Minimum length corridor problem on grids

Minimal partition problems describe the task of partitioning a domain into a set of meaningful regions. Two important examples are image segmentation and 3D reconstruction. They can both be formulated as energy minimization problems requiring minimum boundary length or surface area of the regions. This common prior often leads to the removal of thin or elongated structures. Volume constraints impose an additional prior which can help preserve such structures. There exist a multitude of algorithms to minimize such convex functionals under convex constraints. We systematically compare the recent Primal Dual PD algorithm [1] to the Alternating Direction Method of Multipliers ADMM [2] on volume-constrained minimal partition problems. Our experiments indicate that the ADMM approach provides comparable and often better performance.

Parallel algorithms for the minimum cut and the minimum length tree layout problems

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, no two terminals on the same horizontal or vertical line, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) belong to the axis-parallel grid defined by T and are oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this article, we present a polynomial factor 2-approximation algorithm for the bidirected minimum Manhattan network problem. © 2016 Wiley Periodicals, Inc. NETWORKS, 2016

We present a primal-dual algorithm for solving the minimum average weighted length circuit problem. The algorithm solves the problem by solving a series of subproblems with more combinatorial aspects iteratively. We also prove that the complexity of the algorithm is O(n4 max{tij}) and show that our algorithm is actually a generalization of the Karp–Orlin algorithm. Finally, the relationship between the two algorithms is discussed.

Projection-based algorithms for continuum topology optimization have received considerable attention in recent years due to their ability to control minimum length scale in a computationally efficient manner. This not only provides a means for imposing manufacturing length scale constraints, but also circumvents numerical instabilities of solution mesh dependence and checkerboard patterns. This research aims at embedding the minimum and maximum length scale requirement into the projection methodology used for material distribution approaches to topology optimization. The proposed algorithms for two-phase minimum and solid maximum length scale requirements are demonstrated on benchmark minimum compliance problems and are shown to satisfy the length scale constraints imposed.

A Steiner Minimal Tree (SMT) for a given set P of points is a shortest network interconnecting the points of P whose vertex set may include some additional points in order to get the minimum possible total length in a metric space. When no additional points are allowed the minimum interconnection network is the well-known minimum spanning tree (MST) of P. The Steiner ratio is the greatest lower bound of the ratio of the length of an SMT over that of an MST of P. In this paper we study the Steiner minimal tree problem in which all the edges of SMT have fixed orientations. We call it the SMT problem in the λ-geometry plane, where λ is the number of possible orientations.Here is the summary of our results. 1. We show that the Steiner ratio for ¦P¦≥3 is √3/2 cos(π/2λ), for λ=6m+3 and integer m≥0, and is √3/2, for λ=6k and integer k≥1, disproving a a conjecture of Du et al.[3] that the ratio is √3/2 iff the unit disk in normed planes is an ellipse. 2. We derive the Steiner ratios for ¦P¦ ≤ 4 for all possible λ's and show that for ¦P¦≥3 there exists an SMT whose Steiner points lie in a multi-level Hanan-grid, generalizing a result that holds for rectilinear case, i.e., λ=2. These results show that the Steiner ratio is not a monotonically increasing function of λ, as believed by many researchers. We conjecture that the Steiner ratios obtained above (¦P¦≤4) are actually true for all ¦P¦≥3.KeywordsMinimum Span TreeSteiner TreeRegular PointSteiner PointNormed PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Ship Berthing Problem (SBP) is one of the many problems faced by a port in its daily execution. Ships arriving at the port will have to be assigned specific berthing location based on certain constraints arising from the ship's cargo or physical characteristics. SBP can be solved as a minimization problem, in which the aim is to assign all ships in an arrangement that would result a minimum wharf length. The SBP can be represented using a Directed Acyclic Graph (DAG) where vertices represents ships and edges represent the contemporary relationship between ships. Two ships are contemporary if they are both required to be berthed at a particular instance. The direction of the edges determines the relative positions of the ships and hence the resulting required wharf length. For the ease of manipulation, the DAG is represented by an acyclic list where each node in the list represents a vertex and the relative positions of the nodes reflects the direction of edges in the DAG. A Greedy Local Search (GLS) algorithm is proposed to solve the SBP. The idea is to constantly displace nodes in the acyclic list as long as the resulting required berth length is reduced. A Tabu-Search (TS) post-optimization technique is proposed to improve the result of the GLS, here, a potentially 'bad' displacement is allowed in hope that the move would result in an escape from the local optimal, the performance of the TS can be varied by specifying the maximum number of consecutive 'bad' displacement allowed. Two displacement techniques with different local search neighbourhood were proposed. The initial solution in which the GLS is performed can be obtained by using a Greedy Heuristic, alternatively, a randomly generated initial solution can be used. A different approach in solving the SBP is the use of Genetic Algorithm (GA). Due to the unique nature of the acyclic list, common crossover reproductive methods cannot be used, instead three reproductive methods that would not corrupt the acyclic list were proposed. The performance of the G A can be configured by using a decline factor on the fitness measure, a high decline factor would give rise to a more stringent natural selection by modifing the fitness measure as the generation progresses. The roulette selection method is used to ensure that the solution would not be moving towards the local optimal all the time. We have experimented with a total of 22 different variants of the GLS, TS and GA and found that all techniques possess variants that produces good results for solving the SBP.

In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef et al., FOCS, 2006).We show a polynomial time algorithm that, given an n vertex graph G with minrank k, finds a linear index code for G of length O(nf(k)), where f(k) depends only on k. For example, for k = 3 we obtain f(3) a 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank.At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovasz θ-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

River routing in VLSI

The present article concentrates on the dogleg-free Manhattan model where horizontal and vertical wire segments are positioned on different sides of the board and each net (wire) has at most one horizontal segment. While the minimum width can be found in linear time in the single row routing, apparently there was no efficient algorithm to find the minimum wire length. We show that there is no hope to find such an algorithm because this problem is ${\cal NP}$-complete even if each net has only two terminals. The results on dogleg-free Manhattan routing can be connected with other application areas related to interval graphs. In this paper we define the minimum value interval placement problem. There is given a set of weighted intervals and w rows and the intervals have to be placed without overlapping into rows so that the sum of the interval values, which is the value of a function of the weight and the row number assigned to the interval, is minimum. We show that this problem is ${\cal NP}$-complete. This implies the ${\cal NP}$-completeness of other problems including the minimum wire length routing and the sum coloring on interval graphs.

Minimum length scale control on real and void material phases in topology optimization is an important topic of research with direct implications on numerical stability and solution manufacturability. And it also is a challenge area of research due to serious conflicts of both the solid and the void phase element densities in phase mixing domains of the topologies obtained by existing methods. Moreover, there is few work dealing with controlling distinct minimum feature length scales of real and void phase materials used in topology designs. A new method for solving the minimum length scale controlling problem of real and void material phases, is proposed. Firstly, we introduce two sets of coordinating design variable filters for these two material phases, and two distinct smooth Heaviside projection functions to destroy the serious conflicts in the existing methods (e.g. Guest Comput Methods Appl Mech Eng 199(14):123–135, 2009). Then, by introducing an adaptive weighted 2-norm aggregation constraint function, we construct a coordinating topology optimization model to ensure distinct minimum length scale controls of real and void phase materials for the minimum compliance problem. By adopting a varied volume constraint limit scheme, this coordinating topology optimization model is transferred into a series of coordinating topology optimization sub-models so that the structural topology configuration can stably and smoothly changes during an optimization process. The structural topology optimization sub-models are solved by the method of moving asymptotes (MMA). Then, the proposed method is extended to the compliant mechanism design problem. Numerical examples are given to demonstrate that the proposed method is effective and can obtain a good 0/1 distribution final topology.

Heuristics for optimum binary search trees and minimum weight triangulation problems

In this paper, we introduce an O(nm) time algorithm to determine the minimum length directed cycle (also called the "minimum weight directed cycle") in a directed network with n nodes and m arcs and with no negative length directed cycles. This result improves upon the previous best time bound of O(nm + n2 log log n). Our algorithm first determines the cycle with minimum mean length λ* in O(nm) time. Subsequently, it chooses node potentials so that all reduced costs are λ* or greater. It then solves the all pairs shortest path problem, but restricts attention to paths of length at most nλ*. We speed up the shortest path calculations to O(m) per source node, leading to an O(nm) running time in total. We also carry out computational experiments comparing the performance of the proposed methods and other state-of-the-art methods. Experiments confirmed that it is advantageous to solve the minimum mean cycle problem prior to solving shortest path problems. Analysis of our experiments suggest that the running time to solve the minimum length directed cycle problem was much faster than O(n2) on average.

Hamada ([8]) and Maruta ([17]) proved the minimum length <TEX>$n_3(6,\;d)=g_3(6,\;d)+1$</TEX> for some ternary codes. In this paper we consider such minimum length problem for <TEX>$q{\geq}4$</TEX>, and we prove that <TEX>$n_q(6,\;d)=g_q(6,\;d)+1$</TEX> for <TEX>$d=q^5-q^3-q^2-2q+e$</TEX>, <TEX>$1{\leq}e{\leq}q$</TEX>. Combining this result with Theorem A in [4], we have <TEX>$n_q(6,\;d)=g-q(6,\;d)+1$</TEX> for <TEX>$q^5-q^3-q^2-2q+1{\leq}d{\leq}q^5-q^3-q^2$</TEX> with <TEX>$q{\geq}4$</TEX>. Note that <TEX>$n_q(6,\;d)=g_q(6,\;d)$</TEX> for <TEX>$q^5-q^3-q^2+1{\leq}d{\leq}q^5$</TEX> by Theorem 1.2.

A new suite of computational procedures for stress-constrained continuum topology optimization is presented. In contrast to common approaches for imposing stress constraints, herein it is proposed to limit the maximum stress by controlling the length scale of the optimized design. Several procedures are formulated based on the treatment of the filter radius as a design variable. This enables to automatically manipulate the minimum length scale such that stresses are constrained to the allowable value, while the optimization is driven to minimizing compliance under a volume constraint – without any direct constraints on stresses. Numerical experiments are presented that incorporate the following : 1) Global control over the filter radius that leads to a uniform minimum length scale throughout the design; 2) Spatial variation of the filter radius that leads to local manipulation of the minimum length according to stress concentrations; and 3) Combinations of the two above. The optimized designs provide high-quality trade-offs between compliance, stress and volume. From a computational perspective, the proposed procedures are efficient and simple to implement: essentially, stress-constrained topology optimization is posed as a minimum compliance problem with additional treatment of the length scale.