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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.

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 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.

The minmax regret problem for combinatorial optimization under uncertainty can be viewed as a zero-sum game played between an optimizing player and an adversary, where the optimizing player selects a solution and the adversary selects costs with the intention of maximizing the regret of the player. The existing minmax regret model considers only deterministic solutions/strategies, and minmax regret versions of most polynomial solvable problems are NP-hard. In this paper, we consider a randomized model where the optimizing player selects a probability distribution (corresponding to a mixed strategy) over solutions and the adversary selects costs with knowledge of the player's distribution, but not its realization. We show that under this randomized model, the minmax regret version of any polynomial solvable combinatorial problem becomes polynomial solvable. This holds true for both the interval and discrete scenario representations of uncertainty. Using the randomized model, we show new proofs of existing approximation algorithms for the deterministic model based on primal-dual approaches. Finally, we prove that minmax regret problems are NP-hard under general convex uncertainty.

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.

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.

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.

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.

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

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.

In this lecture, we will: � Present Edmonds� algorithm for computing a maximum matching in a (not necessarily bipartite) graph G. � Use the analysis of the algorithm to derive the Edmonds-Gallai Decomposition Theorem stated in the last lecture.

De�nition 1: A matroid M = (S, I) is a �nite ground set S together with a collection of sets...

Today we will brie�y survey matroid representation and then discuss some problems in matroid optimization and the corresponding applications. The tools we develop will help us answer the following puzzle: Puzzle: A game is played on a graph G(V, E) and has two players, George and Ari. Ari�s moves consist of ��xing� edges e _ E. George�s moves consist of deleting any un�xed edge. The game ends when every edge has been either �xed or deleted. Ari wins if the graph at the end of the game is connected (i.e. if the �xed edges form a spanning tree). Otherwise George wins. Supposing George moves �rst, characterize the graphs in which George has a winning strategy.

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.

In this lecture, we introduce the complementary slackness conditions and use them to obtain a primal-dual method for solving linear programming.

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 {\em risk-averse} or {\em risk-prone} behaviors, and instead we consider a more general objective which is to maximize the {\em 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 {\em additive error} $\epsilon$ for any $\epsilon>0$, if there is a pseudopolynomial time algorithm for the {\em 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.

This paper presents a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning. We focus on the traveling salesman problem (TSP) and train a recurrent network that, given a set of city coordinates, predicts a distribution over different city permutations. Using negative tour length as the reward signal, we optimize the parameters of the recurrent network using a policy gradient method. We compare learning the network parameters on a set of training graphs against learning them on individual test graphs. Despite the computational expense, without much engineering and heuristic designing, Neural Combinatorial Optimization achieves close to optimal results on 2D Euclidean graphs with up to 100 nodes. Applied to the KnapSack, another NP-hard problem, the same method obtains optimal solutions for instances with up to 200 items.

This paper presents a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning. We focus on the traveling salesman problem (TSP) and train a recurrent network that, given a set of city coordinates, predicts a distribution over different city permutations. Using negative tour length as the reward signal, we optimize the parameters of the recurrent network using a policy gradient method. We compare learning the network parameters on a set of training graphs against learning them on individual test graphs. Despite the computational expense, without much engineering and heuristic designing, Neural Combinatorial Optimization achieves close to optimal results on 2D Euclidean graphs with up to 100 nodes. Applied to the KnapSack, another NP-hard problem, the same method obtains optimal solutions for instances with up to 200 items.

tecnologie segrete

The talk focuses on expander graphs in conjunction with the combined use of SDPs and eigenvalue techniques for approximating optimal solutions to combinatorial optimization problems. In the first part of the talk I will explain how to construct cost-effective, expanding networks by using 'local' sparsifiers of graphs that emerge as a solution to a semidefinite program. In the second part of the talk I will show that the Unique Games Conjecture is false when the underlying constraint graph is a (spectral) expander. Namely, I will present a polynomial-time algorithm for Unique Games on expanding instances that finds a good assignment when there exists one. ©2009 Microsoft Corporation. All rights reserved.

The talk focuses on expander graphs in conjunction with the combined use of SDPs and eigenvalue techniques for approximating optimal solutions to combinatorial optimization problems. In the first part of the talk I will explain how to construct cost-effective, expanding networks by using 'local' sparsifiers of graphs that emerge as a solution to a semidefinite program. In the second part of the talk I will show that the Unique Games Conjecture is false when the underlying constraint graph is a (spectral) expander. Namely, I will present a polynomial-time algorithm for Unique Games on expanding instances that finds a good assignment when there exists one. ©2009 Microsoft Corporation. All rights reserved.

The talk focuses on expander graphs in conjunction with the combined use of SDPs and eigenvalue techniques for approximating optimal solutions to combinatorial optimization problems. In the first part of the talk I will explain how to construct cost-effective, expanding networks by using 'local' sparsifiers of graphs that emerge as a solution to a semidefinite program. In the second part of the talk I will show that the Unique Games Conjecture is false when the underlying constraint graph is a (spectral) expander. Namely, I will present a polynomial-time algorithm for Unique Games on expanding instances that finds a good assignment when there exists one. ©2009 Microsoft Corporation. All rights reserved.

The talk focuses on expander graphs in conjunction with the combined use of SDPs and eigenvalue techniques for approximating optimal solutions to combinatorial optimization problems. In the first part of the talk I will explain how to construct cost-effective, expanding networks by using 'local' sparsifiers of graphs that emerge as a solution to a semidefinite program. In the second part of the talk I will show that the Unique Games Conjecture is false when the underlying constraint graph is a (spectral) expander. Namely, I will present a polynomial-time algorithm for Unique Games on expanding instances that finds a good assignment when there exists one. ©2009 Microsoft Corporation. All rights reserved.

We study the moduli space of euclidean structures with cone points on a surface, and describe a decomposition into cells each of which corresponds to a given combinatorial type of Delaunay tessellation. We use some of the ideas to study hyperbolic structures on three-dimensional manifolds

We study the problem of compressing massive tables within the partition-training paradigm introduced by Buchsbaum et al. [SODA'00], in which a table is partitioned by an off-line training procedure into disjoint intervals of columns, each of which is compressed separately by a standard, on-line compressor like gzip. We provide a new theory that unifies previous experimental observations on partitioning and heuristic observations on column permutation, all of which are used to improve compression rates. Based on the theory, we devise the first on-line training algorithms for table compression, which can be applied to individual files, not just continuously operating sources; and also a new, off-line training algorithm, based on a link to the asymmetric traveling salesman problem, which improves on prior work by rearranging columns prior to partitioning. We demonstrate these results experimentally. On various test files, the on-line algorithms provide 35-55% improvement over gzip with negligible slowdown; the off-line reordering provides up to 20% further improvement over partitioning alone. We also show that a variation of the table compression problem is MAX-SNP hard.

Linear programming is a very general and widely used framework. In this thesis we consider several combinatorial optimization problems that can be viewed as classes of linear programming problems with special structure. It is known that polynomial time algorithms exist for the general linear programming problem. It is not known, however, whether any of them are strongly polynomial. In addition, it seems that the general problem is inherently sequential. For problems with special structure, our goals are to develop sequential and parallel algorithms that are faster than those known for general linear programming and to determine whether strongly polynomial algorithms exist. (1) We develop a technique that extends the classes of problems known to have strongly polynomial algorithms, or known to be quickly solvable in parallel. This technique is used to obtain a fast parallel algorithm and a strongly polynomial algorithm for detecting cycles in periodic graphs of fixed dimension. We mention additional applications to parametric extensions of problems where the number of parameters is fixed. (2) We introduce algorithms for solving linear systems where each inequality involves at most two variables. These algorithms improve over the sequential and parallel running times of previous algorithms. These results are combined with additional ideas to yield faster algorithms for some general network flow problems.

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.

The main activities supported by the contract were research support (primarily travel) for C. Chekuri, B. Shepherd and P. Winkler. Funds were also used to run the Bell Labs Pow Wow workshop which addressed various problems in the field of combinatorial optimization algorithm design. The workshop resulted in fruitful research papers and has sparked other collaborations, which may lead to new results in the future. The contract also supported a 2-week visit by Bill Cunningham who worked with G. Wilfong and B. Shepherd on some generalized coloring problems arising from the Clos-Network switch designs. Some new technical results were enabled by the contract in the areas of All or Nothing Problems, Combinatorial Probability and Generalized Coloring and Optical Networking.

Many combinatorial optimization algorithms have no mechanism to capture inter-parameter dependencies. However, modeling such dependencies may allow an algorithm to concentrate its sampling more effectively on regions of the search space which have appeared promising in the past. We present an algorithm which incrementally learns second-order probability distributions from good solutions seen so far, uses these statistics to generate optimal (in terms of maximum likelihood) dependency trees to model these distributions, and then stochastically generates new candidate solutions from these trees. We test this algorithm on a variety of optimization problems. Our results indicate superior performance over other tested algorithms that either (1) do not explicitly use these dependencies, or (2) use these dependencies to generate a more restricted class of dependency graphs.

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.

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.

Research activities discussed: Mathematical methods; mathematical programming and optimal control; decision analysis, statistics and stochastic systems; models and applications; urban and other public systems; traffic and transportation; industrial and management systems; energy; economics; and computation. (Author)

The main goal of this project is to develop fundamental algorithmic techniques that can be applied to problems that arise in the context of high speed communications networks. The emphasis is on efficient algorithms for resource management. This research project has two main components The first is development of well founded algorithmic techniques for online resource allocation where one needs to make decisions based on partial data and without knowledge of the future. The main goal is to develop techniques that lead to provable guarantees on worst case performance and ensures good performance on average. Research on these problems consists of both theoretical analysis and simulation based studies. The second component is design of efficient offline resource allocation algorithms based on multi commodity flow techniques. The difference between this work and existing efforts is that we are designing and tuning the algorithms to produce approximate solutions instead of designing algorithms to produce exact solutions in theory. The result is code that is orders of magnitude faster than existing code and which achieves a precision of better than 1.

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.

In classic robust optimization, it is assumed that a set of possible parameter realizations, the uncertainty set, is modeled in a previous step and part of the input. As recent work has shown, finding the most suitable uncertainty set is in itself already a difficult task. We consider robust problems where the uncertainty set is not completely defined. Only the shape is known, but not its size. Such a setting is known as variable-sized uncertainty. In this work we present an approach how to find a single robust solution, that performs well on average over all possible uncertainty set sizes. We demonstrate that this approach can be solved efficiently for min-max robust optimization, but is more involved in the case of min-max regret, where positive and negative complexity results for the selection problem, the minimum spanning tree problem, and the shortest path problem are provided. We introduce an iterative solution procedure, and evaluate its performance in an experimental comparison.

We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, i.e., its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of MOCO problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions.

We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, i.e., its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of MOCO problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions.

Optimization problems are concerned with the efficient use or allocation of limited resources to meet desired objectives. These problems are characterized by the large number of alternatives that satisfy the basic conditions of each problem. The selection of a particular solution as the best solution to a problem depends on some goal or overall objective. The versatility of the combinatorial model stems from the fact that in many practical problems, activities or resources, such as machines, airplanes, missile target sites, and people are indivisible. Also, many problems have only a finite number of alternative choices and consequently can appropriately be formulated as combinatorial problems. We refer the reader to the following texts and their bibliographical references for further review of some of these important engineering and managerial decision problems: Combinatorial and Integer Programming (Nemhauser and Wolsey), Applied Mathematieal Programming (Bradley, Hax and Magnanti), Principles of Operations Research (Wagner), and Model Building in Mathematical Programming (Williams).

Presented in this paper is a derivation of a 2D catalytic reaction-based model to solve combinatorial optimization problems (COPs). The simulated catalytic reactions, a computational metaphor, occurs in an artificial chemical reactor that finds near-optimal solutions to COPs. The artificial environment is governed by catalytic reactions that can alter the structure of artificial molecular elements. Altering the molecular structure means finding new solutions to the COP. The molecular mass of the elements was considered as a measure of goodness of fit of the solutions. Several data structures and matrices were used to record the directions and locations of the molecules. These provided the model the 2D topology. The Traveling Salesperson Problem (TSP) was used as a working example. The performance of the model in finding a solution for the TSP was compared to the performance of a topology-less model. Experimental results show that the 2D model performs better than the topology-less one.

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.

Let M1 = (S, I1), M2 = (S, I2) be two matroids on common ground set S with rank functions r1 and r2. Many combinatorial optimization problems can be reformulated as the problem of �nding the maximum size common independent set ..

An NP-hard combinatorial optimization problem $\Pi$ is said to have an {\em approximation threshold} if there is some $t$ such that the optimal value of $\Pi$ can be approximated in polynomial time within a ratio of $t$, and it is NP-hard to approximate it within a ratio better than $t$. We survey some of the known approximation threshold results, and discuss the pattern that emerges from the known results.

Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares \emph{etc}. that exhibit scale-free statistics much in the spirit of the logarithmic Ritcher scale for earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs) exhibiting this type of dynamics belong to the finite-width phase (N-phase for brevity) that precedes ordinary chaotic behavior and that is known under such names as noise-induced chaos, self-organized criticality, dynamical complexity \emph{etc.} Within the recently formulated approximation-free supersymemtric theory of stochastics, the N-phase can be roughly interpreted as the noise-induced "overlap" between integrable and chaotic deterministic dynamics. As a result, the N-phase dynamics inherits the properties of the both. Here, we analyze this unique set of properties and conclude that the N-phase DSs must naturally be the most efficient optimizers: on one hand, N-phase DSs have integrable flows with well-defined attractors that can be associated with candidate solutions and, on the other hand, the noise-induced attractor-to-attractor dynamics in the N-phase is effectively chaotic or a-periodic so that a DS must avoid revisiting solutions/attractors thus accelerating the search for the best solution. Based on this understanding, we propose a method for stochastic dynamical optimization using the N-phase DSs. This method can be viewed as a hybrid of the simulated and chaotic annealing methods. Our proposition can result in a new generation of hardware devices for efficient solution of various search and/or combinatorial optimization problems.

This article is from BMC Systems Biology , volume 6 . Abstract Background: The discovery of single-nucleotide polymorphisms (SNPs) has important implications in a variety of genetic studies on human diseases and biological functions. One valuable approach proposed for SNP discovery is based on base-specific cleavage and mass spectrometry. However, it is still very challenging to achieve the full potential of this SNP discovery approach. Results: In this study, we formulate two new combinatorial optimization problems. While both problems are aimed at reconstructing the sample sequence that would attain the minimum number of SNPs, they search over different candidate sequence spaces. The first problem, denoted as SNP - MSP, limits its search to sequences whose in silico predicted mass spectra have all their signals contained in the measured mass spectra. In contrast, the second problem, denoted as SNP - MSQ, limits its search to sequences whose in silico predicted mass spectra instead contain all the signals of the measured mass spectra. We present an exact dynamic programming algorithm for solving the SNP - MSP problem and also show that the SNP - MSQ problem is NP-hard by a reduction from a restricted variation of the 3-partition problem. Conclusions: We believe that an efficient solution to either problem above could offer a seamless integration of information in four complementary base-specific cleavage reactions, thereby improving the capability of the underlying biotechnology for sensitive and accurate SNP discovery.

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

Harmony search algorithm (HSA) is relatively considered as one of the most recent metaheuristic algorithms. HSA is a modern-nature algorithm that simulates the musicians’ natural process of musical improvisation to enhance their instrument’s note to find a state of pleasant (harmony) according to aesthetic standards. Lots of variants of HSA have been suggested to tackle combinatorial optimization problems. They range from hybridizing some components of other metaheuristic approaches (to improve the HSA) to taking some concepts of HSA and utilizing them to improve other metaheuristic methods. This study reviews research pertaining to parameter settings of HSA and its applications to efficiently solve hard combinatorial optimization problems.

A significant progress has been made in the past three decades over the study of combinatorial NP optimization problems and their associated optimization and approximate classes, such as NPO, PO, APX (or APXP), and PTAS. Unfortunately, a collection of problems that are simply placed inside the P-solvable optimization class PO never have been studiously analyzed regarding their exact computational complexity. To improve this situation, the existing framework based on polynomial-time computability needs to be expanded and further refined for an insightful analysis of various approximation algorithms targeting optimization problems within PO. In particular, we deal with those problems characterized in terms of logarithmic-space computations and uniform-circuit computations. We are focused on nondeterministic logarithmic-space (NL) optimization problems or NPO problems. Our study covers a wide range of optimization and approximation classes, dubbed as, NLO, LO, APXL, and LSAS as well as new classes NC1O, APXNC1, NC1AS, and AC0O, which are founded on uniform families of Boolean circuits. Although many NL decision problems can be naturally converted into NL optimization (NLO) problems, few NLO problems have been studied vigorously. We thus provide a number of new NLO problems falling into those low-complexity classes. With the help of NC1 or AC0 approximation-preserving reductions, we also identify the most difficult problems (known as complete problems) inside those classes. Finally, we demonstrate a number of collapses and separations among those refined optimization and approximation classes with or without unproven complexity-theoretical assumptions.

A significant progress has been made in the past three decades over the study of combinatorial NP optimization problems and their associated optimization and approximate classes, such as NPO, PO, APX (or APXP), and PTAS. Unfortunately, a collection of problems that are simply placed inside the P-solvable optimization class PO never have been studiously analyzed regarding their exact computational complexity. To improve this situation, the existing framework based on polynomial-time computability needs to be expanded and further refined for an insightful analysis of various approximation algorithms targeting optimization problems within PO. In particular, we deal with those problems characterized in terms of logarithmic-space computations and uniform-circuit computations. We are focused on nondeterministic logarithmic-space (NL) optimization problems or NPO problems. Our study covers a wide range of optimization and approximation classes, dubbed as, NLO, LO, APXL, and LSAS as well as new classes NC1O, APXNC1, NC1AS, and AC0O, which are founded on uniform families of Boolean circuits. Although many NL decision problems can be naturally converted into NL optimization (NLO) problems, few NLO problems have been studied vigorously. We thus provide a number of new NLO problems falling into those low-complexity classes. With the help of NC1 or AC0 approximation-preserving reductions, we also identify the most difficult problems (known as complete problems) inside those classes. Finally, we demonstrate a number of collapses and separations among those refined optimization and approximation classes with or without unproven complexity-theoretical assumptions.