Mixed Integer Programming

A tool to consider.

By Marco Neumann.

Why

An practical example.

K8s Pod Placement

Cluster

Node

Pod

Solver

Find a solution.

SAT Solver

Variables

boolean (true/false; 1/0)
what we want

Constraints

depending on solver:
- boolean
- linear integers (= pseudo-boolean)
- ...
what we know

Knapsack

	Node 1	Node 2
Pod A	☐	☐	~~∑=0 = 1~~
Pod B	☐	☐	~~∑=0 = 1~~
Pod C	☐	☐	~~∑=0 = 1~~
	∑=0 ≤ 5	∑=0 ≤ 5

Cluster

Node

Knapsack

	Node 1	Node 2
Pod A	☑	☐	∑=1 = 1
Pod B	☐	☐	~~∑=0 = 1~~
Pod C	☐	☐	~~∑=0 = 1~~
	∑=1 ≤ 5	∑=0 ≤ 5

Cluster

Node

Pod A

Knapsack

	Node 1	Node 2
Pod A	☑	☑	∑=2 = 1
Pod B	☐	☐	~~∑=0 = 1~~
Pod C	☐	☐	~~∑=0 = 1~~
	∑=1 ≤ 5	∑=0 ≤ 5

Cluster

Node

Pod A

Knapsack

	Node 1	Node 2
Pod A	☑	☐	∑=0 = 1
Pod B	☑	☐	∑=0 = 1
Pod C	☑	☐	∑=0 = 1
	~~∑=6 ≤ 5~~	∑=0 ≤ 5

Cluster

Node

Pod B

Pod C

Pod A

Knapsack

	Node 1	Node 2
Pod A	☑	☐	∑=1 = 1
Pod B	☑	☐	∑=1 = 1
Pod C	☐	☑	∑=1 = 1
	∑=4 ≤ 5	∑=2 ≤ 5

Cluster

Node

Pod B

Pod C

Pod A

NP-Completeness

Roughly: The problem is so hard that you have to enumerate all solutions.

Math: Exponential complexity.

😱

Knapsack is NP-complete.

Most real instances are NOT.

Solvers are VERY fast.

😅

Over-Constrained

Optimizer

Find best solution.

Inputs

Constraints

equation ≤ constant
linear
quadratic
... depending on optimizer

Target Function

min/max equation:

linear
quadratic
... depending on optimizer

Most solvers are numeric!

Multiple Target Functions

Weight-based

Unit-based

e.g. express cost in €

100 \cdot c_\textrm{nodes} + 25 \cdot c_\textrm{movement}

c_\textrm{nodes}

Cost of running nodes

c_\textrm{movement}

Discruption by moving existing pods

Tricks

c_n = \begin{cases} a &\textrm{ if } |\textrm{pods on node}| > 0 \\ 0 &\textrm{ if } |\textrm{pods on node}| = 0 \end{cases}

c_n = ay_n \\ y_n \textrm{ binary} \\ |\textrm{pods on node}| \leq y_n \left( |\textrm{pods}| + 1 \right)

🔖 AIMMS Modeling Guide - Integer Programming Tricks

Implementation

Actually implementing it.

Code

See

github.com/crepererum/mip-pod-scheduler

YAML

serde

Rust struct

good_lp

problem gen

SCIP

interprete

YAML

serde

Rust struct

⚠️ prototype code ⚠️

Input

nodes:
  n1:
    cpu: 2.0
    memory: 16.0
    cost: 50.0
  n2:
    cpu: 1.0
    memory: 32.0
    cost: 200.0
  n3:
    cpu: 1.0
    memory: 32.0
    cost: 100.0
pods:
  p1:
    cpu: 0.5
    memory: 10.0
    movement_cost: 200.0
    current_node: n2
  p2:
    cpu: 1.0
    memory: 8.0
  p3:
    cpu: 1.0
    memory: 8.0
  p4:
    cpu: 0.5
    memory: 2.0

Demo

Summary

Wrap-up & alternatives.

Approaches

Manual

low barrier
potentially better if sufficiently tuned
most people will end up with semi-optimal heuristics
harder to maintain & extend

Existing Optimizer

provides structured approach
easy to extend
extensive set of optimizer passes
requires modelling knowledge

🔖 Presolve Reductions in Mixed Integer Programming

Choose The Right Solver

Open Source

Proprietary

Choose The Right Tool

Use the "tightest" fitting tool!

Alternatives

Cannot describe solution space
⇒ Genetic Programming, Metaheuristic
Solver only, but more complex data types
⇒ SMT (Satisfiability Modulo Theories) Solver

???