Randomized Strategies for Robust Combinatorial Optimization

In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. Our goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update method. A key technique is to introduce a new concept called $(\eta,\gamma)$-reductions for objective functions with parameters $\eta, \gamma$. We show that our scheme outputs a nearly $\alpha$-approximate solution if there exists an $\alpha$-approximation algorithm for a subproblem defined by $(\eta,\gamma)$-reductions. This improves approximation ratio in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem

General Bounds for Incremental Maximization

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $k\in\mathbb{N}$ that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all $k$ between the incremental solution after $k$ steps and an optimum solution of cardinality $k$. We define a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general $2.618$-competitive incremental algorithm for this class of problems, and show that no algorithm can have competitive ratio below $2.18$ in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be $1.58$-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) ($b$-)matching and a variant of the maximum flow problem. We show that the greedy algorithm has competitive ratio (exactly) $2.313$ for the class of problems that satisfy this relaxed submodularity condition. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems.Comment: fixed typo

General Bounds for Incremental Maximization

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value k in N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after~$k$ steps and an optimum solution of cardinality k. We define a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show that the greedy algorithm has competitive ratio (exactly) 2.313 for the class of problems that satisfy this relaxed submodularity condition. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems

Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods