Linear Arrangement Problem Research Articles

The maximum linear arrangement problem for trees under projectivity and planarity

A linear arrangement is a mapping π from the n vertices of a graph G to n distinct consecutive integers. Linear arrangements can be represented by drawing the vertices along a horizontal line and drawing the edges as semicircles above said line. In this setting, the length of an edge is defined as the absolute value of the difference between the positions of its two vertices in the arrangement, and the cost of an arrangement as the sum of all edge lengths. Here we study two variants of the Maximum Linear Arrangement problem (MaxLA), which consists of finding an arrangement that maximizes the cost. In the planar variant for free trees, vertices have to be arranged in such a way that there are no edge crossings. In the projective variant for rooted trees, arrangements have to be planar and the root of the tree cannot be covered by any edge. In this paper we present algorithms that are linear in time and space to solve planar and projective MaxLA for trees. We also prove several properties of maximum projective and planar arrangements, and show that caterpillar trees maximize planar MaxLA over all trees of a fixed size thereby generalizing a previous extremal result on trees.

A compact quadratic model and linearizations for the minimum linear arrangement problem

Given an undirected connected graph G = ( V , E ) , the minimum linear arrangement problem (MinLA) consists in determining a permutation π ≔ ( π 1 , … , π n ) of the node-set V = { 1 , … , n } , which minimizes the sum of edge costs | π i − π j | over all edges { i , j } ∈ E . We introduce a compact quadratic formulation, prove its correctness, and generate new mixed-integer linear formulations that require a very small number of variables and constraints. The idea behind the way we model permutations allows the development of strong valid constraints to strengthen the new formulations. We also adapt some cutting plane procedures from the literature to one of the new models. Computational experiments with benchmark and new instances are very encouraging and improve state-of-the-art solution approaches from the literature. • We investigate the minimum linear arrangement problem. • We propose three new compact formulations for the problem. • We develop strong constraints to strengthen the new formulations. • Theoretical results can be extended to handle problems with a permutational structure.

Minimum projective linearizations of trees in linear time

The Minimum Linear Arrangement problem (MLA) consists of finding a mapping π from vertices of a graph to distinct integers that minimizes ∑{u,v}∈E|π(u)−π(v)|. In that setting, vertices are often assumed to lie on a horizontal line and edges are drawn as semicircles above said line. For trees, various algorithms are available to solve the problem in polynomial time in n=|V|. There exist variants of the MLA in which the arrangements are constrained. Iordanskii, and later Hochberg and Stallmann (HS), put forward O(n)-time algorithms that solve the problem when arrangements are constrained to be planar (also known as one-page book embeddings). We also consider linear arrangements of rooted trees that are constrained to be projective (planar embeddings where the root is not covered by any edge). Gildea and Temperley (GT) sketched an algorithm for projective arrangements which they claimed runs in O(n) but did not provide any justification of its cost. In contrast, Park and Levy claimed that GT's algorithm runs in O(nlog⁡dmax) where dmax is the maximum degree but did not provide sufficient detail. Here we correct an error in HS's algorithm for the planar case, show its relationship with the projective case, and derive simple algorithms for the projective and planar cases that run without a doubt in O(n) time.

Decorous combinatorial lower bounds for row layout problems

In this paper we consider the Double-Row Facility Layout Problem (DRFLP). Given a set of departments and pairwise transport weights between them the DRFLP asks for a non-overlapping arrangement of the departments along both sides of a common path such that the weighted sum of the center-to-center distances between the departments is minimized. Despite its broad applicability in factory planning, only small instances can be solved to optimality in reasonable time. Apart from this even deriving good lower bounds using existing integer programming formulations and branch-and-cut methods is a challenging problem. We focus here on deriving combinatorial lower bounds which can be computed very fast. These bounds generalize the star inequalities of the Minimum Linear Arrangement Problem. Furthermore we exploit a connection of the DRFLP to some parallel identical machine scheduling problem. Our lower bounds can be further improved by combining them with a new distance-based mixed-integer linear programming model, which is not a formulation for the DRFLP, but can be solved close to optimality quickly. We compare the new lower bounds to some heuristically determined upper bounds on medium-sized and large DRFLP instances. Special consideration is given to the case when all departments have the same length. Furthermore we show that the lower bounds that we derive using adapted variants of our approaches for the Parallel Row Ordering Problem, a DRFLP variant where the row assignment of the departments is given in advance and spaces between neighboring departments are not allowed, are even better with respect to the gaps.

Revealing the political affinity of online entities through their Twitter followers

In this work, we show that the structural features of the Twitter online social network can divulge valuable information about the political affinity of the participating nodes. More precisely, we show that Twitter followers can be used to predict the political affinity of prominent Nodes of Interest (NOIs) they opt to follow. We utilize a series of purely structure-based algorithmic approaches, such as modularity clustering, the minimum linear arrangement (MinLA) problem and the DeGroot opinion update model in order to reveal diverse aspects of the NOIs’ political profile. Our methods are applied to a dataset containing the Twitter accounts of the members of the Greek Parliament as well as an enriched dataset that additionally contains popular news sources. The results confirm the viability of our approach and provide evidence that the political affinity of NOIs can be determined with high accuracy via the Twitter follower network. Moreover, the outcome of an independently performed expert study about the offline political scene confirms the effectiveness of our methods.

Lower and upper bounds for the linear arrangement problem on interval graphs

We deal here with theLinear Arrangement Problem(LAP) onintervalgraphs, any interval graph being given here together with its representation as theintersectiongraph of some collection of intervals, and so with relatedprecedenceandinclusionrelations. We first propose a lower boundLB, which happens to be tight in the case ofunit intervalgraphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with theprecedencerelation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP onintervalgraphs.

Approximation Guarantees for the Minimum Linear Arrangement Problem by Higher Eigenvalues

Given an n -vertex undirected graph G = ( V , E ) and positive edge weights { w e } e∈E , a linear arrangement is a permutation π : V → {1, 2, …, n }. The value of the arrangement is val ( G , π) := 1/n∑ e ={ u, v } ∈ E w e |π( u ) − π ( v )|. In the minimum linear arrangement problem, the goal is to find a linear arrangement π * that achieves val ( G , π * ) = MLA( G ) := min π val ( G , π). In this article, we show that for any ϵ > 0 and positive integer r , there is an n O ( r /ϵ) -time randomized algorithm that, given a graph G , returns a linear arrangement π, such that val ( G , π) ≤ (1 + 2/(1 − ε)λ r ( L )) MLA( G ) + O (√log n / n ∑ e ∈ E w e ) with high probability, where L is the normalized Laplacian of G and λ r ( L ) is the r th smallest eigenvalue of L . Our algorithm gives a constant factor approximation for regular graphs that are weak expanders.

A note on minimum linear arrangement for BC graphs

A linear arrangement is a labeling or a numbering or a linear ordering of the vertices of a graph. In this paper, we solve the minimum linear arrangement problem for bijective connection graphs (for short BC graphs) which include hypercubes, Möbius cubes, crossed cubes, twisted cubes, locally twisted cube, spined cube, [Formula: see text]-cubes, etc., as the subfamilies.

Minimum Linear Arrangements

Let G = (V, E) be a simple undirected graph. Given distinct labels in {1,⋯,|V|} to the vertices of G, we define the weight of an edge uv∈E as the absolute difference between the labels assigned to u and v. The minimum linear arrangement problem (MinLA) consists in finding a labeling of the vertices of G such that the sum of the weights of its edges is minimized. It is an NP-Hard problem whose corresponding polyhedron has a factorial number of extreme points. We propose a quadratic model for MinLA and use it to obtain a novel compact mixed integer linear programming (MILP) formulation for the problem, featuring O(|V|2) variables and O(|V|2) constraints. We show the correctness of the new model and discuss valid inequalities for the problem. Our findings open new insights on the study of effective exact approaches to the problem. Computational experiments show that the new quadratic and mixed linear models performed better than existing ones in the literature for new and benchmark instances of this problem.

New relationships for multi-neighborhood search for the minimum linear arrangement problem

We develop a series of theorems about the graph structure of the classical Minimum Linear Arrangement (MinLA) problem which disclose properties that can be exploited by Multi-Neighborhood Search (MNS) algorithms. As a foundation, we differentiate between swaps of labels attached to adjacent and non-adjacent nodes to create two new neighborhood classes, and show how our theorems yield efficient algorithms for updating key arrays used by local search procedures. In addition, we introduce a class of neighborhoods called set-based neighborhoods supported by a theorem that identifies solutions (labelings) for the MinLA problem in polynomial time that dominate exponential numbers of alternative solutions. The component neighborhoods within this new neighborhood class can be applied in various sequences in conjunction with the first two new neighborhoods introduced. Our results also apply to problems with objectives different than those of MinLA. Finally, our results make it possible to exploit the new neighborhoods according to the user's choice of MNS protocols and alternative local search algorithms.

A Correction on Shiloach's Algorithm for Minimum Linear Arrangement of Trees

More than 30 years ago, Shiloach published an algorithm to solve the minimum linear arrangement problem for undirected trees. Here we fix a small error in the original version of the algorithm and discuss its effect on subsequent literature. We also improve some aspects of the notation.

The scaling of the minimum sum of edge lengths in uniformly random trees

The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial time, they have remained theoretical and have not been implemented in practical contexts to our knowledge. Here we use one of those algorithms to investigate the growth of this sum as a function of the size of the tree in uniformly random trees. We show that this sum is bounded above by its value in a star tree. We also show that the mean edge length grows logarithmically in optimal linear arrangements, in stark contrast to the linear growth that is expected on optimal arrangements of star trees or on random linear arrangements.

Variable neighborhood search for minimum linear arrangement problem

The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. It consists of finding an embedding of the nodes of a graph on the line such that the sum of the resulting edge lengths is minimized. This problem is one among the classical NP-hard optimization problems and therefore there has been extensive research on exact and approximative algorithms. In this paper we present an implementation of a variable neighborhood search (VNS) for solving minimum linear arrangement problem. We use Skewed general VNS scheme that appeared to be successful in solving some recent optimization problems on graphs. Based on computational experiments, we argue that our approach is comparable with the state-of-the-art heuristic.

Layout Optimization for Quantum Circuits with Linear Nearest Neighbor Architectures

This paper is concerned with the physical design of quantum logic circuits. More precisely, it addresses the problem of minimizing the number of required qubit reorderings (achieved by inserting explicit SWAP gates) when mapping a quantum circuit into a linear nearest neighbor quantum archi-tecture. First, an interaction graph that captures the interaction distances among various qubits in the quantum circuit is constructed. The interaction graph is utilized to partition the quantum circuit into a set of subcircuits such that the number of required qubit reoderings within each subcircuit is provably no more than a given threshold. Next, a Minimum Linear Arrangement problem for each subcircuit is formulated and solved to achieve the minimum number of internal qubit reorderings and determine the subcircuit input and output qubit orderings. Finally, a bubble sort algorithm is repeatedly employed to minimize the number of qubit reorderings that are required between the consecutive subcircuits. Experiments done on various quantum Fourier transform circuits as well as various reversible logic circuits demonstrate the effectiveness of the proposed approach.

Boosting the performance of metaheuristics for the MinLA problem using a more discriminating evaluation function

This paper investigates the role of evaluation function used by metaheuristics for solving combinatorial optimization problems. Evaluation function (EF) is a key component of any metaheuristic algorithm and its design directly influences the performance of such an algorithm. However, the design of more discriminating EFs is somewhat overlooked in the literature. We present in this work the first in-depth analysis of the conventional EF for the Minimum Linear Arrangement (MinLA) problem. The results from this study highlighted its potential drawbacks and led to useful insight and information which guided us to design a new more discerning EF. Its practical usefulness was assessed within three different algorithms: a parameter-free Steepest Descent, an Iterated Local Search and a Tabu Search. The analysis of the data produced by these comparisons showed that the performance of the three adopted approaches could be boosted by using the proposed more discriminating EF.

Self-adjusting grid networks to minimize expected path length

Given a network infrastructure (e.g., data-center or on-chip-network) and a distribution on the source-destination requests, the expected path (route) length is an important measure for the performance, efficiency and power consumption of the network. In this work we initiate a study on self-adjusting networks: networks that use local-distributed mechanisms to adjust the position of the nodes (e.g., virtual machines) in the network to best fit the route requests distribution. Finding the optimal placement of nodes is defined as the minimum expected path length (MEPL) problem. This is a generalization of the minimum linear arrangement (MLA) problem where the network infrastructure is a line and the computation is done centrally. In contrast to previous work, we study the distributed version and give efficient and simple approximation algorithms for interesting and practically relevant special cases of the problem. In particular, we consider grid networks in which the distribution of requests is a symmetric product distribution. In this setting, we show that a simple greedy policy of position switching between neighboring nodes to locally minimize an objective function achieves good approximation ratios. We are able to prove this result using the useful notions of expected rank of the distribution and the expected distance to the center of the graph.

D-dimensional arrangement revisited

We revisit the d-dimensional arrangement problem and analyze the performance ratios of previously proposed algorithms based on the linear arrangement problem with d-dimensional cost. The two problems are related via space-filling curves and recursive balanced bipartitioning. We prove that the worst-case ratio of the optimum solutions of these problems is Θ(logn), where n is the number of vertices of the graph. This invalidates two previously published proofs of approximation ratios for d-dimensional arrangement. Furthermore, we conclude that the currently best known approximation ratio for this problem is O(logn).

An optimal time algorithm for minimum linear arrangement of chord graphs

A linear arrangement ϕ of an undirected graph G=(V,E) with ∣V∣=n nodes is a bijective function ϕ:V→{0,…,n−1}. The cost function is cost(G,ϕ)=∑uv∈E|(ϕ(u)-ϕ(v))| and opt(G)=min∀ϕcost(G,ϕ). The problem of finding opt(G) is called minimum linear arrangement (MINLA). The Minimum Linear Arrangement is an NP-hard problem in general. But there are some classes of graphs optimally solvable in polynomial time. In this paper, we show that the label of each node equals to the reverse of binary representation of its id in the optimal arrangement. Then, we design an O(n) algorithm to solve the minimum linear arrangement problem of Chord graphs.

On a binary distance model for the minimum linear arrangement problem

The minimum linear arrangement problem consists of finding an embedding of the nodes of a graph on the line such that the sum of the resulting edge lengths is minimized. The problem is among the classical NP-hard optimization problems and there has been extensive research on exact and approximative algorithms. In this paper, we introduce a new model based on binary variables dijk that are equal to 1 if nodes i and j have distance k in the ordering. We analyze this model and point to connections and differences to a model using integer distance variables. Based on computational experiments, we argue that our model is worth further theoretical and practical investigation and that is has potentials yet to be examined.

Exact solution of the 2-dimensional grid arrangement problem

Given an undirected graph G=(V,E), we consider injective mappings of its vertices to the k-dimensional Cartesian integer grid. Among such embeddings we are interested in those that minimize the sum of the resulting edge lengths, where the length of an edge is defined by the L1-metric. The case k=1 is the well-known Minimum Linear Arrangement Problem. We prove that the general problem is NP-hard for any fixed grid dimension. Our computational study focuses on the case k=2. We present as a first exact optimization algorithm a branch-and-cut scheme for sparse graphs. Several classes of valid inequalities are introduced and analyzed for facet defining properties of two corresponding polyhedra. Finally, computational results from a successful real-world application of the problem at the German Cancer Research Center (DKFZ) are presented.