The Travelling Salesman Problem in symmetric circulant matrices with two stripes

2008 ◽  
Vol 18 (1) ◽  
pp. 165-175 ◽  
Author(s):  
IVAN GERACE ◽  
FEDERICO GRECO
Keyword(s):  
Computational Complexity ◽  
Lower Bound ◽  
Lower Bounds ◽  
Travelling Salesman Problem ◽  
Minimum Cost ◽  
Circulant Matrix ◽  
Upper And Lower Bounds ◽  
Circulant Matrices ◽  
Travelling Salesman ◽  
New Construction

The Symmetric Circulant Travelling Salesman Problem asks for the minimum cost tour in a symmetric circulant matrix. The computational complexity of this problem is not known – only upper and lower bounds have been determined. This paper provides a characterisation of the two-stripe case. Instances where the minimum cost of a tour is equal to either the upper or lower bound are recognised. A new construction providing a tour is proposed for the remaining instances, and this leads to a new upper bound that is closer than the previous one.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

scholarly journals A Novel Cooperative Path Planning for Multi-robot Persistent Coverage with Obstacles and Coverage Period Constraints

Sensors ◽  
2019 ◽  
Vol 19 (9) ◽  
pp. 1994 ◽  
Author(s):  
Guibin Sun ◽  
Rui Zhou ◽  
Bin Di ◽  
Zhuoning Dong ◽  
Yingxun Wang
Keyword(s):  
Path Planning ◽  
Numerical Simulations ◽  
Obstacle Avoidance ◽  
Travelling Salesman Problem ◽  
Region Of Interest ◽  
Minimum Cost ◽  
Closed Path ◽  
Combinatorial Approach ◽  
Travelling Salesman ◽  
Multi Robot

In this paper, a multi-robot persistent coverage of the region of interest is considered, where persistent coverage and cooperative coverage are addressed simultaneously. Previous works have mainly concentrated on the paths that allow for repeated coverage, but ignored the coverage period requirements of each sub-region. In contrast, this paper presents a combinatorial approach for path planning, which aims to cover mission domains with different task periods while guaranteeing both obstacle avoidance and minimizing the number of robots used. The algorithm first deploys the sensors in the region to satisfy coverage requirements with minimum cost. Then it solves the travelling salesman problem to obtain the frame of the closed path. Finally, the approach partitions the closed path into the fewest segments under the coverage period constraints, and it generates the closed route for each robot on the basis of portioned segments of the closed path. Therefore, each robot can circumnavigate one closed route to cover the different task areas completely and persistently. The numerical simulations show that the proposed approach is feasible to implement the cooperative coverage in consideration of obstacles and coverage period constraints, and the number of robots used is also minimized.

scholarly journals When is non-trivial estimation possible for graphons and stochastic block models?‡

2017 ◽  
Vol 7 (2) ◽  
pp. 169-181
Author(s):  
Audra McMillan ◽  
Adam Smith
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Random Networks ◽  
Block Models

Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error ${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$ in the $\delta_2$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp et al.

SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

2014 ◽  
Vol 25 (07) ◽  
pp. 877-896 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER ◽  
MATTHIAS WENDLANDT
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Unary Languages ◽  
Order Of Magnitude ◽  
Simulation Results ◽  
Special Case ◽  
Nondeterministic Finite Automaton

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

scholarly journals Simplified group activity selection with group size constraints

2021 ◽  
Author(s):  
Andreas Darmann ◽  
Janosch Döcker ◽  
Britta Dorn ◽  
Sebastian Schneckenburger
Keyword(s):  
Computational Complexity ◽  
Simple Model ◽  
Lower Bounds ◽  
Group Activity ◽  
Core Stability ◽  
Upper And Lower Bounds ◽  
Size Constraints ◽  
Broad Variety ◽  
Special Case ◽  
Additional Constraints

AbstractSeveral real-world situations can be represented in terms of agents that have preferences over activities in which they may participate. Often, the agents can take part in at most one activity (for instance, since these take place simultaneously), and there are additional constraints on the number of agents that can participate in an activity. In such a setting, we consider the task of assigning agents to activities in a reasonable way. We introduce the simplified group activity selection problem providing a general yet simple model for a broad variety of settings, and start investigating its special case where upper and lower bounds of the groups have to be taken into account. We apply different solution concepts such as envy-freeness and core stability to our setting and provide a computational complexity study for the problem of finding such solutions.

scholarly journals Travelling Santa Problem: Optimization of a Million-Households Tour Within One Hour

2021 ◽  
Vol 8 ◽  
Author(s):  
Tilo Strutz
Keyword(s):  
Computational Complexity ◽  
Optimization Problems ◽  
State Of The Art ◽  
Travelling Salesman Problem ◽  
Limited Time ◽  
Two Dimensional ◽  
Optimal Solutions ◽  
Major Difficulty ◽  
Travelling Salesman ◽  
New Approach

Finding the shortest tour visiting all given points at least ones belongs to the most famous optimization problems until today [travelling salesman problem (TSP)]. Optimal solutions exist for many problems up to several ten thousand points. The major difficulty in solving larger problems is the required computational complexity. This shifts the research from finding the optimum with no time limitation to approaches that find good but sub-optimal solutions in pre-defined limited time. This paper proposes a new approach for two-dimensional symmetric problems with more than a million coordinates that is able to create good initial tours within few minutes. It is based on a hierarchical clustering strategy and supports parallel processing. In addition, a method is proposed that can correct unfavorable paths with moderate computational complexity. The new approach is superior to state-of-the-art methods when applied to TSP instances with non-uniformly distributed coordinates.

scholarly journals ON THE POSSIBLE RANKS AMONG MATRICES WITH A GIVEN PATTERN

2010 ◽  
Vol 02 (03) ◽  
pp. 363-377 ◽  
Author(s):  
CHARLES R. JOHNSON ◽  
YULIN ZHANG
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Null Space ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Given are tight upper and lower bounds for the minimum rank among all matrices with a prescribed zero–nonzero pattern. The upper bound is based upon solving for a matrix with a given null space and, with optimal choices, produces the correct minimum rank. It leads to simple, but often accurate, bounds based upon overt statistics of the pattern. The lower bound is also conceptually simple. Often, the lower and an upper bound coincide, but examples are given in which they do not.

BOUNDS ON THE PRICE OF STABILITY OF UNDIRECTED NETWORK DESIGN GAMES WITH THREE PLAYERS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 1-17 ◽  
Author(s):  
VITTORIO BILÒ ◽  
ROBERTA BOVE
Keyword(s):  
Lower Bound ◽  
Network Design ◽  
Lower Bounds ◽  
Upper Bound ◽  
Cost Allocation ◽  
Upper And Lower Bounds ◽  
Price Of Stability ◽  
Undirected Network ◽  
Basic Case

After almost seven years from its definition,2 the price of stability of undirected network design games with fair cost allocation remains to be elusive. Its exact characterization has been achieved only for the basic case of two players2,7 and, as soon as the number of players increases, the gap between the known upper and lower bounds becomes super-constant, even in the special variants of multicast and broadcast games. Motivated by the intrinsic difficulties that seem to characterize this problem, we analyze the already challenging case of three players and provide either new or improved bounds. For broadcast games, we prove an upper bound of 1.485 which exactly matches a lower bound given in Ref. 4; for multicast games, we show new upper and lower bounds which confine the price of stability in the interval [1.524; 1.532]; while, for the general case, we give an improved upper bound of 1.634. The techniques exploited in this paper are a refinement of those used in Ref. 7 and can be easily adapted to deal with all the cases involving a small number of players.

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