Combinatorial Optimization with Rational Objective Functions: A Communication

Counting with rational generating functions

We consider to identify an unknown Boolean function f by asking an oracle the functional values f(a) for a selected set of test vectors a ε {0,1}n. If f is known to be a positive function of n variables, the algorithm by Gainanov can achieve the goal by issuing O(mn) queries, where m = ¦minT(f)¦ + ¦maxF(f)¦ and minT(f) (resp. maxF(f)) denotes the set of minimal true vectors (resp. maximal false vectors) of f. However, it is not known whether this whole task including the generation of test vectors can be carried out in polynomial time in n and m or not. To partially answer this question, we propose here two algorithms that, given an unknown positive function f of n variables, decide whether f is 2-monotonic or not, and if f is 2-monotonic, output sets min T(f) and max F(f). The first algorithm uses O(nm 2+n 2 m) time and O(nm) queries while the second one uses O(n3m) time and O(n 3 m) queries.

Reversible parallel computation: An evolving space-model

Light types for polynomial time computation in lambda calculus

A graph-based approach to finding near-optimal output phase assignments for PLA minimization is presented. A distinctive feature of the approach is that it exploits the necessary and sufficient conditions to reduce the number of product terms needed for PLA implementation and permits the use of existing graph algorithms to solve the PLA output phase optimization problem. The work is based on the transformation of a PLA into a graph whose vertices are the set of conditions required to reduce each product term of the PLA, and whose edges represent the relations between these conditions. Cliques (completely connected subgraphs) in the graph correspond to the output phase assignments required to reduce the product terms represented by the vertices in the cliques. The optimal PLA output phase assignment problem is then formulated as the well-studied problem in graph theory: finding maximum cliques in graphs. Using an existing algorithm for locating cliques in graphs, a modified greedy algorithm is proposed to compute output phase assignments for logic functions in polynomial time. Experimental results using a number of benchmark functions show that the graph-based approach can achieve optimal or near-optimal output phase assignment for PLA minimization and can lead to PLA's with fewer product terms than achieved using existing approaches.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.

Local search algorithms for combinatorial optimization problems are in general of pseudopolynomial running time and polynomial-time algorithms are often not known for finding locally optimal solutions for NP-hard optimization problems. We introduce the concept of epsilon-local optimality and show that an epsilon-local optimum can be identified in time polynomial in the problem size and 1/epsilon whenever the corresponding neighborhood can be searched in polynomial time, for epsilon > 0. If the neighborhood can be searched in polynomial time for a delta-local optimum, we present an algorithm that produces a (delta+epsilon)-local optimum in time polynomial in the problem size and 1/epsilon. As a consequence, a combinatorial optimization problem has a fully polynomial-time approximation scheme if and only if it has a fully polynomial-time augmentation schem

&lt;abstract&gt;&lt;p&gt;Recent advancements in network and multimedia technologies have facilitated the distribution and sharing of digital videos over the Internet. These long videos contain very complex contents. Additionally, it is very challenging to use as few frames as possible to cover the video contents without missing too much information. There are at least two ways to describe these complex videos contents with minimal frames: the keyframes extracted from the video or the video summary. The former lays stress on covering the whole video contents as much as possible. The latter emphasizes covering the video contents of interest. As a consequence, keyframes are widely used in many areas such as video segmentation and object tracking. In this paper, we propose a keyframe extraction method based on multiple features via a novel combinatorial optimization algorithm. The key frame extraction is modeled as a combinatorial optimization problem. A fast dynamic programming algorithm based on a forward non-overlapping transfer matrix in polynomial time and a 0-1 integer linear programming algorithm based on an overlapping matrix is proposed to solve our maximization problem. In order to quantitatively evaluate our approach, a long video dataset named 'Animal world' is self-constructed, and the segmentation evaluation criterions are introduced. A good result is achieved on 'Animal world' dataset and a public available Keyframe-Sydney KFSYD dataset &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;]&lt;/sup&gt;.&lt;/p&gt;&lt;/abstract&gt;

Intensional Properties of Polygraphs

Contributors

By examining the patterns of solutions obtained for various instances, one can gain insights into the structure and behavior of combinatorial optimization (CO) problems and develop efficient algorithms for solving them. Machine learning techniques, especially Graph Neural Networks (GNNs), have shown promise in parametrizing and automating this laborious design process. The inductive bias of GNNs allows for learning solutions to mixed-integer programming (MIP) formulations of constrained CO problems with a relational representation of decision variables and constraints. The trained GNNs can be leveraged with primal heuristics to construct high-quality feasible solutions to CO problems quickly. However, current GNN-based end-to-end learning approaches have limitations for scalable training and generalization on larger-scale instances; therefore, they have been mostly evaluated over small-scale instances. Addressing this issue, our study builds on supervised learning of optimal solutions to the downscaled instances of given large-scale CO problems. We introduce several improvements on a recent GNN model for CO to generalize on instances of a larger scale than those used in training. We also propose a two-stage primal heuristic strategy based on uncertainty-quantification to automatically configure how solution search relies on the predicted decision values. Our models can generalize on 16x upscaled instances of commonly benchmarked five CO problems. Unlike the regressive performance of existing GNN-based CO approaches as the scale of problems increases, the CO pipelines using our models offer an incremental performance improvement relative to CPLEX. The proposed uncertainty-based primal heuristics provide 6-75% better optimality gap values and 45-99% better primal gap values for the 16x upscaled instances and brings immense speedup to obtain high-quality solutions. All these gains are achieved through a computationally efficient modeling approach without sacrificing solution quality.

By examining the patterns of solutions obtained for various instances, one can gain insights into the structure and behavior of combinatorial optimization (CO) problems and develop efficient algorithms for solving them. Machine learning techniques, especially Graph Neural Networks (GNNs), have shown promise in parametrizing and automating this laborious design process. The inductive bias of GNNs allows for learning solutions to mixed-integer programming (MIP) formulations of constrained CO problems with a relational representation of decision variables and constraints. The trained GNNs can be leveraged with primal heuristics to construct high-quality feasible solutions to CO problems quickly. However, current GNN-based end-to-end learning approaches have limitations for scalable training and generalization on larger-scale instances; therefore, they have been mostly evaluated over small-scale instances. Addressing this issue, our study builds on supervised learning of optimal solutions to the downscaled instances of given large-scale CO problems. We introduce several improvements on a recent GNN model for CO to generalize on instances of a larger scale than those used in training. We also propose a two-stage primal heuristic strategy based on uncertainty-quantification to automatically configure how solution search relies on the predicted decision values. Our models can generalize on 16x upscaled instances of commonly benchmarked five CO problems. Unlike the regressive performance of existing GNN-based CO approaches as the scale of problems increases, the CO pipelines using our models offer an incremental performance improvement relative to CPLEX. The proposed uncertainty-based primal heuristics provide 6-75% better optimality gap values and 45-99% better primal gap values for the 16x upscaled instances and brings immense speedup to obtain high-quality solutions. All these gains are achieved through a computationally efficient modeling approach without sacrificing solution quality.

Logical Definability of NP Optimization Problems

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

We consider combinatorial optimization problems with nonlinear objective functions. Solution approaches for this class of problems proposed so far are either highly problem-specific or they apply generic algorithms for constrained nonlinear optimization, which often does not yield satisfactory results in practice. Our aim is to develop, implement and experimentally evaluate exact algorithms that address the nonlinearity of the objective function and at the same time exploit the underlying combinatorial structure of the problem. To this end we follow two approaches. The first combines good polyhedral descriptions of the objective function and the feasible set in a branch and cut-algorithm. The second approach is based on Lagrangean decomposition. By decomposing the original problem into an unconstrained nonlinear problem and a linear combinatorial problem, we are able to compute strong dual bounds for the optimal value. The computation of lower bounds is then embedded into a branch and bound-algorithm. For many applications there already exist efficient algorithms for the combinatorial subproblem, thus an important aspect of this thesis is the study of the corresponding unconstrained nonlinear subproblems. Both approaches have the advantage that they can easily be adapted to a wide range of nonlinear combinatorial problems.We devise both polyhedral and decomposition- based algorithms for submodular applications from wireless network design and portfolio optimization and evaluate their performance experimentally. Exploiting the equivalence between unconstrained binary quadratic optimization and the maximum cut problem gives rise to a branch and cut-algorithm for quadratic combinatorial problems which we use to compute optimal layouts of tanglegrams, an application from computational biology. Additionally we study the effect of quadratic reformulation of linear constraints, both theoretically and experimentally. The last class of nonlinear combinatorial problems we consider are two-scenario problems. Here we propose a new technique to compute lower bounds in the unconstrained subproblem of the decomposition. Our computational study of the two-scenario minimum spanning tree problem shows that the new Lagrangean decomposition-based algorithm is able to solve significantly larger instances than the standard linearization approach.