Minimum Linear Arrangement Research Articles

Hardness and approximation of submodular minimum linear ordering problems

AbstractThe minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $$f(\cdot )$$ f ( · ) due to an ordering $$\sigma $$ σ of the items (say [n]), i.e., $$\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })$$ min σ ∑ i ∈ [ n ] f ( E i , σ ) , where $$E_{i,\sigma }$$ E i , σ is the set of items mapped by $$\sigma $$ σ to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a $$(2-\frac{1+\ell _{f}}{1+|E|})$$ ( 2 - 1 + ℓ f 1 + | E | ) -approximation for monotone submodular MLOP where $$\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}$$ ℓ f = f ( E ) max x ∈ E f ( { x } ) satisfies $$1 \le \ell _f \le |E|$$ 1 ≤ ℓ f ≤ | E | . Our theory provides new approximation bounds for special cases of the problem, in particular a $$(2-\frac{1+r(E)}{1+|E|})$$ ( 2 - 1 + r ( E ) 1 + | E | ) -approximation for the matroid MLOP, where $$f = r$$ f = r is the rank function of a matroid. We further show that minimum latency vertex cover is $$\frac{4}{3}$$ 4 3 -approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.

Minimum Linear Arrangement of Generalized Sierpinski Graphs

The creation of scientific supercomputers is one of the most pressing issues confronting technology today. Experts in computer science anticipate that future supercomputers will be built on large-scale parallel processing. A system with multiple processors and memories will be used in such a computer. The interconnection network that allows communication between the system’s processors and memories is a critical component of such systems. In the topic of interconnection networks for parallel computer architectures, graph embedding problems have grown in relevance. Network embedding has been recognized as a valuable method for developing efficient algorithms and simulating various architectures in parallel and distributed computing. In this paper, we obtain the maximum subgraph of the generalized Sierpinski graphs $S(n, m), n\geq 2, m\geq 3$, and calculate the minimum linear arrangement of generalized Sierpinski graphs by graph embeddings.

Sequential locality of graphs and its hypothesis testing

The adjacency matrix is the most fundamental and intuitive object in graph analysis that is useful not only mathematically but also for visualizing the structures of graphs. Because the appearance of an adjacency matrix is critically affected by the ordering of rows and columns, or vertex ordering, statistical assessment of graphs together with their vertex sequences is important in identifying the characteristic structures of graphs. In this paper, we propose a hypothesis testing framework that assesses how locally vertices are connected to each other along a specified vertex sequence, which provides a statistical foundation for an optimization problem called envelope reduction or minimum linear arrangement. The proposed tests are particularly suitable for moderately small data and formulated based on a combinatorial approach and a block model with intrinsic vertex ordering.

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.

Minimum Linear Arrangement of the Cartesian Product of Optimal Order Graph and Path

The minimum linear arrangement of an arbitrary graph is the embedding of the vertices of the graph onto the line topology in such a way that the sum of the distances between adjacent vertices in the graph is minimized. This minimization can be attained by finding an optimal ordering of the vertex set of the graph and labeling the vertices of the line in that order. In this paper, we compute the minimum linear arrangement of the Cartesian product of certain sequentially optimal order graphs which include interconnection networks such as hypercube, folded hypercube, complete Josephus cube and locally twisted cube with path and the edge faulty counterpart of sequentially optimal order graphs.

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.

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.

Scheduling series-parallel task graphs to minimize peak memory

We consider a variant of the well-known, NP-complete problem of minimum cut linear arrangement for directed acyclic graphs. In this variant, we are given a directed acyclic graph and we are asked to find a topological ordering such that the maximum number of cut edges at any point in this ordering is minimum. In our variant, the vertices and edges have weights, and the aim is to minimize the maximum weight of cut edges in addition to the weight of the last vertex before the cut. There is a known, polynomial time algorithm (Liu, 1987 [17]) for the cases where the input graph is a rooted tree. We focus on the instances where the input graph is a directed series-parallel graph, and propose a polynomial time algorithm, thus expanding the class of graphs for which a polynomial time algorithm is known. Directed acyclic graphs are used to model scientific applications where the vertices correspond to the tasks of a given application and the edges represent the dependencies between the tasks. In such models, the problem we address reads as minimizing the peak memory requirement in an execution of the application. Our work, combined with Liu's work on rooted trees addresses this practical problem in two important classes of applications.

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.

Tractable Parameterizations for the Minimum Linear Arrangement Problem

The M inimum L inear A rrangement (MLA) problem involves embedding a given graph on the integer line so that the sum of the edge lengths of the embedded graph is minimized. Most layout problems are either intractable or not known to be tractable, parameterized by the treewidth of the input graph. We investigate MLA with respect to three parameters that provide more structure than treewidth. In particular, we give a factor (1 + ε)-approximation algorithm for MLA parameterized by (ε, k ), where k is the vertex cover number of the input graph. By a similar approach, we obtain two FPT algorithms that exactly solve MLA parameterized by, respectively, the max leaf and edge clique cover numbers of the input graph.

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.

The Placement of the Head that Minimizes Online Memory

It is well known that the length of a syntactic dependency determines its online memory cost. Thus, the problem of the placement of a head and its dependents (complements or modifiers) that minimizes online memory is equivalent to the problem of the minimum linear arrangement of a star tree. However, how that length is translated into cognitive cost is not known. This study shows that the online memory cost is minimized when the head is placed at the center, regardless of the function that transforms length into cost, provided only that this function is strictly monotonically increasing. Online memory defines a quasi-convex adaptive landscape with a single central minimum if the number of elements is odd and two central minima if that number is even. We discuss various aspects of the dynamics of word order of subject (S), verb (V) and object (O) from a complex systems perspective and suggest that word orders tend to evolve by swapping adjacent constituents from an initial or early SOV configuration that is attracted towards a central word order by online memory minimization. We also suggest that the stability of SVO is due to at least two factors, the quasi-convex shape of the adaptive landscape in the online memory dimension and online memory adaptations that avoid regression to SOV. Although OVS is also optimal for placing the verb at the center, its low frequency is explained by its long distance to the seminal SOV in the permutation space.