# Controlling the Difficulty of Combinatorial Optimization Problems for Fair Proof-of-Useful-Work-Based Blockchain Consensus Protocol

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## Abstract

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## 1. Introduction

## 2. Related Work

#### 2.1. Controlling the Difficulty in PoW and PoUW Protocols from the Literature

#### 2.1.1. PoW Difficulty Adjustment

#### 2.1.2. PoUW-Based Protocols That Utilize Difficulty Adjustment

#### 2.1.3. PoUW-Based Protocols Considering AI, ML, and DL

#### 2.1.4. Controlling the Difficulty in PoUW

#### 2.2. Difficulty Estimation for CO Problem Instances

## 3. Controlling the Difficulty—Methodology

#### 3.1. Difficulty Estimation Module

#### 3.2. Instance Grouping Module

**if**statement in the Grouping function identifies the instances that have already passed the difficulty estimation module. Specifically, it may happen that the miner did not manage to be the first one to solve all instances from the group and publish a new block. In such a case, all solved instances are transferred to the solution pool and the best found solution for each of them is provided to the client. The unsolved instances are returned to the instances grouping module to be inserted in some new group. For these instances, difficulty estimation is already performed, and it is known that they should be a part of some group. This is because too difficult instances have already been discarded and the ones with ${t}_{i}\approx f$ are sent individually to the instance pool on the first entrance to the Grouping function. Therefore, if for instance i, ${t}_{i}>0$ holds, it is directly passed to the Insert function.

## 4. Case Study

#### 4.1. Difficulty Estimation Module

#### 4.1.1. Experimental Setup

#### 4.1.2. ArcFlow Case Study

#### 4.1.3. GIST Case Study

#### 4.2. Instances Grouping Module Case Study

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notation

Abbreviations | |

ANN | Artificial Neural Network |

AUC | Area Under the Curve |

BC | Blockchain |

BPP | Bin Packing Problem |

BIT | Block Insertion Time |

CNN | Convolutional Neural Network |

CO | Combinatorial Optimization |

COCP | Combinatorial Optimization Consensus Protocol |

CPU | Central processing unit |

CRISP-DM | CRoss Industry Standard Process for Data Mining |

DL | Deep Learning |

DPLS | Doubly Parallel Local Search |

GIST | Greedy Iterative Stochastic Transformation |

JSSP | Job Shop Scheduling Problem |

LSTM | Long Short Term Memory |

MFP | Maximum Flow Problem |

MIP | Mixed Integer Programming |

ML | Machine Learning |

NSGA-II | Non-dominated Sorting Genetic Algorithm II |

$P\left|\right|{C}_{max}$ | Problem of scheduling independent tasks on identical parallel machines |

PCA | Principal Component Analysis |

DLBC | Deep Learning-Based Consensus Protocol |

PoS | Proof-of-Search |

PoUW | Proof-of-Useful-Work |

PoW | Proof-of-Work |

RMSE | Root Mean Squared Error |

RNN | Recurrent Neural Networks |

ROC | Receiver Operating Characteristics |

R-squared | Coefficient of Determination |

SAT | Propositional Satisfiability |

SNN | Shallow Neural Network |

SPP | Strip Packing Problem |

SSP | Subset Sum Problem |

TSP | Traveling Salesman Problem |

Symbols | |

∅ | Empty set |

$\Delta $ | Package size tolerance parameter |

$softmax$ | Type of activation function |

$ReLU$ | Type of activation function |

$sigmoid$ | Type of activation function |

${k}_{TSP}$ | Number of chosen coordinates in TSP instance |

${n}_{TSP}$ | Number of coordinates in TSP instance |

${N}_{TSP}$ | Number of coordinates in hard TSP instance |

${T}_{1}$ | Threshold for good enough solution for TSP instance |

i | Submitted instance |

${t}_{i}$ | Instance score (difficulty) |

${t}_{c}$ | Current time |

f | BIT value |

$\rho $ | Number of insertion cycles an instance is expected to wait in the pool before it is solved |

${G}_{l}$ | Group of instances |

G | Set of groups of instances |

L | Number of groups |

${d}_{i}$ | Deadline for execution of instance |

H | Heap of instances not included in any group yet |

j | Instance from heap |

M | Set of mandatory instances |

${G}_{M}$ | Group formed of mandatory instances |

$SSP$ | Subset sum problem solver function |

n | Number of tasks in $P\left|\right|{C}_{max}$ |

m | Number of processors in $P\left|\right|{C}_{max}$ |

P | Set of tasks in $P\left|\right|{C}_{max}$ |

${p}_{q}$ | Processing time of task q in $P\left|\right|{C}_{max}$ |

y | Time needed for $P\left|\right|{C}_{max}$ problem execution (target variable) |

${I}_{w}$ | w-th example of instance for $P\left|\right|{C}_{max}$ |

Blockchain and machine learning jargon | |

Miners | BC participants who have the computer hardware and appropriate software needed to mine digital currencies or solve complex mathematical problems |

Consensus Protocol | Mechanism to perform BC management without the central authority |

Transaction | A unit measure of data in BC |

Block | Blocks are the basic containers of information in a blockchain, they contain transaction as stored data |

Cryptographic Puzzle | A mathematical puzzle that miners must solve in PoW-based BCs in order to append their blocks |

Proof-of-Work (PoW) | A common mechanism used to validate peer-to-peer transactions and maintain highly secured immutability of the blockchain |

Proof-of-Useful-Work (PoUW) | Energy efficient consensus protocol that re-purposes the computational effort required to maintain protocol security to solve complex real-world problems |

Proof-of-Learning (PoLe) | PoW that exploit the computation power of miners for training ML models as a useful work in consensus protocol |

Proof-of-Search (PoS) | Combines BC consensus formation with solving optimization problems |

Instance | An example of a problem with all the inputs needed to compute a solution to the problem |

Feature | Individual measurable property or characteristic of an instance |

Solution Space | The set of all possible solutions for the combinatorial optimization problem |

Randomized Algorithm | An algorithm that employs a degree of randomness as a part of its logic or procedure |

Parameter | The configuration variable that is internal to the model and whose value can be estimated from the given data |

Hyperparameter | The explicitly specified parameter that controls the training process |

Training (Train) Dataset | A dataset of examples used during the learning process to fit the parameters |

Validation Dataset | A dataset of examples used to tune the hyperparameters |

Test Dataset | A dataset that is independent of the training dataset, but follows the same probability distribution |

Data Normalization | The organization of data to appear similar across all records and fields |

Data Standardization | The process of converting data to a common format suitable for analysis by users |

Data Preprocessing | Data mining technique used to transform the raw data in a useful and efficient format |

Bias | Describes how well a model matches the training set |

Confusion Matrix | A table that is used to define the performance of a classification algorithm |

Transfer Learning | Taking the relevant parts of a pre-trained ML model and applying it to a new problem |

Depth | Number of layers in ML model |

Group | Group of instances that the miner should solve before publishing a new block |

Heap | All instances that are not included in any group yet |

Package | Successfully created group |

Model | A decision process in an abstract manner |

Framework | A tool that provides ready-made components or solutions that are customized in order to speed up development |

Cloud Service | Refers to a wide range of services delivered on demand to companies and customers over the internet |

Relative Bound | Difference between upper and lower bound of a solution relative to the upper bound value |

Scaling | A technique to standardize the independent features present in the data in a fixed range |

Oversampling | A technique that creates synthetic samples by randomly sampling the characteristics from occurrences in the minority class |

Neuron | A connection point in an ANN |

Activation Function | Decides whether a neuron should be activated or not |

Layer | In a DL model a structure or network topology in the model’s architecture |

Hidden Layer | A layer in between input layer and output layer |

Dropout | A regularization method that approximates training a large number of ANNs with different architectures in parallel |

Output Layer | The last layer of neurons that produces given outputs for the ANN |

Input Layer | Brings the initial data into the ANN for further processing by subsequent layers of artificial neurons |

Regression | A technique for investigating the relationship between independent variables or features and a dependent variable or outcome |

Classification | A supervised learning concept which basically categorizes a set of data into classes |

Pool | Place to store unprocessed objects |

Hierarchical Model | A model in which lower levels are sorted under a hierarchy of successively higher-level units |

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**Figure 9.**ANN regression predictions (without classifier applied beforehand) for instances with execution time in range [0, 10] s on the left and [10, 1000] s on the right.

**Figure 10.**ANN regression predictions (with classifier applied beforehand)—for instances from test set with execution time in [0, 10] s.

**Figure 11.**ANN regression predictions (with classifier applied beforehand)—for instances from test set with execution time in [10, 1000] s.

**Figure 12.**ANN regression predictions (with classifier applied beforehand)—for instances from the test set with execution time in [1000, 4510] s.

**Figure 15.**The first generated package and the heap content after analyzing four CO problem instances.

Feature Notation | Explanation of Feature |
---|---|

n | Cardinality of set $P,$ that is, number of tasks |

m | Number of processors |

av.length | Average length of elements in set P |

median | Median value of elements in set P |

std.dev | Standard deviation of set P |

max | Maximum value in set P |

min | Minimum value in set P |

range | Difference between maximum and minimum value in set P |

k | Number of different values of elements in set P |

$rel.bound$ | Difference between a solution upper and lower bound relative to the upper bound |

$n/m$ | Average number of tasks per processor |

${(n/m)}^{2}$ | Polynomial feature of $n/m$ |

${(n/m)}^{3}$ | Polynomial feature of $n/m$ |

$m/n$ | Polynomial feature of $n/m$ |

${(m/n)}^{2}$ | Polynomial feature of $n/m$ |

${(m/n)}^{3}$ | Polynomial feature of $n/m$ |

$subtype$ | Instances with same subtype have the same $n/m$ value |

$class$ | Probability distribution of random generator for elements of P |

$index$ | Index of instance in dataset with same subtype and class [1–10] |

Feature | Correlation with Target y |
---|---|

y | 1.000000 |

std.dev | 0.539327 |

max | 0.538683 |

av.length | 0.538287 |

median | 0.536837 |

k | 0.529941 |

range | 0.528752 |

min | 0.495868 |

$n/m$ | 0.423243 |

${(n/m)}^{2}$ | 0.420624 |

${(n/m)}^{3}$ | 0.412105 |

subtype | 0.386236 |

$m/n$ | 0.377794 |

$class$ | 0.364556 |

n | 0.356450 |

${(m/n)}^{2}$ | 0.337697 |

${(m/n)}^{3}$ | 0.300864 |

$rel.bound$ | 0.246362 |

m | 0.100150 |

$index$ | 0.005231 |

Feature | Correlation with Target y |
---|---|

y | 1.000000 |

$rel.bound$ | 0.476547 |

m | 0.442684 |

$subtype$ | 0.430256 |

$m/n$ | 0.418171 |

${(m/n)}^{2}$ | 0.411345 |

${(m/n)}^{3}$ | 0.382118 |

$n/m$ | 0.340878 |

av.length | 0.306203 |

median | 0.305065 |

min | 0.302683 |

max | 0.297263 |

${(n/m)}^{2}$ | 0.289090 |

range | 0.283401 |

std.dev | 0.253432 |

${(n/m)}^{3}$ | 0.250783 |

k | 0.215968 |

$class$ | 0.211151 |

n | 0.160885 |

$index$ | 0.007393 |

i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

${t}_{i}$ | 0.000013 | 0.0001 | 0.000105 | 0.000231 | 0.000471 | 0.000607 | 0.001301 | 0.275697 | 1.420326 | 4.651514 |

$o\left(i\right)$ | 10 | 20 | 6 | 19 | 17 | 3 | 13 | 7 | 12 | 11 |

i | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

${t}_{i}$ | 4.905763 | 4.928929 | 5.492491 | 5.833624 | 6.058383 | 6.132013 | 9.387262 | 11.366228 | 15.392948 | 19.907111 |

$o\left(i\right)$ | 8 | 4 | 9 | 14 | 2 | 18 | 15 | 1 | 5 | 16 |

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**MDPI and ACS Style**

Maleš, U.; Ramljak, D.; Jakšić Krüger, T.; Davidović, T.; Ostojić, D.; Haridas, A.
Controlling the Difficulty of Combinatorial Optimization Problems for Fair Proof-of-Useful-Work-Based Blockchain Consensus Protocol. *Symmetry* **2023**, *15*, 140.
https://doi.org/10.3390/sym15010140

**AMA Style**

Maleš U, Ramljak D, Jakšić Krüger T, Davidović T, Ostojić D, Haridas A.
Controlling the Difficulty of Combinatorial Optimization Problems for Fair Proof-of-Useful-Work-Based Blockchain Consensus Protocol. *Symmetry*. 2023; 15(1):140.
https://doi.org/10.3390/sym15010140

**Chicago/Turabian Style**

Maleš, Uroš, Dušan Ramljak, Tatjana Jakšić Krüger, Tatjana Davidović, Dragutin Ostojić, and Abhay Haridas.
2023. "Controlling the Difficulty of Combinatorial Optimization Problems for Fair Proof-of-Useful-Work-Based Blockchain Consensus Protocol" *Symmetry* 15, no. 1: 140.
https://doi.org/10.3390/sym15010140