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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.

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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.

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.

There are well-established connections between combinatorial optimization, optimal transport theory and Hydrodynamics, through the linear assignment problem in combinatorics, the Monge-Kantorovich problem in optimal transport theory and the model of inviscid, potential, pressure-less fluids in Hydrodynamics. Here, we consider the more challenging quadratic assignment problem (which is NP, while the linear assignment problem is just P) and find, in some particular case, a correspondence with the problem of finding stationary solutions of Euler's equations for incompressible fluids. For that purpose, we introduce and analyze a suitable "gradient flow" equation. Combining some ideas of P.-L. Lions (for the Euler equations) and Ambrosio-Gigli-Savar\'e (for the heat equation), we provide for the initial value problem a concept of generalized "dissipative" solutions which always exist globally in time and are unique whenever theyare smooth.

There are well-established connections between combinatorial optimization, optimal transport theory and Hydrodynamics, through the linear assignment problem in combinatorics, the Monge-Kantorovich problem in optimal transport theory and the model of inviscid, potential, pressure-less fluids in Hydrodynamics. Here, we consider the more challenging quadratic assignment problem (which is NP, while the linear assignment problem is just P) and find, in some particular case, a correspondence with the problem of finding stationary solutions of Euler's equations for incompressible fluids. For that purpose, we introduce and analyze a suitable "gradient flow" equation. Combining some ideas of P.-L. Lions (for the Euler equations) and Ambrosio-Gigli-Savar\'e (for the heat equation), we provide for the initial value problem a concept of generalized "dissipative" solutions which always exist globally in time and are unique whenever theyare smooth.

This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems.

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 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.

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 continue the discussion of how a 2k-edge-connected graph can be oriented so that the resulting digraph is k-arc-connected. Last time we have seen that this can be achieved using submodular �ows. Today we present a di_erent approach, which relates the problem to matroid intersection.

In this lecture we will cover: 1. Topics related to Edmonds-Gallai decompositions 2. Factor critica-raphs and ear-decompositions.

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of $\alpha$-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statistical-mechanical analyses reveal for $\alpha=2$ that the LP optimal solution is typically equal to that given by the IP below the critical average degree $c=e$ in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result $c=1$, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with $\alpha\geq 3$, minimum vertex covers on $\alpha$-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree $c=e/(\alpha-1)$ where the replica symmetry is broken.

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of $\alpha$-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statistical-mechanical analyses reveal for $\alpha=2$ that the LP optimal solution is typically equal to that given by the IP below the critical average degree $c=e$ in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result $c=1$, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with $\alpha\geq 3$, minimum vertex covers on $\alpha$-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree $c=e/(\alpha-1)$ where the replica symmetry is broken.

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of $\alpha$-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statistical-mechanical analyses reveal for $\alpha=2$ that the LP optimal solution is typically equal to that given by the IP below the critical average degree $c=e$ in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result $c=1$, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with $\alpha\geq 3$, minimum vertex covers on $\alpha$-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree $c=e/(\alpha-1)$ where the replica symmetry is broken.

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of $\alpha$-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statistical-mechanical analyses reveal for $\alpha=2$ that the LP optimal solution is typically equal to that given by the IP below the critical average degree $c=e$ in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result $c=1$, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with $\alpha\geq 3$, minimum vertex covers on $\alpha$-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree $c=e/(\alpha-1)$ where the replica symmetry is broken.

Database query optimization plays a crucial role in the efficiency of database management systems (DBMS). As data complexity increases, optimizing query performance becomes even more critical. This paper investigates the application of combinatorial mathematics in query optimization, particularly focusing on the algorithms and techniques that aid in determining optimal query execution plans. The role of combinatorial algorithms in solving problems related to join ordering, query enumeration, and graph traversal is examined. By leveraging combinatory, database systems can process queries more efficiently, improving resource management, response time, and overall system performance.

Database query optimization plays a crucial role in the efficiency of database management systems (DBMS). As data complexity increases, optimizing query performance becomes even more critical. This paper investigates the application of combinatorial mathematics in query optimization, particularly focusing on the algorithms and techniques that aid in determining optimal query execution plans. The role of combinatorial algorithms in solving problems related to join ordering, query enumeration, and graph traversal is examined. By leveraging combinatory, database systems can process queries more efficiently, improving resource management, response time, and overall system performance.

Database query optimization plays a crucial role in the efficiency of database management systems (DBMS). As data complexity increases, optimizing query performance becomes even more critical. This paper investigates the application of combinatorial mathematics in query optimization, particularly focusing on the algorithms and techniques that aid in determining optimal query execution plans. The role of combinatorial algorithms in solving problems related to join ordering, query enumeration, and graph traversal is examined. By leveraging combinatory, database systems can process queries more efficiently, improving resource management, response time, and overall system performance.

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.

Last lecture we covered matroid intersection, and de�ned matroid union. In this lecture we review the de�nitions of matroid intersection, and then show that the matroid intersection polytope is TDI. This is Chapter 41 in Schrijver�s book. Next we review matroid union, and show that unlike matroid intersection, the union of two matroids is again a matroid. This material is largely contained in Chapter 42 in Schrijver�s book. We leave testing independence in the union matroid for the next lecture.

By now Artificial Intelligence (AI), Theoretical Computer Science (CS theory) and Operations Research (OR) have investigated a variety of search and optimization problems. However, methods from these scientific areas use different problem descriptions, models, and tools. They also address problems with particular efficiency requirements. For example, approaches from CS theory are mainly concerned with the worst-case scenarios and are not focused on empirical performance. A few efforts have tried to apply methods across areas. Usually a significant amount of work is required to make different approaches "talk the same language," be successfully implemented, and, finally, solve the actual same problem with an overall acceptable efficiency. This thesis presents a systematic approach that attempts to advance the state of the art in the transfer of knowledge across the above mentioned areas. In this work we investigate a number of problems that belong to or are close to the intersection of areas of interest of AI, OR and CS theory. We illustrate the advantages of considering knowledge available in different scientific areas and of applying algorithms across distinct disciplines through successful applications of novel hybrid algorithms that utilize benefitial features of known efficient approaches. Testbeds for such applications in this thesis work include both open theoretical problems and ones of significant practical importance. We introduce a representation change that enables us to question the relation between the Pigeonhole Principle and Linear Programming Relaxation. We show that both methods have exactly the same bounding power. Furthermore, even stronger relation appears to be between the two methods: The Pigeonhole Principle is the Dual of Linear Programming Relaxation. Such a relation explains the "hidden magic" of the Pigeonhole Principle, namely its power in establishing upper bounds and its effectiveness in constructing optimal solutions.

A linear program consists of linear constraints with the goal of maximizing or minimizing a linear objective function subject to the constraints

The Matroid matching Problem: Given a matroid M = (S, I), let E be a set of pairs on S.

This next theorem of Edmonds states that these three conditions determine the perfect matching polytope of any graph.

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).

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.

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.

This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0-1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function. The performance and utility of the proposed algorithms are demonstrated with the ON/OFF control problems of interdependent refrigeration systems.

Firefly algorithms belong to modern meta-heuristic algorithms inspired by nature that can be successfully applied to continuous optimization problems. In this paper, we have been applied the firefly algorithm, hybridized with local search heuristic, to combinatorial optimization problems, where we use graph 3-coloring problems as test benchmarks. The results of the proposed memetic firefly algorithm (MFFA) were compared with the results of the Hybrid Evolutionary Algorithm (HEA), Tabucol, and the evolutionary algorithm with SAW method (EA-SAW) by coloring the suite of medium-scaled random graphs (graphs with 500 vertices) generated using the Culberson random graph generator. The results of firefly algorithm were very promising and showed a potential that this algorithm could successfully be applied in near future to the other combinatorial optimization problems as well.

We consider the assortment optimization problem with disjoint-cardinality constraints under two-level nested logit model. To solve this problem, we first identify a candidate set with $O(mn^2)$ assortments and show that at least one optimal assortment is included in this set. Based on this observation, a fast algorithm, which runs in $O(m n^2 \log mn)$ time, is proposed to find an optimal assortment.

In this paper we develop two approaches to find minmax robust efficient solutions for multi-objective combinatorial optimization problems with cardinality-constrained uncertainty. First, we extend an algorithm of Bertsimas and Sim (2003) for the single-objective problem to multi-objective optimization. We propose also an enhancement to accelerate the algorithm, even for the single-objective case, and we develop a faster version for special multi-objective instances. Second, we introduce a deterministic multi-objective problem with sum and bottleneck functions, which provides a superset of the robust efficient solutions. Based on this, we develop a label setting algorithm to solve the multi-objective uncertain shortest path problem. We compare both approaches on instances of the multi-objective uncertain shortest path problem originating from hazardous material transportation.

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.

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.

Modern approaches to the search of Relative and Global minima of potential energy function of Biomacromolecular structures include techniques of combinatorial optimization like the study of Steiner Points and Steiner Trees. These methods have been successfully applied to the problem of modelling the configurations of the average atomic positions when they are disposed in the usual sequence of evenly spaced points along right circular helices. In the present contribution, we intend to show how these methods can be adapted for explaining the advantages of introducing the concept of a Steiner Ratio Function (SRF). We also show how this new concept is adequate for fitting the results obtained by computing experiments and for providing an improvement to these results if we use the restriction of working with Full Steiner Trees.

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.

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.

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.

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.

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.

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.

We improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set $K\subset \mathbb{R}^n$ contained in a box of radius $R$, we show how to either find a point in $K$ or prove that $K$ does not contain a ball of radius $\epsilon$ using an expected $O(n\log(nR/\epsilon))$ oracle evaluations and additional time $O(n^3\log^{O(1)}(nR/\epsilon))$. This matches the oracle complexity and improves upon the $O(n^{\omega+1}\log(nR/\epsilon))$ additional time of the previous fastest algorithm achieved over 25 years ago by Vaidya for the current matrix multiplication constant $\omega

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.

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.