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Submodular maximization is one of the central topics in combinatorial optimization. It has found numerous applications in the real world. In the past decades, a series of algorithms have been proposed for this problem. However, most of the state-of-the-art algorithms are randomized. There remain non-negligible gaps with respect to approximation ratios between deterministic and randomized algorithms in submodular maximization. In this paper, we propose deterministic algorithms with improved approximation ratios for non-monotone submodular maximization. Specifically, for the matroid constraint, we provide a deterministic 0.283 − o (1) approximation algorithm, while the previous best deterministic algorithm only achieves a 1/4 approximation ratio. For the knapsack constraint, we provide a deterministic 1/4 approximation algorithm, while the previous best deterministic algorithm only achieves a 1/6 approximation ratio. For the linear packing constraints with large widths, we provide a deterministic 1 / 6 − ϵ approximation algorithm. To the best of our knowledge, there is currently no deterministic approximation algorithm for the constraints. [ABSTRACT FROM AUTHOR]

In an inverse combinatorial optimization problem, a feasible solution is given and is not optimal under the current parameters. The aim is to make the feasible solution optimal by modifying the current parameters as little as possible. The modification cost can be measured by different norms, such as weighted l 1 norm, weighted l 2 norm, weighted l ∞ norm, weighted Hamming distance and so on. In this paper, we focus on the constrained inverse minimum flow problems under the weighted Hamming distance. Three different models are considered: the general problem under the weighted bottleneck-type Hamming distance and two cases under the mixed of sum-type and bottleneck type. Strongly polynomial algorithms are presented for the models we studied. [ABSTRACT FROM AUTHOR]

We propose a dynamic programming framework for an exact multi-stage (lexicographic) combinatorial optimization. Unlike conventional algorithms of dynamic programming that return one optimal solution, two dynamic programming algorithms proposed in this paper are coping with the whole set of optimal solutions or with its essential part. We describe the set of elements for optimization by a labeled directed acyclic graph, which in some sense, is similar to the structure of subproblems of the considered problem. For a given cost function (objective), the first algorithm constructs a subgraph of this graph that describes the whole set of optimal elements or its essential part. This algorithm can be used for multi-stage optimization of elements relative to a sequence of cost functions. The second algorithm counts elements before the optimization and after each optimization step. The considered labeled directed acyclic graph is a kind of circuit. This circuit builds the set of elements for optimization from one-element sets attached to input nodes. It uses the operation of union of sets attached to unifying nodes and functional operations attached to functional nodes. The algorithms for optimization and counting elements are defined for an arbitrary circuit without repetitions in which each element is generated exactly one time. For a considered problem with a known conventional dynamic programming solution algorithm, usually, it is easy to construct a corresponding circuit without repetitions. Once the circuit and cost functions are defined, our framework provides the correctness proofs and detailed time complexity analysis for the proposed algorithms. To make this approach more intuitive, we consider an illustrative example related to the maximum subarray problem. We tested our approach on the following nine combinatorial optimization problems: matrix chain multiplication, global sequence alignment, optimal paths in directed graphs, binary search trees, optimal bitonic tour, segmented least squares, convex polygon triangulation, one-dimensional clustering, and line breaking (text justification). We consider the last three problems in detail: construct a circuit without repetitions, describe at least two cost functions, evaluate a number of operations and time required by the algorithms, discuss an example, and consider the results of computer experiments with randomly generated instances of the problem. [ABSTRACT FROM AUTHOR]

Partitioning a network into k pieces is a fundamental problem in network science. A simple measure of partitioning a network is provided by the Max k -Uncut problem. Given an n -vertex undirected graph G with nonnegative weights defined on edges, and a positive integer k , the Max k -Uncut problem asks to find a k -partition of the vertices of G to maximize the total weight of edges that are not in the cut. This problem is the complement of the classic Min k -Cut problem, and has close relation to many combinatorial optimization problems, including the famous Densest k -Subgraph problem. In this paper, we propose a greedy approximation algorithm for the Max k -Uncut problem with performance ratio 1 − 2 (k − 1) n. The algorithm is very simple, which consists of only k − 1 min cut computations. The algorithm has fast running time O (k n 2) and is hence implementable. The experimental results show that the algorithm has excellent practical performance. [ABSTRACT FROM AUTHOR]

Popular matchings have been intensively studied recently as a relaxed concept of stable matchings. By applying the concept of popular matchings to branchings in directed graphs, Kavitha et al. introduced popular branchings. Let G = (V G , E G) be a directed graph, where each vertex has preferences over its incoming arcs, and B and B ′ be branchings in G. A vertex v ∈ V G prefers B to B ′ if v prefers its incoming arc of B to that of B ′ , where having an arbitrary incoming arc is preferred to having no incoming arc. We say that B is more popular than B ′ if the number of vertices preferring B to B ′ is greater than the number of vertices preferring B to B ′. A branching B is called a popular branching if there is no branching more popular than B. Kavitha et al. proposed an algorithm for finding a popular branching when the preferences of each vertex are given by a strict partial order. The correctness of this algorithm is proved by utilizing classical theorems on the duality of weighted arborescences. In this paper, we generalize popular branchings to weighted popular branchings in vertex-weighted directed graphs, in the same manner as weighted popular matchings by Mestre. We give an algorithm for finding a weighted popular branching for the case where the preferences of each vertex are given by a total preorder and the weights satisfy certain conditions. Our algorithm is based on Kavitha et al.'s algorithm, while it includes elaborated procedures resulting from the vertex-weights. Its correctness also builds upon the duality of weighted arborescences. [ABSTRACT FROM AUTHOR]