# Questions tagged [combinatorial-optimization]

For questions about optimization over a discrete solution space.

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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 ...
3k views

### 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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### 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, my follow-up question is: With all ...
807 views

### 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 ...
814 views

### 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 ...
970 views

### 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 ...
737 views

### 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 ...
553 views

### 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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### 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 ...
952 views

### 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)....
863 views

### 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 ...
281 views

### 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 ...
138 views

### 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 ...
462 views

### 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. ...
148 views

### 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 ...
348 views

### 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 ...
173 views

### 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 ...
94 views

### 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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### 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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### 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 ...
475 views

### 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. ...
424 views

### 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\\...
399 views

### 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 ...
151 views

### 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 ...
108 views

### 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 ...
370 views

### 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 ...
519 views

### 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 ...
157 views

### 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 ...
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 $N>M$ ...