponding problems. Population sizing is one of the most promising among these approaches; it is expected to be applicable to virtually any memetic algorithm.