In recent years, research on the spectral extremal problems of fractional [a,b]-factors has attracted the attention of many scholars. The main content of our research is the generalization of fractional [a,b]-factors, namely fractional (a,b,l)-critical graph. Let a and b be two positive integers. A graph G is called a fractional (a,b,l)-critical graph if after deleting any l vertices of G the remaining graph of G has a fractional [a,b]-factor. In this paper, we present spectral radius and size conditions for a graph to be fractional (a,b,l)-critical.
The atom-bond sum-connectivity (ABS) index of a graph is a variant of several well-known chemical topological indices, such as the randi´c index, the sum-connectivity index and the atom-bond connectivity index. For a graph G = (V (G), E(G)), the ABS index of G is defined as ABS(G) = ∑uv∈E(G)√1− 2dG(u)+dG(v) , where dG(u) denotes the degree of the vertex u in G. A cactus is a connected graph in which each block is either an edge or a cycle. For two integers n ≥ 2 and k ≥ 0, let G(n, k) be the set of cacti of order n and with k cycles. Obviously, G(n, 0) is the set of all trees of order n and G(n, 1) is the set of all unicyclic graphs of order n. We will determine the maximum ABS index of graphs among G(n, k), and also characterize the corresponding extremal graphs.
The independence number a of a graph is the cardinality of a largest set of vertices that are pairwise independent. Calculating this number is NP-hard. Various attempts have been made to better understand the structure of a graph in the hopes of yielding, for instance, computational improvements (more efficient algorithms). In this paper we show that two of these, a classical linear programming approach and a more recent purely graph-theoretic approach, are in fact the same. We conclude with a new characterization of the well-studied class of K & odblac;nig-Egerv & aacute;ry graphs.
We consider online speed-robust scheduling on a multiprocessor consisting of m uniformly related machines. Jobs of arbitrary sizes arriving one by one are to be assigned to a fixed number of M=m bags immediately and without any information on machine speeds. When the sequence of jobs is completed, the list of speeds is provided to the algorithm, and it assigns its entire collection of bags to the machines. For each bag, all jobs assigned to this bag are assigned to the same machine. In the offline variant of the problem, the sequence of jobs is given as a set, but the speeds are also revealed only after the bags are created. The quality of algorithms is measured similarly to approximation algorithms and online algorithms, and we study two objectives which are makespan minimization and maximization of the minimum load.Even though we allow jobs to arrive one by one, we match the best known result for offline speed-robust makespan minimization. For most cases that we study, we prove results that are significantly better than those known for the corresponding standard online problems, where each new job is assigned to a machine irrevocably. However, we show that not all cases allow such an improved result. Some of our results are extended to the situation where one can use M>m bags.Finally, we consider the offline problem on two machines and three classes of jobs that were studied in previous work. We show that arbitrary job sizes differ from unit job sizes for two machines and two bags, even though the known result for unit job sizes is the same as that of infinitesimal jobs. Moreover, we show that the best possible results for the three job classes are all different for three bags. This is the first variant for which there is evidence that the case of jobs of arbitrary sizes is harder than that of unit sizes.
The football pool problem asks for the minimum number of bets on the result of n football matches ensuring that some bet correctly predicts the outcome of at least n-1 of them. This combinatorial problem has proven to be extremely difficult and is open for n >= 6. Integer programming techniques have been applied to this problem in the past but, in order to tackle the open cases, a deep knowledge of the polytopes associated with the integer programs modeling this problem may be required. In this work, we address this issue by defining and studying the football pool polytope in connection with a natural integer programming formulation of the football pool problem. We explore the basic properties of this polytope, present several classes of facet-inducing valid inequalities over natural combinatorial structures in the original problem, and introduce a lifting lemma for this polytope.
We consider the red-blue-yellow matching problem: given two natural numbers k_R, k_B and a graph G whose edges are colored red, blue or yellow, the goal is to find a matching of G that contains exactly k_R red edges and exactly k_B blue edges, and is of maximum cardinality subject to these constraints. This is a natural generalization of the well known red-blue matching problem, whose complexity status is unknown: although a randomized polynomial-time algorithm exists, a deterministic algorithm has remained elusive for nearly four decades. The best known deterministic approach to the red-blue matching problem, due to Yuster (2012), gives an additive approximation. In this paper, we show a similar result for the red-blue-yellow matching problem, giving a polynomial-time deterministic algorithm that, under natural assumptions, finds a matching satisfying the color requirements almost exactly and has cardinality within 3 of the optimal solution. Our algorithm is a mix of classic linear programming techniques and ad hoc existence results on restricted classes of graphs such as paths and cycles. As a key ingredient, we prove a curious topological property of plane curves, which is a strengthened version of a result by Grandoni and Zenklusen (2010) in the related context of budgeted matchings.
A graph G is H-saturated if G does not contain a copy of the graph H, but adding any missing edge e of G will create a copy of H in G + e. The saturation number of H, denoted by sat(n, H), is the minimal number of edges among all H-saturated graphs with n vertices. In general, it is difficult to determine the exact value of the saturation number. In this paper, we focus on the saturation number sat(n,S-t+1,S-t+1), where S-t+1,S-t+1 is called a balanced double star obtained by adding an edge between the centers of two stars St+1. We prove the new bounds tn/2 + & varepsilon; <= sat(n,S-t+1,S-t+1) <= tn/2 + (t + 1)/2 where & varepsilon; = 0 if n equivalent to 0 mod t +1 and & varepsilon; = 1 otherwise. We also establish the graphs achieving the new upper bound. Furthermore, we determine the exact values for sat(n, S-t+1,S-t+1)when t= 3,4 and n equivalent to 1 mod (t+ 1), where n >= (t+2) (t+3)(2)+6t-5/4
Let G be a graph with adjacency matrix A(G) and degree diagonal matrix D(G). In 2017, Nikiforov (2017) defined the matrix A alpha(G) = alpha D(G) + (1-alpha)A(G) for any real alpha E [0,1]. The largest eigenvalue of A(G) is called the spectral radius of G, while the largest eigenvalue of A alpha(G) is called the A alpha spectral radius of G. Let & ccedil;n,i be the set of graphs of order n with independence number i. Recently, for all graphs in & ccedil;n,i having the minimum or the maximum of A, Q and A alpha spectral radius where i E {1, 2, & LeftFloor; n2 & RightFloor; n2 + 1, n-3, n-2, n-1} there are some results given by Xu, Li and Sun et al., respectively. In 2022, Luo and Guo (2022) determined all graphs in & ccedil;n,n-4 having the minimum spectral radius. In this paper, we characterize the graphs in alpha n,n-4 having the minimum and the maximum A spectral radius for alpha E [ 12,1), respectively.
Let G be an undirected graph. An independent result of Maurras (Maurras, 1975), and Gr & ouml;tschel and Padberg (Gr & ouml;tschel and Padberg, 1979b) implies a characterization of the facets of the subtour polytope when G is complete. In this paper, we generalize this result to arbitrary graphs.
For a graph G, we denote by sigma(G) the number of independent sets, including the empty set, in G. A Halin graph is a plane graph which consists of a plane embedding of a tree T of order at least 4 without a vertex of degree 2 and a cycle C connecting all leaves of T. In this paper we characterize the maximum general Halin graphs and maximum cubic Halin graphs, respectively, of order n with respect to the number of independent sets. Moreover, the existence of asymptotic lower bounds is provided on the number of independent sets of general and cubic Halin graphs, respectively. Also two open problems are provided for future research.
The manufacturer’s pallet loading problem asks for a maximum-sized axis-parallel packing of identical rectangles within an enclosing axis-parallel rectangular pallet. This problem has been widely studied in the literature, and most instances coming from practical settings have been solved with optimality. Contrary to the situation in other classical combinatorial optimization problems, integer programming techniques are not at the core of the most successful exact procedures for this problem, which are based on combinatorial exhaustive search coupled with sophisticated heuristics. In this work we are interested in evaluating whether it is possible to extend the reach of integer programming techniques at solving this problem. To this end, we evaluate two different column generation procedures for this problem, one of them based on a clustering idea by Ribeiro and Lorena, and the other based on a simultaneous row and column generation procedure proposed by Feillet et al. We show that this second procedure is effective and allows to solve with optimality many instances that were, until now, open.
An isolating set in a graph G is a set S of vertices such that removing S and its neighborhood leaves no edge. The isolation number of G (also known as the vertex-edge domination number), denoted ι(G), is the minimum size among all isolating sets of G. We provide a technique for proving upper bounds on this parameter for graphs with a given minimum degree. For example, we show that if G has order n and minimum degree at least 4, then ι(G)≤13n/41, and if G is also triangle-free, then ι(G)≤3n/10.
This paper introduces a novel class of valid inequalities for the Time-Indexed Non-Preemptive Single Machine Scheduling Problem (T-SMSP). Under some assumptions on the length of the planning horizon, these inequalities are proven to define facets for the convex hull of a relaxation of T-SMSP that is tighter than the one considered in the literature. Furthermore, it is shown that the proposed set of valid inequalities are not dominated by certain existing valid inequalities for this problem and they either dominate or are equivalent to some of these existing valid inequalities. The computational performance of these inequalities, when integrated into a cutting-plane algorithm, shows significant promise.
Given a connected graph G, a set of vertices X C V(G) is a weak k-resolving set of G if for each two vertices y,z is an element of V(G), the sum of the values |dG(y, x)-dG(z, x)| over all x is an element of X is at least k, where dG(u, v) stands for the length of a shortest path between u and v. The cardinality of a smallest weak k-resolving set of G is the weak k-metric dimension of G, and is denoted by wdimk(G). In this paper, wdimk(Kn square Kn) is determined for every n >= 3 and every 2 <= k <= 2n. An improvement of a known integer linear programming formulation for this problem is developed and implemented for the graphs Kn square Km. Conjectures regarding these general situations are posed.
We explore the inverse of integer programs (IPs) by studying the inverse of their Gomory corner relaxations (GCRs). We propose a linear programming (LP) formulation for solving any inverse GCR problem under the L1 and L infinity norms by reformulating the inverse GCR problem as the inverse of a shortest path problem. We show that the minimum objective of the inverse GCR across all feasible bases of the LP relaxation yields an upper bound on the optimal value of the inverse IP that is at least as tight as the optimal value of the inverse of the LP relaxation. We provide conditions under which this upper bound is exactly equal to the optimal value of the inverse IP.
We study the settings where we are given a function of n variables defined in a given box of integers. We show that in many cases we can replace the given objective function by a new function with a much smaller domain (though still exponential in n). Our approach allows us to transform a family of weakly polynomial upper bounds on analysis measures of algorithms into strongly polynomial ones. To prove our results we use standard tools of linear programming (LP) of a simple LP formulation of the problem of finding an equivalent function of minimum domain size. Our method allows us to derive specific improved bounds for classes of functions and we demonstrate it here on four natural classes of functions. Our bounds are constructive and can be applied both for cases where the goal is to define the equivalent function explicitly or if the goal is to obtain a bound on the domain of a function with equal comparison oracle (as the original function). We note that for the cases considered also in earlier works, we provide here improved upper bounds on the domain of the resulting equivalent function.
For a finite, simple, and undirected graph G with n vertices and average degree d, Nikiforov introduced the degree deviation of G as s = & sum;(u is an element of V(G))||d(G)(u-d|. Provided that G has largest eigenvalue lambda, minimum degree atleast delta, and maximum degree at most Delta, where 0 <= delta < d ={d(2)n/root d(2)n(2)-s(2), if s <= dn/root 2, 2s/n, if s <= > dn/root 2 Our results are based on a smoothing technique relating the degree deviation and the largest eigen value to low-dimensional non-linear optimization problems.
The length polyhedron QP of an interval order P is the convex hull of integral vectors representing the interval lengths in interval representations of P. This polyhedron has been studied by various authors, including Fishburn and Isaak. Notably, QP forms a pointed affine cone, a property inherited from being a projection of the representation polyhedron, a structure explored also by Doignon and Pauwels. The apex of the length polyhedron corresponds to Greenough’s minimal endpoint representation, which is, in fact, the length vector of the canonical interval representation—an interval representation that minimizes the sum of the interval lengths.Building on a combinatorial perspective of canonical representations, we refine Isaak’s graph-theoretical model by introducing a new and simpler directed graph. From directed cycles of this key digraph, we extract a linear system of inequalities that precisely characterizes the length polyhedron QP. This combinatorial approach also reveals the unique Hilbert basis of the polyhedron, which consists of binary rays. We prove that the intersection graph of the sets corresponding to these binary rays are Berge graphs; therefore they are perfect graphs. As a result, for interval orders with bounded width, the length polyhedron has a polynomial-sized Hilbert basis, which can be computed in polynomial time. We also provide an example of a family of interval orders with a Hilbert basis of exponential size. In a companion paper we determine the Schrijver system for the length polyhedron. We conclude with open problems.
In this article, we introduce the parametric matroid 8-interdiction problem, where 8 E N is a fixed number of elements allowed to be interdicted. Each element of the matroid's ground set is assigned a weight that depends linearly on a real parameter from a given interval. The goal is to compute, for each possible parameter value, a set of 8-most vital elements with corresponding objective value the deletion of which causes a maximum increase of the weight of a minimal basis. We show that such a set, which of course depends on the parameter, can only change polynomially often if the parameter varies. We develop several exact algorithms to solve the problem that have polynomial running times if an independence test can be performed in polynomial time.
We consider the coordinated vehicle platooning problem on a tree network with time constraints while the routes of vehicles are given. The problem is to coordinate the departure time of each vehicle to enable platoon formation hence maximizing the total fuel saving. For this problem setting, relative time windows can be defined for all vehicles to which an efficient time discretization can be applied. This property leads to a tight mixed-integer linear program reformulation as compared to the continuous-time formulation involving big-M coefficients proposed in our previous work. It is demonstrated by systematic numerical experiments that the reformulation outperforms the continuous-time formulation for this family of problem instances. Our study hence extends upon the results of Boysen et al. (2018) from the case of a single-path network to a tree network.
