Maximum Minimal Set Cover is the max-min variant of Set Cover, and consists of finding a minimal set cover of maximum cardinality. It generalizes Max Min Vertex Cover and Upper Domination. Besides that, (Maximum Minimal) Set Cover and (Maximum Minimal) Hitting Set can be considered similar problems depending on the complexity aspect analyzed. However, when the size of the universe, say n, is considered as a parameter, these families of problems may behave differently. An O*(2n)-time algorithm for (Maximum Minimal) Hitting Set is trivial, while exhaustively trying all possible ways to cover the elements takes O*(2m) time, where m is the number of sets. Since either m or n may be significantly larger than the other, it is natural to ask for the best running time that one can achieve in terms of n (the universe size) to Maximum Minimal Set Cover. In this regime, we show that such a problem can be solved in O*(2n) time, but it cannot be solved in O*(2o(n)), unless ETH fails. We also present an O*(2k)-time algorithm for Maximum Minimal Set Cover, where k is the vertex cover number of the incidence graph of the input. This is an improvement of the O*(2n)-time algorithm, since k ≤ min {n, m}. Finally, we show that Maximum Minimal Set Cover cannot be solved in O*(1.1892k) time, unless SETH fails.
Given a budget and an undirected graph with a cost and a profit function on the edges, the Edge Orienteering Problem asks for a closed walk in the graph that collects the most profit while the sum of costs does not exceed the budget. While the profit is collected only once per edge, the cost is incurred each time the edge is traversed by the walk. In this paper, we present a (4+ε)-approximation algorithm for the Edge Orienteering Problem.
Inspired by the FLIP stream cipher introduced by M & eacute;aux et al. at Eurocrypt 2016, Boolean functions with good cryptographic properties when their inputs are limited to specific subsets of F(2)n are well studied. Weightwise perfectly balanced (WPB) functions are a class of Boolean functions that remain balanced across all subsets of F(2)n with a fixed Hamming weight ranging from 1 to n-1. This paper introduces a class of novel iterative constructions of WPB functions based on a family of 2(m)-variable Boolean functions with degree 2d-1, which is also a WPB function for d = m. The obtained WPB functions have high algebraic degree, weightwise nonlinearity and algebraic immunity.
A k-tuple common substring (abbr. k-CSS) is a common subsequence of two or more strings including at most k common substrings. In this paper, we present an algorithm for finding a longest k-CSS of two strings of lengths m and n in O(kmn) time and O(m + kn) space. Using this algorithm as a subroutine, a longest k-CSS of two circular strings is retrieved in O(kmn2) time and O(m + kn) space. Extending the idea in this algorithm, a longest common subsequence with the minimum number of common substrings of two strings is retrieved in O(mn) time and O(m + n) space.
We consider the quantum query complexity of local search as a function of graph geometry. Given a graph G=(V,E) with n vertices and black box access to a function f:V→R, the goal is to find a vertex v that is a local minimum, i.e. with f(v) ≤ f(u) for all (u, v) ∈ E.We show the quantum query complexity of local search on G is Ω(n34g), where g is the vertex congestion of the graph. For a β-expander with maximum degree Δ, this implies a lower bound of Ω(βn14Δlogn). The bounds follow by applying the strong weighted adversary method to a construction in [1].As a corollary, we get a lower bound of Ω(n14logn) for constant-degree regular expanders, improving upon the best prior quantum bound of Ω(n18logn) from [2]. In contrast to the classical case, a gap remains between our lower bound and the best-known quantum upper bound of O(n13) for such graphs.
