Integer and Combinatorial Optimization Solutions: 24 Examples with Detailed Explanations

The IPCO conference is under the auspices of the Mathematical Optimization Society and is held every year, except for those in which the International Symposium on Mathematical Programming takes place. The conference is a forum for researchers and practitioners working on various aspects of integer programming and combinatorial optimization. The aim is to present recent developments in theory, computation, and applications in these areas.

integer and combinatorial optimization solution 24

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The IPCO conference is a forum for researchers and practitioners working on various aspects of integer programming and combinatorial optimization. The aim is to present recent developments in theory, computation, and applications. The scope of IPCO is viewed in a broad sense, to include algorithmic and structural results in integer programming and combinatorial optimization as well as revealing computational studies and novel applications of discrete optimization to practical problems.

Amitabh Basu is a professor in the Dept. of Applied Mathematics and Statistics at Johns Hopkins University. He received his Ph.D. from Carnegie Mellon University in 2010 and did postdoctoral work in the Dept. of Mathematics at the University of California, Davis from 2010-2013. His main research interests are in mathematical optimization and its applications, with an emphasis on problems with a combinatorial or discrete flavor. He serves on the editorial boards of Mathematics of Operations Research, Discrete Optimization, MOS-SIAM Series on Optimization, Mathematical Programming, and SIAM Journal on Optimization. His work has been recognized by the NSF Career award and the Egon Balas Prize from the INFORMS Optimization Society.

Domenico Salvagnin received his degree in Computer Science Engineering (cum laude) at the University of Padova, Italy, in 2005, and his PhD degree in Computational Mathematics (Operations Research) at the University of Padova, Italy, in 2009. He is associate professor in Operations Research at DEI, University of Padova, Italy since 2018, and got the National Academic Qualification as full professor in 2018. He was lead development scientist in IBM ILOG CPLEX team in 2015-2017, and is currently scientific consultant for FICO XPRESS.His research interests include theory and algorithms for linear and mixed integer linear programming, constraint programming, and hybrid methods for optimization. His awards include: winner of the 11th DIMACS Implementation Challenge for the best computer codes for Steiner Tree problems, Computational Optimization and Applications 2016 Best Paper Award, CPAIOR 2019 Distinguished Paper Award and ICAPS 2019 Best Paper Award.

Grading Policy: There will be 2 to 3 homeworks and areading/project in the second half. Course projects could involveresearch on a specific problem or topic, a survey of several papers ona topic (summarized in a report and/or talk), or an application ofcombinatorial optimization to some applied area of interest. I alsoexpect students to scribe one or two lectures in latex.

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects,[1] where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.

Some research literature[2] considers discrete optimization to consist of integer programming together with combinatorial optimization (which in turn is composed of optimization problems dealing with graph structures), although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.[clarification needed]

There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization. A considerable amount of it is unified by the theory of linear programming. Some examples of combinatorial optimization problems that are covered by this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Perhaps the most universally applicable[weasel words] approaches are branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman (decision) problem,[7] this is expected unless P=NP.

For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure m 0 \displaystyle m_0 . For example, if there is a graph G \displaystyle G which contains vertices u \displaystyle u and v \displaystyle v , an optimization problem might be "find a path from u \displaystyle u to v \displaystyle v that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from u \displaystyle u to v \displaystyle v that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

The field of approximation algorithms deals with algorithms to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is then more naturally characterized as an optimization problem.[8]

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions.[9] Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.[10]

An international Symposium on Combinatorial Optimization will be held in Hammamet, Tunisia, March 24-26, 2010. The symposium aims to bring together researchers from combinatorial optimization, and from related fields of operations research and linear and integer programming.

It is intended to be a forum for the exchange of recent scientific developments and for the discussion of new trends. The scope of the conference includes all aspects of combinatorial optimization from fundamental research to numerical applications.

OR Problems are formulated as integer constrained optimization, i.e., with integral or binary variables (called decision variables). While not all such problems are hard to solve (e.g., finding the shortest path between two locations), we concentrate on Combinatorial (NP-Hard) problems. NP-Hard problems are impossible to solve optimally at large scales as exhaustively searching for their solutions is beyond the limits of modern computers. The Travelling Salesman Problem (TSP) and the Minimum Spanning Tree Problem (MST) are two of the most popular examples for such problems defined using graphs.

Linear programming provides a method to mathematically represent and addressoptimization problems. We are typically given a set of continuous variables thatare associated with costs or profits, and we wish to minimize the total cost ormaximize the total profit. The solution must satisfy a set of linearinequalities which model the specific restrictions of each problem.

Imagine, now, that instead of grain, we wish to transport machine parts. Itmakes no sense to take fractional amounts of parts, so we must restrict thevariables to be integers. Note that it does not suffice to simply round theoriginal solution. In our example, rounding gives $(x, y) = (3, 0)$, with profit$\$1500.00$. However, the optimal solution is $(x, y) = (2, 2)$, with profit$\$1600.00$.

In this work, we present an attention-based encoder-decoder model to approximately solve the team orienteering problem with multiple depots (TOPMD). The TOPMD instance is an NP-hard combinatorial optimization problem that involves multiple agents (or autonomous vehicles) and not purely Euclidean (straight line distance) graph edge weights. In addition, to avoid tedious computations on dataset creation, we provide an approach to generate synthetic data on the fly for effectively training the model. Furthermore, to evaluate our proposed model, we conduct two experimental studies on the multi-agent reconnaissance mission planning problem formulated as TOPMD. First, we characterize the model based on the training configurations to understand the scalability of the proposed approach to unseen configurations. Second, we evaluate the solution quality of the model against several baselines--heuristics, competing machine learning (ML), and exact approaches, on several reconnaissance scenarios. The experimental results indicate that training the model with a maximum number of agents, a moderate number of targets (or nodes to visit), and moderate travel length, performs well across a variety of conditions. Furthermore, the results also reveal that the proposed approach offers a more tractable and higher quality (or competitive) solution in comparison with existing attention-based models, stochastic heuristic approach, and standard mixed-integer programming solver under the given experimental conditions. Finally, the different experimental evaluations reveal that the proposed data generation approach for training the model is highly effective. 2ff7e9595c