In this short note, we present a new class of GAPN power functions of the type over finite fields with p odd and (up to EA-equivalence).
In this short note, we present a new class of GAPN power functions of the type over finite fields with p odd and (up to EA-equivalence).
Let be a set of points and be an integer. Dasgupta et al. [1] considered the problem of building a partition of into k groups, induced by an axis-aligned decision tree with k leaves. The motivation is obtaining partitions that are simple to explain. We study the computational complexity of this problem for k-means, k-medians and the k-center cost-functions. We prove that the optimization problems induced by these cost-functions are hard to approximate.
A directed graph is semi-transitive if and only if it is acyclic and for any directed path , , either there is no edge from to or all edges exist for . An undirected graph is semi-transitive if it admits a semi-transitive orientation. Recognizing semi-transitive orientability of a graph is an NP-complete problem.
A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Semi-transitive orientability of split graphs was recently studied in a series of papers in the literature. The main result in this paper is proving that recognition of semi-transitive orientability of split graphs can be done in a polynomial time.
Piccolo is a lightweight block cipher based on a 16-bit word 4-line generalized Feistel structure. Piccolo adopts byte-wise round permutation (RP) instead of the typical word-based RP to improve diffusion. In this paper, we explore the optimality of byte-based RP from the viewpoint of security. We evaluate the security of differential, linear, impossible differential, and integral attacks for all byte-wise RPs using mixed integer linear programming (MILP). We show that the RP of Piccolo is optimal in terms of the number of rounds required to guarantee security against such attacks. In addition, we introduce new two classes of RPs that require 7 rounds for security against impossible differential attacks, which is one round less than required by Piccolo. These new classes require 7/9 and 8/8 rounds to guarantee security against differential/linear attacks, respectively, which is more rounds than required by Piccolo.
We show that if the arc-connectivity of a directed graph D is at most and the reorientation of an arc set F in D results in a k-arc-connected directed graph then we can reorient one arc of F without decreasing the arc-connectivity of D. This improves a corollary of a result of Fukuda, Prodon, Sakuma [2] and one of Ito et al. [3] for .
We consider the problem of computing an -hypernetwork in an acyclic F-hypergraph. This is a fundamental computational problem arising in directed hypergraphs, and is a foundational step in tackling problems of reachability and redundancy. This problem was previously explored in the context of general directed hypergraphs (containing cycles), where it is NP-hard, and acyclic B-hypergraphs, where a linear time algorithm can be achieved. In a surprising contrast, we find that for acyclic F-hypergraphs the problem is NP-hard, which also implies the problem is hard in BF-hypergraphs. This is a striking complexity boundary given that F-hypergraphs and B-hypergraphs would at first seem to be symmetrical to one another. We provide the proof of complexity and explain why there is a fundamental asymmetry between the two classes of directed hypergraphs.
We consider a dynamic situation in the weighted bipartite matching problem: edge weights in the input graph are repeatedly updated and we are asked to maintain an optimal matching at any moment. A trivial approach is to compute an optimal matching from scratch each time an update occurs. In this paper, we show that if each update occurs locally around a single vertex, then a single execution of Dijkstra's algorithm is sufficient to preserve optimality with the aid of a dual solution. As an application of our result, we provide a faster implementation of the envy-cycle procedure for finding an envy-free allocation of indivisible items. Our algorithm runs in time, while the known bound of the original one is , where n and m denote the numbers of agents and items, respectively.
Rankin's constant is an important lattice constant which has applications in many fields including cryptography and communications. In spite of its importance, few of exact values of Rankin's constant are known. In this paper, we develop a lower bound on Rankin's constant which corresponds to the half volume problem. Compared with the previous best lower bound developed by Wen et al., ours is more than times better. This improved lower bound on Rankin's constant directly leads to a sharper lower bound on Schnorr's constant and helps to better understand the intrinsic limitations of the 2k-block-Rankin reduction.
Binary polynomial optimization (BPO) is the problem of maximizing a polynomial function on the Boolean domain. This problem can be formulated by using a hypergraph, and various properties of the input hypergraph have been investigated from the viewpoint of polynomial-time solvability. In this paper, we especially focus on β-acyclic hypergraphs. For BPO over β-acyclic hypergraphs, Del Pia and Di Gregorio proposed a polynomial-time algorithm. We prove that the algorithm proposed by Del Pia and Di Gregorio can be extended to a more general optimization problem in β-acyclic hypergraphs.
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