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

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

In Combinatorial Public Projects, there is a set of projects that may be undertaken, and a set of self-interested players with a stake in the set of projects chosen. A public planner must choose a subset of these projects, subject to a resource constraint, with the goal of maximizing social welfare. Combinatorial Public Projects has emerged as one of the paradigmatic problems in Algorithmic Mechanism Design, a field concerned with solving fundamental resource allocation problems in the presence of both selfish behavior and the computational constraint of polynomial-time. We design a polynomial-time, truthful-in-expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental variant of combinatorial public projects. Our results apply to combinatorial public projects when players have valuations that are 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, assuming P != NP. Ours is the first mechanism that achieves a constant factor approximation for a natural NP-hard variant of combinatorial public projects.

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.

In this paper the approach to solving several combinatorial optimization problems using the local search and the genetic algorithm techniques is proposed. Initially this approach was developed in purpose to overcome some difficulties inhibiting the application of above mentioned techniques to the problems of the Questionnaire Theory. But when the algorithms were developed it became clear that them could be successfully applied also to the Minimum Set Cover, the 0-1-Knapsack and probably to other combinatorial optimization problems.

The random cost problem is the problem of finding the minimum in an exponentially long list of random numbers. By definition, this problem cannot be solved faster than by exhaustive search. It is shown that a classical NP-hard optimization problem, number partitioning, is essentially equivalent to the random cost problem. This explains the bad performance of heuristic approaches to the number partitioning problem and allows us to calculate the probability distributions of the optimum and sub-optimum costs.

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.

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).

Estimation of Distribution Algorithms (EDAs) require flexible probability models that can be efficiently learned and sampled. Generative Adversarial Networks (GAN) are generative neural networks which can be trained to implicitly model the probability distribution of given data, and it is possible to sample this distribution. We integrate a GAN into an EDA and evaluate the performance of this system when solving combinatorial optimization problems with a single objective. We use several standard benchmark problems and compare the results to state-of-the-art multivariate EDAs. GAN-EDA doe not yield competitive results - the GAN lacks the ability to quickly learn a good approximation of the probability distribution. A key reason seems to be the large amount of noise present in the first EDA generations.

In this lecture, we continue with more results on matroid union, as well as tie together some loose ends from the past couple of lectures.

Consider a planar graph G = (V,E) and a set of terminal pairs R = {(si,ti): i = 1,k}. Assume G is planar, (V, E _ R) is Eulerian, and all terminals lie on the outer face of G. In this lecture, we will cover the following results.

In this lecture, we will be concerned with graph coverings by a collection of paths or cycles. The goal will be to cover all the vertices by a small number of either paths or cycles, and this number will be bounded by the independence number _(G). (_(G) is the maximum size of an independent, or stable, set, i.e. a set of vertices inducing no edges.) For a directed graph D, _(D) refers to the corresponding undirected graph. Let�s start with the following statement, proved by Gallai and Milgram in 1960.

This lecture is about jump systems. While they are brie�y discussed in chapter 41 of Schrijver�s book, they are not covered extensively.

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.

In this lecture, we investigate the relationship between total dual integrality and integrality of polytopes. We then use a theorem on total dual integrality to provide a new proof of the Tutte-Berge formula.

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

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

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

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 solving 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 a large space structure 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 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.

This project focused on designing fast algorithms for basic combinational optimization problems, including maximum flow, matching, multicommodity flow, and generalized flow. Many important applications are naturally stated as variants of these problems, and hence improved algorithms for these problems immediately lead to improved algorithms for a wide variety of applications. Our goal was to improve both sequential and parallel complexity. In many applications, solving a multicommodity or a generalized flow problem is only a first step in approximately solving an NP-complete problem; in the majority of such cases there is no need to have an exact solution of the problem. One of the focuses of the project was design of efficient approximation algorithms.

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.

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.

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.

The main activities supported under this grant are research and support for C. Chekuri, B. Shepherd and P. Winkler. Funds also supported one summer intern, Andrew McGregory from U-Penn, who worked with Shepherd on recognizing Hilbert Bases and other theoretical topics in Math Programming. Visits from scientists include a 2-week visit from Gianpaolo Oriolo (Rome), which resulted in new joint work on robust network design, a week visit from Seffi Naor (Technician) and a visit from Anreas Sebo who spoke on a new result of Bessy and Thomasse that solves an old conjecture of Gallai. Research highlights this year include: proof that in planar graphs with all capacities at least 2, the integrality gap for edge-disjoint paths is polylogarithmic (the paper was invited for the selected papers issue devoted to FOCS 2005); a first result showing the hardness of the robust network design and introduction of the single-source hose model for robust networks; and an unlikely question: is it hard to determine whether the rows of 0,1 matrix form a Hilbert Basis? Conferences attended were the 2004 APPROX/RANDOM (Chekuri), CORC 4th Optimization Day, INOC, Aussois workshop (Shepherd), and a workshop in Bertinoro, Italy (Chekuri & Shepherd).

Last time, we saw that the matching polytope was de�ned by: ...

Contents: Introduction; Preliminaries; Matching and Vertex Cover in Bipartite Graphs; Spanning Trees; Matroids; Arborescence and Rooted Connectivity ;Submodular Flows and Applications ; Network Matrices; Matchings ; Network Design; Constrained Optimization Problems;Cut Problems; Iterative Relaxation: Early and Recent Examples. Lecture Notes Collection FreeScience.info ID2817 Obtained from http://research.microsoft.com/en-us/um/people/mohits/book/book.pdf http://www.freescience.info/go.php?pagename=books&id=2817

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.

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 ..

Pattern self-assembly tile set synthesis (PATS) is a combinatorial optimization problem which aim at minimizing a rectilinear tile assembly system (RTAS) that uniquely self-assembles a given rectangular pattern, and is known to be NP-hard. PATS gets practically meaningful when it is parameterized by a constant c such that any given pattern is guaranteed to contain at most c colors (c-PATS). We first investigate simple patterns and properties of minimum RTASs for them. Then based on them, we design a 59-colored pattern to which 3SAT is reduced, and prove that 59-PATS is NP-hard.

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.

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.

The Lovasz splitting-o_ lemma - Lovasz�s splitting-o_ lemma states the following.

A directed graph is a graph in which every edge has a direction. A capacity is the maximum flow allowed on an edge, and is represented by ci,j, where the edge connects the vertices i and j in the direction from i to j.

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.

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

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.

We analyze combinatorial optimization problems over a pair of random point sets of equal cardinal. Typical examples include the matching of minimal length, the traveling salesperson tour constrained to alternate between points of each set, or the connected bipartite r-regular graph of minimal length. As the cardinal of the sets goes to infinity, we investigate the convergence of such bipartite functionals.

We analyze combinatorial optimization problems over a pair of random point sets of equal cardinal. Typical examples include the matching of minimal length, the traveling salesperson tour constrained to alternate between points of each set, or the connected bipartite r-regular graph of minimal length. As the cardinal of the sets goes to infinity, we investigate the convergence of such bipartite functionals.

Libraries of liquid-phase catalyst precursor solutions were printed onto iridium-coated silicon substrates and evaluated for their effectiveness in catalyzing the growth of multi-walled carbon nanotubes (MWNTs) by chemical vapor deposition (CVD). The catalyst precursor solutions were composed of inorganic salts and a removable tri-block copolymer (EO)20(PO)70(EO)20 (EO = ethylene oxide, PO = propylene oxide) structure-directing agent (SDA), dissolved in ethanol/methanol mixtures. Sample libraries were quickly assayed using scanning electron microscopy after CVD growth to identify active catalysts and CVD conditions. Composition libraries and focus libraries were then constructed around the active spots identified in the discovery libraries to understand how catalyst precursor composition affects the yield, density, and quality of the nanotubes. Successful implementation of combinatorial optimization methods in the development of highly active, carbon nanotube catalysts is demonstrated, as well as the identification of catalyst formulations that lead to varying densities and shapes of aligned nanotube towers.

Libraries of liquid-phase catalyst precursor solutions were printed onto iridium-coated silicon substrates and evaluated for their effectiveness in catalyzing the growth of multi-walled carbon nanotubes (MWNTs) by chemical vapor deposition (CVD). The catalyst precursor solutions were composed of inorganic salts and a removable tri-block copolymer (EO)20(PO)70(EO)20 (EO = ethylene oxide, PO = propylene oxide) structure-directing agent (SDA), dissolved in ethanol/methanol mixtures. Sample libraries were quickly assayed using scanning electron microscopy after CVD growth to identify active catalysts and CVD conditions. Composition libraries and focus libraries were then constructed around the active spots identified in the discovery libraries to understand how catalyst precursor composition affects the yield, density, and quality of the nanotubes. Successful implementation of combinatorial optimization methods in the development of highly active, carbon nanotube catalysts is demonstrated, as well as the identification of catalyst formulations that lead to varying densities and shapes of aligned nanotube towers.

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.

Evolutionary algorithms based on modeling the statistical dependencies (interactions) between the variables have been proposed to solve a wide range of complex problems. These algorithms learn and sample probabilistic graphical models able to encode and exploit the regularities of the problem. This paper investigates the effect of using probabilistic modeling techniques as a way to enhance the behavior of MOEA/D framework. MOEA/D is a decomposition based evolutionary algorithm that decomposes a multi-objective optimization problem (MOP) in a number of scalar single-objective subproblems and optimizes them in a collaborative manner. MOEA/D framework has been widely used to solve several MOPs. The proposed algorithm, MOEA/D using probabilistic Graphical Models (MOEA/D-GM) is able to instantiate both univariate and multi-variate probabilistic models for each subproblem. To validate the introduced framework algorithm, an experimental study is conducted on a multi-objective version of the deceptive function Trap5. The results show that the variant of the framework (MOEA/D-Tree), where tree models are learned from the matrices of the mutual information between the variables, is able to capture the structure of the problem. MOEA/D-Tree is able to achieve significantly better results than both MOEA/D using genetic operators and MOEA/D using univariate probability models, in terms of the approximation to the true Pareto front.