scholarly journals Constraint and Satisfiability Reasoning for Graph Coloring

2020 ◽  
Vol 69 ◽  
pp. 33-65
Author(s):  
Emmanuel Hebrard ◽  
George Katsirelos
Keyword(s):  
Combinatorial Optimization ◽  
Lower Bound ◽  
Lower Bounds ◽  
Branch And Bound ◽  
Graph Coloring ◽  
Upper Bounds ◽  
Scheduling Problems ◽  
Marginal Costs ◽  
Free Tree ◽  
Pruning Techniques

Graph coloring is an important problem in combinatorial optimization and a major component of numerous allocation and scheduling problems. In this paper we introduce a hybrid CP/SAT approach to graph coloring based on the addition-contraction recurrence of Zykov. Decisions correspond to either adding an edge between two non-adjacent vertices or contracting these two vertices, hence enforcing inequality or equality, respectively. This scheme yields a symmetry-free tree and makes learnt clauses stronger by not committing to a particular color. We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; a branching heuristic emulating Br´elaz’ heuristic on the Zykov tree; and dedicated pruning techniques relying on marginal costs with respect to the bound and on reasoning about transitivity when unit propagating learnt clauses. The combination of these techniques in both a branch-and-bound and in a bottom-up search outperforms other SAT-based approaches and Dsatur on standard benchmarks both for finding upper bounds and for proving lower bounds.

scholarly journals Clause Learning and New Bounds for Graph Coloring

2019 ◽  
Author(s):  
Emmanuel Hebrard ◽  
George Katsirelos
Keyword(s):  
Lower Bound ◽  
Graph Coloring ◽  
Upper Bounds ◽  
Complete Graphs ◽  
Scheduling Problems ◽  
Original Graph ◽  
Branch And Bound Search ◽  
Clause Learning ◽  
Free Tree ◽  
Different Color

Graph coloring is a major component of numerous allocation and scheduling problems. We introduce a hybrid CP/SAT approach to graph coloring based on exploring Zykov’s tree: for two non-neighbors, either they take a different color and there might as well be an edge between them, or they take the same color and we might as well merge them. Branching on whether two neighbors get the same color yields a symmetry-free tree with complete graphs as leaves, which correspond to colorings of the original graph. We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; and a branching heuristic emulating Brelaz on the Zykov tree. The combination of these techniques in a branch- and-bound search outperforms Dsatur and other SAT-based approaches on standard benchmarks both for finding upper bounds and for proving lower bounds.

A LOWER BOUND FOR ORDER-PRESERVING BROADCAST IN THE POSTAL MODEL

1993 ◽  
Vol 03 (04) ◽  
pp. 313-320 ◽  
Author(s):  
PHILIP D. MACKENZIE
Keyword(s):  
Communication Network ◽  
Lower Bound ◽  
Lower Bounds ◽  
Message Passing ◽  
Upper Bounds ◽  
Communication Latency ◽  
Multiple Messages ◽  
Postal Model ◽  
Parameter Ranges

In the postal model of message passing systems, the actual communication network between processors is abstracted by a single communication latency factor, which measures the inverse ratio of the time it takes for a processor to send a message and the time that passes until the recipient receives the message. In this paper we examine the problem of broadcasting multiple messages in an order-preserving fashion in the postal model. We prove lower bounds for all parameter ranges and show that these lower bounds are within a factor of seven of the best upper bounds. In some cases, our lower bounds show significant asymptotic improvements over the previous best lower bounds.

The Heine-Borel theorem in extended basic logic

1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Fundamental Theorem ◽  
Upper Bounds ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

The Unrestricted Black-Box Complexity of Jump Functions

2016 ◽  
Vol 24 (4) ◽  
pp. 719-744 ◽  
Author(s):  
Maxim Buzdalov ◽  
Benjamin Doerr ◽  
Mikhail Kever
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Black Box ◽  
Problem Instance ◽  
Theory Approach ◽  
Function Classes ◽  
Fitness Value ◽  
Jump Function ◽  
Jump Sizes

We analyze the unrestricted black-box complexity of the Jump function classes for different jump sizes. For upper bounds, we present three algorithms for small, medium, and extreme jump sizes. We prove a matrix lower bound theorem which is capable of giving better lower bounds than the classic information theory approach. Using this theorem, we prove lower bounds that almost match the upper bounds. For the case of extreme jump functions, which apart from the optimum reveal only the middle fitness value(s), we use an additional lower bound argument to show that any black-box algorithm does not gain significant insight about the problem instance from the first [Formula: see text] fitness evaluations. This, together with our upper bound, shows that the black-box complexity of extreme jump functions is [Formula: see text].

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals A lower bound for general t-stack sortable permutations via pattern avoidance

2020 ◽  
Vol 22 ◽  
Author(s):  
Pranav Chinmay
Keyword(s):  
Lower Bound ◽  
Recurrence Relation ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Pattern Avoidance ◽  
Lower And Upper Bounds

There is no formula for general t-stack sortable permutations. Thus, we attempt to study them by establishing lower and upper bounds. Permutations that avoid certain pattern sets provide natural lower bounds. This paper presents a recurrence relation that counts the number of permutations that avoid the set (23451,24351,32451,34251,42351,43251). This establishes a lower bound on 3-stack sortable permutations. Additionally, the proof generalizes to provide lower bounds for all t-stack sortable permutations.

scholarly journals A new lower bound for the smallest singular value

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2711-2723
Author(s):  
Ksenija Doroslovacki ◽  
Ljiljana Cvetkovic ◽  
Ernest Sanca
Keyword(s):  
Lower Bound ◽  
Boundary Value Problems ◽  
Lower Bounds ◽  
Large Scale ◽  
Block Structure ◽  
Upper Bounds ◽  
Singular Value ◽  
Ecological Modelling ◽  
Infinity Norm ◽  
Smallest Singular Value

The aim of this paper is to obtain new lower bounds for the smallest singular value for some special subclasses of nonsingularH-matrices. This is done in two steps: first, unifying principle for deriving new upper bounds for the norm 1 of the inverse of an arbitrary nonsingular H-matrix is presented, and then, it is combined with some well-known upper bounds for the infinity norm of the inverse. The importance and efficiency of the results are illustrated by an example from ecological modelling, as well as on a type of large-scale matrices posessing a block structure, arising in boundary value problems.

Chapter 23. MaxSAT, Hard and Soft Constraints

2021 ◽  
Author(s):  
Chu Min Li ◽  
Felip Manyà
Keyword(s):  
Combinatorial Optimization ◽  
Lower Bounds ◽  
Branch And Bound ◽  
Data Structures ◽  
Optimization Problems ◽  
Soft Constraints ◽  
Inference Rules ◽  
Combinatorial Optimization Problems ◽  
Definition Of ◽  
Hard And Soft Constraints

MaxSAT solving is becoming a competitive generic approach for solving combinatorial optimization problems, partly due to the development of new solving techniques that have been recently incorporated into modern MaxSAT solvers, and to the challenge problems posed at the MaxSAT Evaluations. In this chapter we present the most relevant results on both approximate and exact MaxSAT solving, and survey in more detail the techniques that have proven to be useful in branch and bound MaxSAT and Weighted MaxSAT solvers. Among such techniques, we pay special attention to the definition of good quality lower bounds, powerful inference rules, clever variable selection heuristics and suitable data structures. Moreover, we discuss the advantages of dealing with hard and soft constraints in the Partial MaxSAT formalims, and present a summary of the MaxSAT Evaluations that have been organized so far as affiliated events of the International Conference on Theory and Applications of Satisfiability Testing.

A SIMPLE LOWER BOUND FOR TOTAL COMPLETION TIME MINIMIZATION IN A TWO-MACHINE FLOWSHOP

2005 ◽  
Vol 22 (03) ◽  
pp. 391-407 ◽  
Author(s):  
B. M. T. LIN ◽  
J. M. WU
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Branch And Bound ◽  
Completion Time ◽  
Computational Study ◽  
Total Completion Time ◽  
Optimal Solutions ◽  
Branch And Bound Algorithms ◽  
Time Minimization ◽  
Input Instance

The purpose of this study is to present a simple lower bound to facilitate the development of branch-and-bound algorithms for the minimization of total completion time in a two-machine flowshop. The studied problem is known to be strongly NP-hard. In the literature, several lower bounds have been proposed. The bounding technique addressed in this paper is based upon a concept about rearrangement of the parameters of the input instance. The technique is intrinsically simple for computer implementations. We conduct computational experiments for problems with 10–65 jobs. Numerical results from our computational study indicate that the new scheme is very effective in reducing the execution time needed for composing optimal solutions.

scholarly journals Online Graph Coloring Against a Randomized Adversary

2018 ◽  
Vol 29 (04) ◽  
pp. 551-569 ◽  
Author(s):  
Elisabet Burjons ◽  
Juraj Hromkovič ◽  
Rastislav Královič ◽  
Richard Královič ◽  
Xavier Muñoz ◽  
...  
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Online Algorithms ◽  
Graph Coloring ◽  
Competitive Ratio ◽  
Upper And Lower Bounds ◽  
Coloring Problem ◽  
Single Instance ◽  
Graph Coloring Problem

We consider an online model where an adversary constructs a set of [Formula: see text] instances [Formula: see text] instead of one single instance. The algorithm knows [Formula: see text] and the adversary will choose one instance from [Formula: see text] at random to present to the algorithm. We further focus on adversaries that construct sets of [Formula: see text]-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in [Formula: see text] and matching upper and lower bound are given for a specific set of algorithms that we call “minimalistic online algorithms”.

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