Minimum sum coloring problem: Upper bounds for the chromatic strength

We consider an undirected graph G?=?(V, E), the minimum sum coloring problem (MSCP) asks to find a valid vertex coloring of G, using natural numbers (1,2,...), the aim is to minimize the total sum of colors. In this paper we are interested in the elaboration of an approximate solution for the minimum sum coloring problem (MSCP), more exactly we try to give a lower bound for MSCP by looking for a decomposition of the graph based on the metaheuristic of ant colony optimization (ACO). We test different instances to validate our approach.

Given an undirected graph G = (V,E), the minimum sum coloring problem (MSCP) is to find a legal assignment of colors (represented by natural numbers) to each vertex of G such that the total sum of the colors assigned to the vertices is minimized. In this paper, we present Breakout Local Search (BLS) for MSCP which combines some essential features of several well-established metaheuristics. BLS explores the search space by a joint use of local search and adaptive perturbation strategies. Tested on 27 commonly used benchmark instances, our algorithm shows competitive performance with respect to recently proposed heuristics and is able to find new record-breaking results for 4 instances.Keywordsminimum sum coloringadaptive perturbation strategyheuristiccombinatorial optimization

Evolutionary algorithms are very popular in solving combinatorial optimization problems. Their efficiency is basically related to the appropriate choice of genetic operators, especially the crossover. The performance of this operator depends on the problem definition, the instance structure and the fitness function. The problem of interest in this work is the minimum sum coloring problem (MSCP). In this paper, several genetic operators are studied by varying the instances and the performance measures. Results provide a relevant idea about the effectiveness of the tested operators and show the well suited for the MSCP among them.

On the chromatic index of multigraphs and a conjecture of Seymour (I)

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

Tight bounds on the minimum mean square error for the additive Gaussian noise channel are derived, when the input distribution is constrained to be ε-close to a Gaussian reference distribution in terms of the Kullback-Leibler divergence. The distributions that attain the bounds are shown be Gaussian whose means are identical to that of the reference distribution and whose covariance matrices are defined implicitly via systems of matrix equations. The estimator that attains the upper bound is identified as a minimax optimal estimator that is robust against deviations from the assumed prior. The lower bound is shown to provide a potentially tighter alternative to the Cramér-Rao bound. Both properties are illustrated with numerical examples.

This repository provides data for the problem and scripts for the models. The main folders are 'data', 'scripts', and 'results'. 'data': This folder includes minimum sum graph coloring data sets. 'src': Our metaheuristic solver Q-Card can be found in this folder. 'scripts': This folder provides a running script. 'results': This folder provides the results.

The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $G$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph $G$, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending on who starts. These invariants could be infinite, thus we start by giving sufficient conditions to have infinite game distinguishing numbers. We also show that for graphs with cyclic automorphisms group of prime odd order, both game invariants are finite. After that, we define a class of graphs, the involutive graphs, for which the game distinguishing number can be quadratically bounded above by the classical distinguishing number. The definition of this class is closely related to imprimitive actions whose blocks have size $2$. Then, we apply results on involutive graphs to compute the exact value of these invariants for hypercubes and even cycles. Finally, we study odd cycles, for which we are able to compute the exact value when their order is not prime. In the prime order case, we give an upper bound of $3$.

Rainbow connection number of a connected graph G is the minimum number of colors needed to color the edges of G, so that every pair of vertices is connected by at least one path whose edges have distinct colors. In this paper, we propose an upper bound to the size of the rainbow connection number in Recursive Transpose-Connected 4-Cycles (RTCC) pyramids.KeywordsGraph theoryrainbow connection numberrainbow coloringRTCC pyramids

On the complexity of rainbow coloring problems

List strong edge-coloring of graphs with maximum degree 4

This paper proposes an upper bound for mine productivity (useful for long-term planning) and also a simple truck dispatch rule (useful for short-term operations) that demonstrates how tight the upper bound can be using a simulation. It also proposes a greedy search to approximate the productivity upper bound, which is faster and often exact. Uncertainty is added to the simulation in order to verify how the productivity responds to it. Typically, the productivity’s upper bound is less tight close to its saturation point as a function of the number of trucks, where adding more trucks only increases queues. Furthermore, more uncertainty in the model typically leads to a less tight upper bound. The results conducted using real data from an open pit mine in Brazil show that the gap between the productivity upper bound and the productivity realization using the proposed dispatch rule for a homogeneous fleet can be less than 2%, but it can be as large as 12% near the productivity saturation point without uncertainty. Even though this gap seems to become arbitrarily small as the number of trucks and the simulation horizon increase, the productivity upper bound is never violated, which validates it as an upper bound and induces optimality for the dispatch rule.

In this letter, we obtain lower and upper bounds on the minimum distance $d_{\min }$ of low-density parity-check (LDPC) codes. The bounds are derived by categorizing the non-zero code words of an LDPC code into two categories of elementary and non-elementary. The first category contains code words whose induced subgraph has only degree-2 check nodes. We propose an efficient search algorithm that can find the elementary code words of an LDPC code with weight less than a certain value $a_{\max }$ , exhaustively. We also derive a lower bound $L_{ne}$ on the weight of non-elementary code words. By performing the search with $a_{\max } = L_{ne}$ , we either obtain an elementary code word with the smallest weight $d_{\min }$ , or establish the lower bound of $L_{ne}$ on $d_{\min }$ . For the upper bound, we modify our search algorithm to reach elementary codewords of larger weights at the cost of being non-exhaustive. Once such a codeword is found, its weight acts as an upper bound on $d_{\min }$ . We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, $d_{\min }$ . Finding $d_{\min }$ , or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm.

In Zheng et al. (J Comb Optim 30(2):360–369, 2015) modelled a surgery problem by the one-dimensional bin packing, and developed a semi-online algorithm to give an efficient feasible solution. In their algorithm they used a buffer to temporarily store items, having a possibility to lookahead in the list. Because of the considered practical problem they investigated the 2-parametric case, when the size of the items is at most 1 / 2. Using an NF-based online algorithm the authors proved an ACR of $$13/9 = 1.44\ldots $$ for any given buffer size not less than 1. They also gave a lower bound of $$4/3=1.33\ldots $$ for the bounded-space algorithms that use NF-based rules. Later, in Zhang et al. (J Comb Optim 33(2):530–542, 2017) an algorithm was given with an ACR of 1.4243, and the authors improved the lower bound to 1.4230. In this paper we present a tight lower bound of $$h_\infty (r)$$ for the r-parametric problem when the buffer capacity is 3. Since $$h_\infty (2) = 1.42312\ldots $$ , our result—as a special case—gives a tight bound for the algorithm-class given in 2017. To prove that the lower bound is tight, we present an NF-based online algorithm that considers the r-parametric problem, and uses a buffer with capacity of 3. We prove that this algorithm has an ACR that is equal to the lower bounds for arbitrary r.

We study the problem of determining the size of the smallest tile set in which a given target shape uniquely self-assembles in Winfree’s abstract Tile assembly Model (aTAM), an elegant combinatorial model of DNA tile self-assembly. This problem is also known as the directed tile complexity problem. We work in a variant of the aTAM, restricted to having the minimum binding strength threshold (temperature) set to 1 but mildly generalized to allow self-assembly to take place in a just-barely 3D setting, where unit cubes are allowed to be placed in the $$z=0$$ and $$z=1$$ planes. Furcy, Summers and Withers recently proved lower and upper bounds on the directed tile complexity of a just-barely 3D $$k \times N$$ rectangle at temperature-1 of $$\varOmega \left( N^{\frac{1}{k}}\right) $$ and $$O\left( N^{\frac{1}{k-1}}+\log N\right) $$ , respectively. However, their upper bound does not hold for $$k=2$$ . We close this gap for $$k=2$$ by proving an asymptotically tight bound of $$\varTheta (N)$$ on the directed tile complexity of a just-barely 3D $$2 \times N$$ rectangle at temperature-1. The proof of our lower bound is based on an algorithm that uses a novel projection of a given just-barely 3D assembly onto an equivalent, 2D assembly.