Class Of Interval Graphs Research Articles

Online coloring of short intervals

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in [1,σ]. For σ=1 it is the class of unit interval graphs and for σ=∞ the class of all interval graphs. Our focus is on intermediary classes. We present a (1+σ)-competitive algorithm, which beats the state of the art for 1<σ<2, and proves that the problem we study can be strictly easier than online coloring of general interval graphs. On the lower bound side, we prove that no algorithm is better than 5/3-competitive for any σ>1, nor better than 7/4-competitive for any σ>2, and that no algorithm beats the 5/2 asymptotic competitive ratio for all, arbitrarily large, values of σ. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than 2. Moreover, we prove that the natural algorithm FirstFit is no better than 5/2 for any σ>3, no better than 11/4 for any σ>12, and finally for every ɛ>0 FirstFit is no better than (5−ɛ)-competitive for any σ>σɛ (where σɛ is some finite length depending on ɛ). We also give some upper bounds for FirstFit for σ∈[5,13].

Interval Count of Interval Graphs

The interval count problem is that of determining the smallest number of distinct interval lengths that is sufficient to represent an interval model of a given interval graph. The class of interval graphs is well known, with several applications. This article briefly summarizes my doctorate thesis done on the subject with supervision of Prof. Jayme Szwarcfiter, in his honor, on the occasion of his 80th birthday.

ОПТИМИЗАЦИЯ ОРГАНИЗАЦИОННЫХ СТРУКТУР И СОСТАВА ПОДРАЗДЕЛЕНИЙ ТРАНСПОРТНОЙ БЕЗОПАСНОСТИ НА ЖЕЛЕЗНОДОРОЖНОМ ТРАНСПОРТЕ

The article implements the task of maximizing the technical and economic efficiency and itsapplication for the timely and complete implementation of tasks to ensure the sustainable and safefunctioning of the transport unit. The goal is to find the minimum number of human resourcesrequired for transportation. Guided by the main provisions of lean production, the task of optimizing the number of labor resources was implemented. A mathematical optimization model of thestaffing table for the distribution of tasks to staff-resource units for the management of transportationprocesses is constructed. The optimization problem was solved using a class of intervalgraphs with a maximum load per color. To implement this task, a software tool was created thatallows it to be used for implementation in production, making calculations for finding the optimalamount of labor resources under given boundary conditions, with full system load at maximumload of certain colors in the class of interval graphs. The presented results can be used both directlyby the subjects of transport infrastructure, and in the management systems of enterprisesand organizations that carry out their activities in the field of transport security, in freight aggregatorsand logistics departments of various companies. It is possible to build various optimal variantsof organizational structures and departments of transport security with minimal costs for theorganization of transportation processes.

The Steiner cycle and path cover problem on interval graphs

The Steiner path problem is a common generalization of the Steiner tree and the Hamiltonian path problem, in which we have to decide if for a given graph there exists a path visiting a fixed set of terminals. In the Steiner cycle problem we look for a cycle visiting all terminals instead of a path. The Steiner path cover problem is an optimization variant of the Steiner path problem generalizing the path cover problem, in which one has to cover all terminals with a minimum number of paths. We study those problems for the special class of interval graphs. We present linear time algorithms for both the Steiner path cover problem and the Steiner cycle problem on interval graphs given as endpoint sorted lists. The main contribution is a lemma showing that backward steps to non-Steiner intervals are never necessary. Furthermore, we show how to integrate this modification to the deferred-query technique of Chang et al. to obtain the linear running times.

On low rank-width colorings

We introduce the concept of low rank-width colorings, generalizing the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez (2008). We say that a class C of graphs admits low rank-width colorings if there exist functions N:N→N and Q:N→N such that for all p∈N, every graph G∈C can be vertex colored with at most N(p) colors such that the union of any i≤p color classes induces a subgraph of rank-width at most Q(i).Graph classes admitting low rank-width colorings strictly generalize graph classes admitting low tree-depth colorings and graph classes of bounded rank-width. We prove that for every graph class C of bounded expansion and every positive integer r, the class {Gr:G∈C} of rth powers of graphs from C admits low rank-width colorings. On the negative side, we show that the classes of interval graphs and permutation graphs do not admit low rank-width colorings. As interesting side properties, we prove that every hereditary graph class admitting low rank-width colorings has the Erdős–Hajnal property and is χ-bounded.

New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, respectively total k-dominating set, in a given graph, are referred to as k-domination, respectively total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from previous works by Bui-Xuan et al. (2013) [8] and by Belmonte and Vatshelle (2013) [3] that these two families of problems are solvable in time O(|V(G)|3k+4) in the class of interval graphs. We develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs, by means of reduction to a single shortest path computation in a derived directed acyclic graph with O(|V(G)|2k) nodes and O(|V(G)|4k) arcs. We show that a suitable implementation, which avoids constructing all arcs of the digraph, leads to a running time of O(|V(G)|3k). The algorithms are also applicable to the weighted case.

Characterizing interval graphs which are probe unit interval graphs

A graph G is a probe unit interval graph if its vertex set can be partitioned into a set P of probe vertices and a stable set N of nonprobe vertices, so that a unit interval graph can be obtained by adding a set of edges whose endpoints belong to N. A partitioned graph is a graph having a prescribed partition into P and N. In this article we present structural characterizations for those partitioned interval graphs and unpartitioned interval graphs which are probe unit interval graphs, in terms of certain characteristics of their interval models. These characterizations lead to characterizations of probe unit interval graphs within the class of interval graphs by minimal forbidden induced subgraphs.

The load-balanced multi-dimensional bin-packing problem

The bin-packing problem is one of the most investigated and applicable combinatorial optimization problems. In this paper we consider its multi-dimensional version with the practical extension of load balancing, i.e. to find the packing requiring the minimum number of bins while ensuring that the average center of mass of the loaded bins falls as close as possible to an ideal point, for instance, the center of the bin. We formally describe the problem using mixed-integer linear programming models, from the simple case where we want to optimally balance a set of items already assigned to a single bin, to the general balanced bin-packing problem. Given the difficulty for standard solvers to deal even with small size instances, a multi-level local search heuristic is presented. The algorithm takes advantage of the Fekete–Schepers representation of feasible packings in terms of particular classes of interval graphs, and iteratively improves the load balancing of a bin-packing solution using different search levels. The first level explores the space of transitive orientations of the complement graphs associated with the packing, the second modifies the structure itself of the interval graphs, the third exchanges items between bins repacking proper n-tuples of weakly balanced bins. Computational experiments show very promising results on a set of 3D bin-packing instances from the literature.

Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker--Boland Subgraphs

Tucker characterized the minimal forbidden submatrices of binary matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find such a minimal one in any binary matrix that does not have the consecutive-ones property. Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find such a minimal one in any graph that is not an interval graph.

L-Recursion and a new Logic for Logarithmic Space

We extend first-order logic with counting by a new operator that allows it to formalise a limited form of recursion which can be evaluated in logarithmic space. The resulting logic LREC has a data complexity in LOGSPACE, and it defines LOGSPACE-complete problems like deterministic reachability and Boolean formula evaluation. We prove that LREC is strictly more expressive than deterministic transitive closure logic with counting and incomparable in expressive power with symmetric transitive closure logic STC and transitive closure logic (with or without counting). LREC is strictly contained in fixed-point logic with counting FPC. We also study an extension LREC= of LREC that has nicer closure properties and is more expressive than both LREC and STC, but is still contained in FPC and has a data complexity in LOGSPACE. Our main results are that LREC captures LOGSPACE on the class of directed trees and that LREC= captures LOGSPACE on the class of interval graphs.

Tree loop graphs

Many problems involving DNA can be modeled by families of intervals. However, traditional interval graphs do not take into account the repeat structure of a DNA molecule. In the simplest case, one repeat with two copies, the underlying line can be seen as folded into a loop. We propose a new definition that respects repeats and define loop graphs as the intersection graphs of arcs of a loop. The class of loop graphs contains the class of interval graphs and the class of circular-arc graphs. Every loop graph has interval number 2. We characterize the trees that are loop graphs. The characterization yields a polynomial-time algorithm which given a tree decides whether it is a loop graph and, in the affirmative case, produces a loop representation for the tree.

A Polynomial Time Algorithm for Obtaining a Minimum Vertex Ranking Spanning Tree in Outerplanar Graphs

The minimum vertex ranking spanning tree problem is to find a spanning tree of G whose vertex ranking is minimum. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of interval graphs. This paper proposes a polynomial time algorithm for solving the minimum vertex ranking spanning tree problem on outerplanar graphs.

Tree Loop Graphs

Abstract Many problems involving DNA can be modelled by interval families. However, traditional interval graphs do not take into account the repeat structures of a DNA molecule. We propose a new definition that respects repeats and define loop graphs as the intersection graphs of arcs of a loop. The class of loop graphs contains the class of interval graphs, and the class of circular-arc graphs. Every loop graph has interval number 2. We characterize the trees that are loop graphs. The characterization yields a polynomial-time algorithm which given a tree decides whether it is a loop graph and in the affirmative case, it produces a loop representation for the tree.

Distance labeling schemes for well-separated graph classes

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices can be inferred from inspecting their labels. It is shown in this paper that the classes of interval graphs and permutation graphs enjoy such a distance labeling scheme using O ( log 2 n ) bit labels on n-vertex graphs. Towards establishing these results, we present a general property for graphs, called well- ( α , g ) -separation, and show that graph classes satisfying this property have O ( g ( n ) log n ) bit labeling schemes. In particular, interval graphs are well- ( 2 , log n ) -separated and permutation graphs are well- ( 6 , log n ) -separated.

On the Sum Coloring Problem on Interval Graphs

In this paper we study the following NP-complete problem: given an interval graph G = (V,E) , find a node p -coloring $ \langle V_1, V_2, . . ., V_p\rangle $ such that the cost $ \xi (\langle V_1, V_2, . . ., V_p \rangle ) = \sum^p_{i=1} i|V_i| $ is minimal, where $ \langle V_1, V_2, . . ., V_p \rangle $ denotes a partition of V whose subsets are ordered by nonincreasing cardinality. We present an O(m χ (G) + n log n) time e -approximate algorithm (e < 2) to solve the problem, where n , m , and χ #(G) are the number of nodes of the interval graph, its number of cliques, and its chromatic number, respectively. The algorithm is shown to solve the problem exactly on some classes of interval graphs, namely, the proper and the containment interval graphs, and the intersection graphs of sets of ``short'' intervals. The problem of determining the minimum number of colors needed to achieve the minimum $\xi (\langle V_1, V_2, . . ., V_p \rangle )$ over all p -colorings of G is also addressed.

Chordal graphs, interval graphs, and wqo

Let precedes, equal to be the induced-minor relation. It is shown that, for every t, all chordal graphs of clique number at most t are well-quasi-ordered by precedes, equal to. On the other hand, if the bound on clique number is dropped, even the class of interval graphs is not well-quasi-ordered by precedes, equal to. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 105–114, 1998

Chordal graphs, interval graphs, and wqo

Let precedes, equal to be the induced-minor relation. It is shown that, for every t, all chordal graphs of clique number at most t are well-quasi-ordered by precedes, equal to. On the other hand, if the bound on clique number is dropped, even the class of interval graphs is not well-quasi-ordered by precedes, equal to. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 105–114, 1998

Disjoint paths in circular arc graphs

Arikati, Pandu Rangan, and Manacher (BIT, 31 (1991) 182-193) developed an O(m+n) algorithm to find two vertex disjoint paths in a circular arc graph on n vertices and m edges. We provide an improved solution to this problem: the algorithm presented here is both faster (O(n) time complexity) and simpler than the previous algorithm. The method involves reductions to interval graphs. In an interval graph, the critical notions are unordered paths (vertex disjoint paths from s1, s2 to t1, t2 in either order) and interchangeable paths (existence of both pairs of vertex disjoint paths). We also prove a theorem (which is best possible, in a sense), that guarantees existence of vertex disjoint paths, if arcs are sufficiently dense. Finally, we show that the more general problem of determining the existence of k vertex disjoint paths (from s1,.., sk to t1,..., tk) is NP-complete, where k is part of the input, even restricted co the class of interval graphs.

Algorithmic aspects of intersection graphs and representation hypergraphs

Let ℛ be a family of sets. The intersection graph of ℛ is obtained by representing each set in ℛ by a vertex and connecting two vertices by an edge if and only if their corresponding sets intersect. Of primary interest are those classes of intersection graphs of families of sets having some specific topological or other structure. The “grandfather” of all intersection graphs is the class of interval graphs, that is, the intersection graphs of intervals on a line.

Classes of interval graphs under expanding length restrictions

AbstractLet C(α) denote the finite interval graphs representable as intersection graphs of closed real intervals with lengths in [1, α]. The points of increase for C are the rational α ≥ 1. The set D(α) = [∩β&gt;αC(β)]\C(α) of graphs that appear as soon as we go past α is characterized up to isomorphism on the basis of finite sets E(α) of irreducible graphs for each rational α. With α = p/q and p and q relatively prime, ∣E(α)∣ is computed for all (p,q) with q ⩽ 2 and p = q + 1. When q = 1, E(p) contains only the bipartite star K1, p+2. A lowr bound on ∣E(α)∣ is given for all rational α.