Questions tagged [combinatorial-optimization]
For questions about optimization over a discrete solution space.
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Optimization terminology: "Exact" v. "Approximate"
In optimization literature, I frequently see solution methods termed "exact" or "approximate". (I use the term "method" here because I suspect exactness, or its lack, is a function of both algorithm ...
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When are Decision Diagrams the right way to model and solve a problem?
Decision Diagrams are a relatively new approach to solving difficult combinatorial optimization problems. See http://www.andrew.cmu.edu/user/vanhoeve/mdd/ for some information on this approach. Are ...
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Why is open source operations research software so far behind open source statistics and machine learning software?
Like many who participate in this site, I work on projects in both operations research (OR) and statistics/machine learning (ML). The different states of open source software in these fields are often ...
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What's the current status of the Vehicle Routing Problem in the logistics industry?
After a bit of reading I think I've been able to conclude that state-of-the-art VRP can get solutions for 100~500 stops.
My question is around how this actually affects logistics (like Amazon for ...
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Combinatorial problem in my daughter’s class
In Denmark, a rather substantial amount of work and effort has gone into reducing bullying in the Danish public schools. Many initiatives, which purposes are to strengthen the unity and solidarity in ...
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Combinatorial Optimization: Metaheuristics, CP, IP -- "versus" or "and"?
"Recently" someone asked on Twitter whether "people still use genetic algorithms for integer programs". The "majority answer", i.e., 1 out of 1, was: "Yes" .
So,...
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Variable fixing based on a good feasible solution
Suppose you have a combinatorial optimization problem that is formulated as a mixed integer linear program (minimization). The problem size is denoted $n$ and the expected $n$ is around $100$. The ...
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What are some real-world applications of QUBO?
QUBO (Quadratic Unconstrained Binary Optimization) is the minimization of a quadratic function of binary variables.
It has been used for computer vision, Ramsey numbers, factoring numbers, the ...
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Duality in mixed integer linear programs
I know that the standard duality theory for the linear programming problem does not hold for mixed integer linear programming problems. I was wondering why an integer program does not have a dual ...
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Bin Packing with Relational Penalization
There are $ N $ bins with equal capacity $ C $. In addition, there are $ N $ objects $x_1, x_2, \dots, x_N $ that need to be accommodated using the least amount of bins. Each object $x_i$ has a volume ...
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Optimization models for portfolio optimization
What are the mainstream models for portfolio optimization? We have Markowitz mean-variance model and CVaR-based models (e.g., max return subject to a CVaR constraint). What else is out there in terms ...
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State-of-the-art algorithms for solving linear programs
Průša and Werner (2019) show that the general linear programming problem reduces in nearly linear time to the LP relaxations of many classical NP-hard problems (assuming sparse encoding of instances)....
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Why does the design of heuristics require considerable domain knowledge?
I am from a machine learning (ML) background and am interested in how ML is applied to Combinatorial Optimisation. As such, as I have been reading around the area and have come across the statement ...
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A variant of the Multiple Traveling Salesman Problem
I am trying to find a reference (or a reformulation) of a variant of the multiple Traveling Salesman Problem, where multiple agents need to visit each vertex in a graph with minimal cost.
Most of the ...
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What are the examples (applications) of the MIPs in which the objective function has nonzero coefficients for only continuous variables?
I'm specifically looking for real applications of the following form of MIP:
$$\max\,Cx$$
subject to:
\begin{align}Ax +By &= D\\Ax &= E\\By &= F\\ x &\ge 0\\ y &\in \mathbb{...
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Are there any efficient algorithms to solve the longest path problem in networks with cycles?
I have a directed social network and as a preprocessing step I need to calculate the longest path lengths for each node. Longest path problem is NP-hard as far as I know but I've seen dynamic ...
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Branch and Price algorithm is exact?
I know that the Column Generation algorithm delivers an exact solution when you are solving a linear programming optimization problem. I want to know that, does this column generation approach deliver ...
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Benchmark problems for combinatorial multi-objective optimisation
Does anyone know of any good benchmark problems for combinatorial multi-objective optimisation? Something where pareto frontiers are known for example would be very useful.
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Restoring a list from differences
Given a list of (absolute valued) pair differences ordered and with duplicates removed, how can we recover/reconstruct the list that generated these differences? We do not know anything about the ...
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Two-stage $k$-means clustering
The problem I am facing is clustering problem, needed for a Vehicular Routing Problem (VRP) I'm tackling. It is a heterogeneous VRP with Time Window (TW) and a capacity utilization constraint, i.e. a ...
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Finding an optimal set without forbidden subsets
Given $n$ items, I want to select a set items $S\subseteq\{1,2,\dots,n\}$ that maximize profit. The profit of item $i\in\{1,2,\dots,n\}$ is given by $p_i$ and may be assumed to be non-negative.
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What is the difference between job shop scheduling and resource constrained project scheduling?
I read here https://slideplayer.com/slide/3353960/ that RCPS is a generalized version of job shop scheduling. I'm new to this area and I'm trying to classify a specific variation of these types of ...
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Why is Discrete Optimization "Difficult'?
I have the following question about why Combinatorial and Discrete Optimization Problems are Considered to be "Difficult":
For continuous optimization problems (e.g. Loss Functions in ...
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Are Metaheuristics and Evolutionary Algorithms the "Gold Standard" for the Traveling Salesman Problem?
Are Metaheuristics and Evolutionary Algorithms the "Gold Standard" for the Traveling Salesman Problem?
I am interested in learning more about how we have been able to solve the (famous) ...
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Modeling the Choose function
In statistics, one often encounters the choose function ${x \choose y}$ which encodes the number of ways of choosing $y$ items from a set of $x$ items. How would one go about modeling a choose ...
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Current Issues of Interest
What are some current issue of interest in Operations Research? I am interested in current topics that experts in the field are interested in researching.
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Relationship between the Assignment Problem and the Stable Marriage Problem
Suppose I'm solving a minimum-weight matching problem in a bipartite graph with sets $\mathcal{I}$ and $\mathcal{J}$, where $w_{ij}$ denotes the weight of matching item $i$ to $j$. I can model the ...
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Optimal set order to maximize stochastic reward
You have a ticket allowing you to visit up to $n$ of $M$ carnival booths offering games of chance. At each booth you have probability $p_{i}$ of winning a reward with average value $r_{i}$. Each booth ...
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Heuristic methods for optimising complex black box function over permutations/ranks?
Suppose I have a set $S=\{1,2,\dots,500\}$ and some function $f(\sigma)$ from the permutations $\operatorname{Perm}(S) \rightarrow \mathbb{R}$ to be minimized.
The function is complex (simulation ...
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Algorithmic gap for Hochbaum's (greedy) algorithm for (metric) uncapacitated facility location
In Jain et al. (2003)1, at the bottom of page 801, they construct an instance of (metric) uncapacitated facility location for which they claim the greedy (Hochbaum's) algorithm has gap $\Omega\left(\...
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How to get solver time from CPLEX when using the NEOS server through Pyomo?
I've been using CPLEX on the NEOS server, via Pyomo, to solve a binary program I'm working on.
NEOS is amazing, but the documentation is somewhat lacking on the Pyomo side, so I haven't been able to ...
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Equipment replacement problem
I have a question on the Equipment Replacement Problem, where the following is taken (with some syntactic modifications) from IB2070 Mathematical Programming II (MP2), Warwick Business School.
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How to formulate a MIP that can minimize the costs with a combination of subsets given a set?
I am trying to solve the following problem. I have a set $\{1,2,3\}$, which gives the following subsets with its costs:
$\{1\}=8$, $\{2\}=9$, $\{3\}=7$, $\{1,2\}=9$, $\{1,3\}=18$, $\{2,3\}=15$ and $\{...
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CVRP With Unconstrained Fleet Size: Upper Bound on Optimal Fleet Size
Given a CVRP where the number of trucks is not constrained, is there an upper bound on the number of trucks used in an optimal solution in terms of number of customers, some distances, capacities, and ...
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Equivalence of formulations
I have a simple model such as:
\begin{align}\max&\quad z=X_1+X_2+X_3+X_4\\\text{s.t.}&\quad y_1+y_2+y_3+y_4=2\\&\quad X_1 \leq y_1\\&\quad X_2 \leq y_1+y_2\\&\quad X_3 \leq y_2+...
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Order in which you Stack Groceries in your Car's Trunk : An Optimization Problem?
A situation that we are all familiar with : How do you decide which order to place groceries in your car or which order to place food in your fridge?
At first glance, it appears that the "order&...
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Can I use 'SCIP' solver for PYOMO?
I have an MINLP problem to solve where I was initially using 'ipopt' solver but the solution was not sticking to 'binary/boolean/integer' domain type for a variable. I am not sure which free solver ...
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Is there a greedy heuristic approach to the MILP problem?
I have the following optimization problem which is an MILP. I can solve it with an MILP solver.
\begin{alignat}{1}\max_{x_n,t}\,&\quad t\quad\\\text{s.t.}&\quad\sum_{n=1}^{N} x_n \,&= M\\...
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Finding minimum time for vehicle to reach to its destination
Given a set of Vehicles with source and destination I need to find the minimum time of travel for all the vehicles, there are also some charging stations and its necessary for vehicles to charge 1 ...
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Heuristic Search Planning Tree Leading to Worse TSP Solutions than Naive Greedy
I'm doing a Traveling Salesman Problem (TSP) homework for a coursera optimization course. My first attempt was a regular naive greedy approach, from each point, moving to the closest node (that hadn't ...
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What class of scheduling problem models jobs which require multiple machines simultaneously?
In the Flow/Job Shop problems, and other related scheduling problems, a common assumption is that at any given time, a particular job will be being processed on at most one machine (usually... none).
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What's the name of a finite-capacity bin packing problem trying to minimize the weight of the heaviest bin?
I have a fixed number of bins which are themselves weightless. Each bin can hold only a fixed amount of weight. Not all bins have the same capacity.
I also have a fixed number of objects each of which ...
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Solving Logic puzzles through optimization
I have the following "logic puzzle" (I think this is considered as a "scheduling problem"):
In this problem, there are 5 basketball players - provided some clues about their ...
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How can I linearize or convexify this binary quadratic optimization problem?
I have an optimization problem as below. I am having a hard time with the last constraint.
$\max \eta$
subject to
${\bf U}(:,m)^T{\bf A}{\bf U}(:,m)=0,m=1,2,\cdots,M$
here
$\bf{A}$ is a Binary ...
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Learning local search operator selection
I'm just reading [1]. The authors use a neural network to solve capacitated vehicle routing problems through iterative generation of tours by solving a price-collecting traveling salesman problem in ...
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Does this game that I invented correspond to a valid optimization problem?
Recently, I thought of the following "game" that I would like to frame as an optimization problem:
Assume there are five baskets. The first basket has five discrete objects (e.g., apples), ...
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Square packing variant
I saw the following problem and was looking for references about the problem. The problem is stated as
“The green field is the empty area, the dark green 2x2 blocks are
trees and the grey area is the ...
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Can ALL Optimization Problems be Classified as "P" vs "NP"?
In the context of Computer Science and Optimization, I have heard that different problems can be classified using the "P vs NP" framework. Essentially, there is a hierarchy of problems based ...
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Strong MIP formulations for a large-scale mixed-integer nonlinear feasibility problem
I'm trying to construct a strong MIP formulation for the following integer nonlinear feasibility problem.
Informally:
We have a $m \times n$ decision matrix of binary variables
Each row of the matrix ...
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How to reformulate (linearize/convexify) a budgeted assignment problem?
I have a scheduling problem at hand. In my system, there is a service station with $M$ service outlets, therefore, the service station can serve $M$ users at a time. But, there are $N$ users $N>M$ ...
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