Overview of Submodular Optimization and Examples of Applications and Implementations

2023.06.22
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Submodular Optimization Overview

Submodular Optimization, a type of combinatorial optimization, is a technique for solving the problem of maximizing or minimizing a submodular function, a function with certain properties. A submodular function has the following properties for a subset of a set

  • Diminishing Returns: Each time an element is added to the set, the function increases by a decreasing amount. That is, the addition of a new element is not as effective as the previous addition.
  • Monotonicity: For a set A and its subset B, adding an element of A to B does not decrease the increase in the value of the function.

Submodular optimization has been used in a variety of applications, including data subset selection, feature selection, sensor placement, ad placement, and social network analysis.

The general problem of submodular optimization becomes a maximization problem as follows.

maximize f(S)
subject to S ⊆ V, |S| ≤ k

where f is the submodular function, V is the set of originals, S is the subset to be maximized, and k is the constraint condition, the upper bound on the number of elements.

Although submodular optimization is a type of combinatorial optimization and an NP-hard problem, there may be efficient algorithms by utilizing the properties of the submodular function, and appropriate submodular optimization methods can be selected to solve the problem according to the field of application.

Next, we discuss the algorithms used for columnar submodular functions.

On the algorithm used for submodular optimization

Various algorithms have been proposed to solve the submodular optimization problem. They are described below.

  • Greedy Algorithm: The greedy algorithm is a method to find an efficient solution by exploiting the properties of the submodular function. The basic greedy algorithm uses the “effective addition” property to construct a solution by selecting the most effective element at each step, one by one. Due to the nature of the submodular function, the solution obtained by the greedy algorithm is known to perform well as an approximate solution.
  • Laplacian Relaxation: Laplacian Relaxation is a method that extends the recessive modular function to the continuous domain and solves it as a convex optimization problem. Specifically, the Laplace matrix is used to approximate the underdetermined function, and optimization is performed in the continuous domain. Laplace relaxor is an effective method for solving problems, especially applied to semantic segmentation and clustering problems such as the graph cut problem.
  • Cardinality-Constrained Optimization: In submodular optimization, it is common to restrict the subset size (cardinality) as a constraint, and for problems with such cardinality constraints, constrained optimization algorithms are used. These algorithms search for the optimal solution that maximizes (or minimizes) the submodular function while satisfying the constraints. Specific algorithms include Branch and Bound, Dynamic Programming, and Integer Linear Programming.
  • Approximation Algorithms: Submodular optimization problems are generally NP-hard and exact solutions are not always efficiently obtained. Therefore, approximation algorithms are widely used. Approximation Algorithms are methods to efficiently find approximate solutions instead of exact solutions. Typical approximation algorithms include Greedy Algorithms, Random Sampling, Linear Relaxation, etc. The implementation of these algorithms requires the use of the following

The following libraries and platforms are commonly used to implement these algorithms.

About Libraries and Platforms Available for Submodular Optimization

There are several libraries and platforms available to implement submodular optimization. The following is a list of some of the most popular ones.

  • pyQUBO: pyQUBO is a Python library for solving submodular optimization problems. It defines a submodular function, models the problem as a qubit variable, and then uses techniques such as quantum annealing to find the optimal solution. pyQUBO is a powerful tool for solving submodular optimization problems based on quantum algorithms.
  • submodlib: submodlib is a Python library for submodular optimization. With support for defining submodular functions, running optimization algorithms, and analyzing problems based on the properties of submodular functions, submodlib is a powerful tool that can be used for a wide range of submodular optimization applications.
  • CVXPY: CVXPY is a library for solving convex optimization problems in Python. CVXPY can handle a variety of constraints and objective functions and is applicable to submodular optimization problems as well.
  • OpenAI Gym: OpenAI Gym is a platform that provides an environment for reinforcement learning. Submodular optimization problems may be treated as part of reinforcement learning, and OpenAI Gym can be used to integrate and solve submodular optimization problems into a reinforcement learning framework.
Application examples of the submodular problem

The following are some examples of applications of the submodular problem.

  • Selection problem: This problem can be applied to the selection of elements that maximize the undermodular function in order to obtain the maximum effect or benefit under budget and resource constraints. Specific applications include the selection of advertising slots in an advertising campaign and the placement problem in a sensor network.
  • Clustering: Inferior Modular Optimization can be used for the problem of selecting a characteristic subset of a dataset in order to effectively cluster it. Examples include clustering sensor networks and selecting representative sentences for summary generation.
  • Information gathering: Submodular optimization can be applied to the problem of selecting the most informative information when gathering information with limited resources. Examples include information gathering for sensor networks and optimization of the display order of online advertisements.
  • Minimum coverage problem: The problem of minimizing a submodular function can be applied to the minimum coverage problem. Examples include the set coverage problem and the construction of minimum spanning trees.

To implement the submodular problem for these tasks, the following procedure is used.

Implementation Procedure for the Submodular Problem
  1. Mathematical formulation of the problem:
    • Definition of the objective function: In a submodular problem, it is usually expressed as a set function. In addition, the goal of the task is chosen as either minimization or maximization.
    • Definition of constraints: If specific constraints are given, they are defined.
  2. Selection of the optimization algorithm:
    • Considering the characteristics of the submodular problem, select an appropriate optimization algorithm among the algorithms as described above. Typically, a Greedy algorithm or an approximation algorithm is used to optimize the submodular function.
  3. Algorithm Implementation:
    • Implement the selected optimization algorithm in the program. Depending on the programming language and environment, libraries and platforms such as those mentioned above are used to select the appropriate data structure and algorithm implementation method.
  4. Input problem instances:
    • Input problem instances to the optimization algorithm. The problem instance is concrete data that embodies the objective function and constraints.
  5. Find the optimal solution:
    • The implemented algorithm is executed to find the optimal solution. Some algorithms may require iterative steps to find the optimal solution.
  6. Evaluate the results: Evaluate the obtained optimal solution.
    • The obtained optimal solution is evaluated to check if it satisfies the constraints of the problem. The quality of the optimization can also be evaluated by checking the value of the objective function.

In the following, we describe the implementation in a specific task following these procedures.

Example implementation in python of the sensor network placement problem

Below is a simple example implementation of optimizing a sensor network placement problem in Python using a submodular function. This example uses the greedy sensor placement algorithm.

import numpy as np

def distance(p1, p2):
    # 2つの点の距離を計算する関数
    return np.sqrt((p1[0] - p2[0])**2 + (p1[1] - p2[1])**2)

def uncovered_area(sensors, area):
    # Function to calculate the area of unmonitored area in an area
    total_area = area[0] * area[1]
    covered_area = 0
    for x in range(area[0]):
        for y in range(area[1]):
            covered = False
            for sensor in sensors:
                if distance((x, y), sensor) <= sensor_range: covered = True break if not covered: covered_area += 1 return total_area - covered_area def greedy_sensor_placement(area, sensor_count, sensor_range): sensors = [] for i in range(sensor_count): best_sensor = None best_area = -np.inf for x in range(area[0]): for y in range(area[1]): if (x, y) not in sensors: sensors.append((x, y)) new_area = uncovered_area(sensors, area) if new_area > best_area:
                        best_area = new_area
                        best_sensor = (x, y)
                    sensors.remove((x, y))
        if best_sensor is not None:
            sensors.append(best_sensor)
    return sensors

# Parameter Setting
area_size = (10, 10)  # Area Size
sensor_count = 3  # Number of Sensors
sensor_range = 2  # Sensor communication range
# Seek optimal sensor network placement
sensors = greedy_sensor_placement(area_size, sensor_count, sensor_range)

# Display Results
print("Optimal sensor network placement:")
for sensor in sensors:
    print(sensor)

In this implementation, the distance between two points is calculated with the distance function, the area of unobserved area in the area is calculated with the uncovered_area function, and the optimal sensor network placement is obtained with the greedy_sensor_placement function. The optimal placement results are stored in a list called sensors and displayed at the end.

Example implementation in python of the ad space selection problem in an ad campaign

The ad space selection problem in an ad campaign is the problem of achieving a certain objective (e.g., maximizing the click rate or revenue) by selecting a specific ad space. Below is an example of a Python implementation that optimizes the ad space selection problem using a submodular function.

import numpy as np

def click_rate(selected_slots):
    # Function to calculate the click rate for a selected ad space
    total_clicks = sum(slot["clicks"] for slot in selected_slots)
    total_views = sum(slot["views"] for slot in selected_slots)
    return total_clicks / total_views

def revenue(selected_slots):
    # Function to calculate the revenue for the selected ad space
    total_revenue = sum(slot["revenue"] for slot in selected_slots)
    return total_revenue

def greedy_slot_selection(slots, budget, objective):
    selected_slots = []
    remaining_budget = budget
    while remaining_budget > 0 and slots:
        best_slot = None
        best_value = -np.inf if objective == "click_rate" else 0
        for slot in slots:
            if slot["cost"] <= remaining_budget: selected_slots.append(slot) if objective == "click_rate": new_value = click_rate(selected_slots) else: new_value = revenue(selected_slots) if new_value > best_value:
                    best_value = new_value
                    best_slot = slot
                selected_slots.pop()
        if best_slot is not None:
            selected_slots.append(best_slot)
            remaining_budget -= best_slot["cost"]
            slots.remove(best_slot)
    return selected_slots

# List of ad spaces (each ad space is represented as a dictionary)
slots = [
    {"id": 1, "cost": 10, "clicks": 100, "views": 1000, "revenue": 50},
    {"id": 2, "cost": 20, "clicks": 120, "views": 1500, "revenue": 60},
    {"id": 3, "cost": 15, "clicks": 80, "views": 800, "revenue": 40},
    {"id": 4, "cost": 12, "clicks": 90, "views": 1200, "revenue": 45},
    {"id": 5, "cost": 18, "clicks": 110, "views": 1000, "revenue": 55},
]

# Parameter Setting
budget = 40  # Budget for ad space
objective = "click_rate"  # Selection of the objective function ("click_rate" or "revenue")

# Seek to select the most appropriate ad space
selected_slots = greedy_slot_selection(slots, budget, objective)

# Display Results
print("Selecting the optimal ad space:")
for slot in selected_slots:
    print(slot["id"])

In this implementation, the click_rate and revenue functions calculate the click rate and revenue, respectively, and the greedy_slot_selection function selects the optimal ad slot using the greedy method. The optimal selection results are stored in a list called selected_slots and displayed at the end.

Example python implementation of a typical sentence selection problem in summary generation

The problem of selecting representative sentences in summary generation is the problem of generating a summary by selecting important sentences from a given sentence. Below is an example of a Python implementation that optimizes the representative sentence selection problem using a submodular function.

import numpy as np
from nltk.tokenize import sent_tokenize

def importance_score(sentence):
    # Function to calculate the importance score of a sentence
    # Here, as an example, the number of words in a sentence is used as the importance score, but please define the appropriate importance score according to the actual question
    return len(sentence.split())

def summary_selection(document, summary_length):
    sentences = sent_tokenize(document)  # Split a sentence into sentences

    # Calculate Importance Score
    importance_scores = [importance_score(sentence) for sentence in sentences]

    # Sort sentences based on importance score
    sorted_sentences = [sentence for _, sentence in sorted(zip(importance_scores, sentences), reverse=True)]

    selected_sentences = []
    remaining_length = summary_length
    for sentence in sorted_sentences:
        if len(sentence.split()) <= remaining_length:
            selected_sentences.append(sentence)
            remaining_length -= len(sentence.split())
        if remaining_length <= 0:
            break

    return selected_sentences

# input document
document = """
Natural Language Processing (NLP) is a generic term for technologies for computer processing of natural language (e.g., Japanese and English) used by humans in everyday life. NLP is used in applications such as machine translation, information retrieval, and document summarization, Summary generation is one of the most important tasks in NLP. In summary generation, the selection of representative sentences is important to generate a short summary from a given sentence. """

# Length of summary (maximum number of words in a sentence to be selected)
summary_length = 20

# Select representative sentences and generate a summary
selected_sentences = summary_selection(document, summary_length)

# Display Results
summary = " ".join(selected_sentences)
print("summary:")
print(summary)

In this implementation, the importance score of a sentence is calculated with the importance_score function, and the summary_selection function generates a summary by selecting important sentences using the greedy method. The summary is stored in a list called selected_sentences and displayed at the end.

Example python implementation of set cover problem with submodular functions

The set cover problem is the problem of covering a given set of elements with some subset (cover set) to satisfy certain conditions. Below is an example of a Python implementation that optimizes the set cover problem using a submodular function.

def uncovered_elements(subsets, universe):
    # Function to return uncovered elements
    uncovered = set(universe)
    for subset in subsets:
        uncovered -= set(subset)
    return uncovered

def greedy_set_cover(subsets, universe):
    selected_subsets = []
    remaining_elements = set(universe)
    while remaining_elements:
        best_subset = None
        best_coverage = -1
        for subset in subsets:
            coverage = len(set(subset) & remaining_elements)
            if coverage > best_coverage:
                best_coverage = coverage
                best_subset = subset
        if best_subset is not None:
            selected_subsets.append(best_subset)
            remaining_elements -= set(best_subset)
    return selected_subsets

# List of cover sets
subsets = [
    [1, 2, 3, 4],
    [2, 4, 6],
    [1, 3, 5],
    [4, 5, 6],
    [1, 2, 5, 6]
]

# Set of all elements (universe)
universe = [1, 2, 3, 4, 5, 6]

# Seek minimum cover set
selected_subsets = greedy_set_cover(subsets, universe)

# Display Results
print("Minimum cover set:")
for subset in selected_subsets:
    print(subset)

In this implementation, the uncovered_elements function calculates the uncovered elements and the greedy_set_cover function selects the minimum cover set using the greedy method. The minimum cover set is stored in a list called selected_subsets and displayed at the end.

Reference Information and Books

More details on submodular optimization can be found in “Submodular Optimization and Machine Learning“. Please also refer to this page.

“Submodular Functions and Optimization” is available as a reference.

and “Submodular Rate Region Models for Multicast Communication in Wireless Networks“

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