Leader-following Consensus of Multi-agent Systems over Finite Fields

The leader-following consensus problem of multi-agent systems over finite fields ${\mathbb F}_p$ is considered in this paper. Dynamics of each agent is governed by a linear equation over ${\mathbb F}_p$, where a distributed control protocol is utilized by the followers.Sufficient and/or necessary conditions on system matrices and graph weights in ${\mathbb F}_p$ are provided for the followers to track the leader

Planar kernel and grundy with d≤3, dout≤2, din≤2 are NP-complete

AbstractIt is proved that the questions whether a finite diagraph G has a kernel K or a Sprague—Grundy function g are NP-complete even if G is a cyclic planar digraph with degree constraints dout(u)≤2, din(u)≤2 and d(u)≤3. These results are best possible (if P ≠ NP) in the sense that if any of the constraints is tightened, there are polynomial algorithms which either compute K and g or show that they do not exist. The proof uses a single reduction from planar 3-satisfiability for both problems

On the Complexity of Solving Quadratic Boolean Systems

A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in $4\log_2 n\,2^n$ operations. We give an algorithm that reduces the problem to a combination of exhaustive search and sparse linear algebra. This algorithm has several variants depending on the method used for the linear algebra step. Under precise algebraic assumptions on the input system, we show that the deterministic variant of our algorithm has complexity bounded by $O(2^{0.841n})$ when $m=n$, while a probabilistic variant of the Las Vegas type has expected complexity $O(2^{0.792n})$. Experiments on random systems show that the algebraic assumptions are satisfied with probability very close to~1. We also give a rough estimate for the actual threshold between our method and exhaustive search, which is as low as~200, and thus very relevant for cryptographic applications.Comment: 25 page

Traveling salesmen in the presence of competition

AbstractWe propose the “competing salesmen problem” (CSP), a two-player competitive version of the classical traveling salesman problem. This problem arises when considering two competing salesmen instead of just one. The concern for a shortest tour is replaced by the necessity to reach any of the customers before the opponent does.In particular, we consider the situation where players take turns, moving along one edge at a time within a graph G=(V,E). The set of customers is given by a subset VC⊆V of the vertices. At any given time, both players know of their opponent's position. A player wins if he is able to reach a majority of the vertices in VC before the opponent does.We prove that the CSP is PSPACE-complete, even if the graph is bipartite, and both players start at distance 2 from each other. Furthermore, we show that the starting player may not be able to avoid losing the game, even if both players start from the same vertex. However, for the case of bipartite graphs, we show that the starting player always can avoid a loss. On the other hand, we show that the second player can avoid to lose by more than one customer, when play takes place on a graph that is a tree T, and VC consists of leaves of T. It is unclear whether a polynomial strategy exists for any of the two players to force this outcome. For the case where T is a star (i.e., a tree with only one vertex of degree higher than two) and VC consists of n leaves of T, we give a simple and fast strategy which is optimal for both players. If VC consists not only of leaves, we point out that the situation is more involved

Theory of annihilation games—I

AbstractPlace tokens on distinct vertices of an arbitrary finite digraph with n vertices which may contain cycles or loops. Each of two players alternately selects a token and moves it from its present position u to a neighboring vertex v along a directed edge which may be a loop. If v is occupied, and u ≠ v, both tokens get annihilated and phase out of the game. The player first unable to move is the loser, the other the winner. If there is no last move, the outcome is declared a draw. An O(n6) algorithm for computing the previous-player-winning, next-player-winning and draw positions of the game is given. Furthermore, an algorithm is given for computing a best strategy in O(n6) steps and winning—starting from a next-player-winning position—in O(n5) moves