Discrete Optimization最新文献

In this paper we present and study a new algorithm for the Maximum Satisfiability (Max Sat) problem. The algorithm is based on the Method of Conditional Expectations (MOCE, also known as Johnson’s Algorithm) and applies a greedy variable ordering to MOCE. Thus, we name it Greedy Order MOCE (GO-MOCE). We also suggest a combination of GO-MOCE with CCLS, a state-of-the-art solver. We refer to this combined solver as GO-MOCE-CCLS.

We conduct a comprehensive comparative evaluation of GO-MOCE versus MOCE on random instances and on public competition benchmark instances. We show that GO-MOCE reduces the number of unsatisfied clauses by tens of percents, while keeping the runtime almost the same. The worst case time complexity of GO-MOCE is linear. We also show that GO-MOCE-CCLS improves on CCLS consistently by up to about 80%.

We study the asymptotic performance of GO-MOCE. To this end, we introduce three measures for evaluating the asymptotic performance of algorithms for Max Sat. We point out to further possible improvements of GO-MOCE, based on an empirical study of the main quantities managed by GO-MOCE during its execution.

In many fault detection problems, we want to identify defective items from a set of $n$ items using the minimum number of tests. Group testing is for the scenario where each test is on a subset of items, and tells whether the subset contains at least one defective item or not. In practice, the number $d$ of defective items is often unknown in advance. In this paper, we improve the previously best algorithm for a central problem in combinatorial group testing with unknown number of defectives (Cheng et al., 2014), and prove that the number of tests used by our new algorithm is no more than $dlog\frac{n}{d}+(5-log5)d+O\left({log}^{2}d\right)$, where $log$ is of base 2.

This paper considers the uncapacitated $r$-allocation $p$-hub maximal covering problem (UrApHMCP), which represents a generalization of the well-known $p$-hub maximal covering problem, as it allows each non-hub node to send and receive flow via at most $r$ hubs, $r\le p$. Two coverage criteria are considered in UrApHMCP — binary and, for the first time in the literature, partial coverage. Novel mathematical formulations of UrApHMCP for both coverage criteria are proposed. As the considered UrApHMCP is an NP-hard optimization problem, two efficient heuristic methods are proposed as solution approaches. The first one is a variant of General Variable Neighborhood Search (GVNS), and the second one is based on combining a Greedy Randomized Adaptive Search Procedure (GRASP) with Variable Neighborhood Descent (VND), resulting in a hybrid GRASP-VND method. Computational study is performed over the set of CAB and AP benchmark instances with up to 25 and 200 nodes, respectively, on TR instances including 81 nodes, as well as on the challenging USA423 and URAND hub instances with up 423 and 1000 nodes, respectively. Optimal or feasible solutions are obtained by CPLEX solver for instances with up to 50 nodes, while instances of larger dimensions are out of reach for CPLEX solver. On the other hand, both GVNS and GRASP-VND reach optimal solutions or improve lower bounds provided by CPLEX in short CPU times. In addition, both heuristics quickly return solutions on problem instances of large dimensions, thus indicating their potential to solve effectively large, realistic sized problem instances. The conducted non-parametric statistical tests confirm robustness of the proposed GVNS and GRASP-VND and demonstrate that the these two metaheuristics outperform other tested algorithms for UrApHMCP.

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation.

In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig–Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.

The benefits of transmission line switching are well-known in terms of reducing operational cost and improving system reliability of power systems. However, finding the optimal power network configuration is a challenging task due to the combinatorial nature of the underlying optimization problem. In this work, we identify a certain “node-based” set that appears as substructure of the optimal transmission switching problem and then conduct a polyhedral study of this set. We construct an extended formulation of the integer hull of this set and present the inequality description of the integer hull in the original space in some cases. These inequalities in the original space can be used as cutting-planes for the transmission line switching problem. Finally, we present the results of our computational experiments using these cutting-planes on difficult test cases from the literature.

Given a graph $G$, a $k$-sparse $j$-set is a set of $j$ vertices inducing a subgraph with maximum degree at most $k$. A $k$-dense $i$-set is a set of $i$ vertices that is $k$-sparse in the complement of $G$. As a generalization of Ramsey numbers, the $k$-defective Ramsey number $R_k^G(i,j)$ for the graph class $G$ is defined as the smallest natural number $n$ such that all graphs on $n$ vertices in the class $G$ have either a $k$-dense $i$-set or a $k$-sparse $j$-set. In this paper, we examine $R_k^G(i,j)$ where $G$ represents various graph classes. For forests and cographs, we give exact formulas for all defective Ramsey numbers. For cacti, bipartite graphs and split graphs, we derive defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.

We present coloring-based algorithms for tree augmentation and use them to construct convex combinations of 2-edge-connected subgraphs. This classic tool has been applied previously to the problem, but our algorithms illustrate its flexibility, which – in coordination with the choice of spanning tree – can be used to obtain various properties (e.g., 2-vertex connectivity) that are useful in our applications.

We use these coloring algorithms to design approximation algorithms for the 2-edge-connected multigraph problem (2ECM) and the 2-edge-connected spanning subgraph problem (2ECS) on two well-studied types of LP solutions. The first type of points, half-integer square points, belong to a class of fundamental extreme points, which exhibit the same integrality gap as the general case. For half-integer square points, the integrality gap for 2ECM is known to be between $\frac{6}{5}$ and $\frac{4}{3}$. We improve the upper bound to $\frac{9}{7}$. The second type of points we study are uniform points whose support is a 3-edge-connected graph and each entry is $\frac{2}{3}$. Although the best-known upper bound on the integrality gap of 2ECS for these points is less than $\frac{4}{3}$, previous results do not yield an efficient algorithm. We give the first approximation algorithm for 2ECS with ratio below $\frac{4}{3}$ for this class of points.

A natural and important generalization of submodularity – $k$-submodularity – applies to set functions with $k$ arguments and appears in a broad range of applications, such as infrastructure design, machine learning, and healthcare. In this paper, we study maximization problems with $k$-submodular objective functions. We propose valid linear inequalities, namely the $k$-submodular inequalities, for the hypograph of any $k$-submodular function. This class of inequalities serves as a novel generalization of the well-known submodular inequalities. We show that maximizing a $k$-submodular function is equivalent to solving a mixed-integer linear program with exponentially many $k$-submodular inequalities. Using this representation in a delayed constraint generation framework, we design the first exact algorithm, that is not a complete enumeration method, to solve general $k$-submodular maximization problems. Our computational experiments on the multi-type sensor placement problems demonstrate the efficiency of our algorithm in constrained nonlinear $k$-submodular maximization problems for which no alternative compact mixed-integer linear formulations are available. The computational experiments show that our algorithm significantly outperforms the only available exact solution method—exhaustive search. Problems that would require over 13 years to solve by exhaustive search can be solved within ten minutes using our method.

Multimarginal Optimal Transport ($\mathrm{MOT}$) is the problem of linear programming over joint probability distributions with fixed marginals. A key issue in many applications is the complexity of solving $\mathrm{MOT}$: the linear program has exponential size in the number of marginals $k$ and their support sizes $n$. A recent line of work has shown that $\mathrm{MOT}$ is $\mathrm{poly}(n,k)$-time solvable for certain families of costs that have $\mathrm{poly}(n,k)$-size implicit representations. However, it is unclear what further families of costs this line of algorithmic research can encompass. In order to understand these fundamental limitations, this paper initiates the study of intractability results for $\mathrm{MOT}$.

Our main technical contribution is developing a toolkit for proving $\mathrm{NP}$-hardness and inapproximability results for $\mathrm{MOT}$ problems. This toolkit reduces proving intractability of $\mathrm{MOT}$ problems to proving intractability of more amenable discrete optimization problems. We demonstrate this toolkit by using it to establish the intractability of a number of $\mathrm{MOT}$ problems studied in the literature that have resisted previous algorithmic efforts. For instance, we provide evidence that repulsive costs make $\mathrm{MOT}$ intractable by showing that several such problems of interest are $\mathrm{NP}$-hard to solve—even approximately.

In the storable good monopoly problem, a monopolist sells a storable good by announcing a price in each time period. Each consumer has a unitary demand per time period with an arbitrary valuation. In each period, consumers may buy none, one, or more than one good (in which case the extra goods are stored for future consumption incurring in a linear storage cost). We compare the performance of two important monopoly pricing optimization mechanisms: price optimization using pre-announced prices and price optimization without commitments (contingent mechanism). In pre-announced pricing the prices in each time period are stated in advance; in a price contingent mechanism each price is stated at the start of the time period, and these prices are dependent upon past purchases. We prove that monopolist can earn up to $O(logT+logN)$ times more profit by using a pre-announced pricing mechanism rather than a price contingent mechanism. Here $T$ denotes the number of time periods and $N$ denotes the number of consumers. This bound is tight; examples exist where the monopolist would earn a factor $\Omega (logT+logN)$ more by using a pre-announced pricing mechanism.