This release is a pre-release and may not be stable for production use.
🆕 What's new
- 2026/10/19: v0.11.0rc1 added the Rust and hybrid solvers:
lapjvr,lapjvh,lapjvr_batch, andlapjvh_batch.
- 2026/09/07: v0.10.0rc1 added Python 3.15 support andlapmod_batch. It improved type hints, stability, and performance.
- 2025/10/31: v0.9.0 improved stability and performance across all solvers.
- 2025/10/27: v0.8.0 addedlapjvx_batch,lapjvxa_batch,lapjvs_batch,lapjvsa_batchandlapjvsa.
- 2025/10/21: v0.7.0 addedlapjvs.
- 2025/10/16: v0.6.0 addedlapjvx,lapjvxa, andlapjvc.
- See GitHub releases for all release notes.
lapx started as a project to maintain Tomas Kazmar's lap, a Jonker-Volgenant solver package.
It now provides additional solver functions. See the usage section for all available functions.
Read the algorithm background
lapx provides linear assignment problem solvers for dense (LAPJV ¹) and sparse (LAPMOD ²) matrices.
The original lapjv and lapmod implementations come from Tomas Kazmar's lap project,
based on papers ¹˒² and A. Volgenant's public domain Pascal implementation ³.
¹ R. Jonker and A. Volgenant, "A Shortest Augmenting Path Algorithm for Dense and Sparse Linear Assignment Problems", Computing 38, 325-340 (1987)
² A. Volgenant, "Linear and Semi-Assignment Problems: A Core Oriented Approach", Computer Ops Res. 23, 917-932 (1996)
³ http://www.assignmentproblems.com/LAPJV.htm | [archive.org]
💽 Installation
Install from PyPI:
python -m pip install lapx
Published wheels cover Python 3.9–3.15 on most platforms and architectures. See the package files for details.
musllinux ARMv7 wheels include all pre-existing solvers but omit lapjvr, lapjvh,
lapjvr_batch, and lapjvh_batch because native Rust build tools are unavailable.
🛠️ Other installation options
Python 3.7 and 3.8 (source builds only):
Python 3.7 and 3.8 no longer receive prebuilt wheels or automated testing. Source installation remains available with compatible build dependencies and the required C++ and current Rust toolchains:
python -m pip install --no-binary=lapx lapx
Install from GitHub (requires C++ and current Rust toolchains):
python -m pip install git+https://github.com/rathaROG/lapx.git
Build and install (requires C++ and current Rust toolchains):
git clone https://github.com/rathaROG/lapx.git
cd lapx
python -m pip install .
cd ..
Run examples outside the checkout so Python loads the installed package.
🆕 lapjvr and lapjvh families
Source builds include both families by default. Set
LAPX_BUILD_RUST=0to build only the other solvers without a Rust toolchain.See the R/H API and build guide and the historical comparisons with JVX, JVS, and SciPy.
⚡ Extra performance
Since v0.9.1,
lapxenables safe base optimizations by default. For source builds, environment variables control these optimizations and optional compiler flags. The optional flags might improve performance:
LAPX_BASEOPTS=0disables the base optimization flags that LAPX adds and link-time optimization (LTO). Other compiler defaults may still apply. This switch is available since v0.9.2.LAPX_FASTMATH=1enables fast-math. This option may change numerical results and the treatment of NaN or infinity. R/H retains strict floating-point rules.LAPX_NATIVE=1enables CPU-specific tuning for GCC/Clang and Rust. Do not distribute these builds.LAPX_LTO=0disables LTO, including inherited LTO defaults.The build enables LTO by default when the compiler and linker support it.
LAPX_BASEOPTS=0does not guarantee an unoptimized or debug build. Fast-math and native tuning are independent options. Both options are disabled by default. Keep both options disabled for portable release wheels.
See setup.py for more details.
🧪 Usage
Install the package as lapx and use import lap.
Show additional notes
- Do not install both
lapandlapxat the same time. Both packages provide the import namelap. The package that you install last replaces the other package's files.- If you only need
lapjvandlapmod, the originallapis sufficient. Chooselapxfor additional fixes, batch processing, output formats, and solvers. It also improves stability and performance.
ℹ️ Cost values
The caller must check cost values for lapjv, lapjvx, lapjvxa, lapjvs, lapjvsa, lapjvr, lapjvh, and their batch variants.
These solvers omit this check to reduce overhead for repeated calls.
Before you call these solvers, prepare costs without NaN (not a number) or negative infinity.
Results with these values are undefined. Positive infinity (np.inf) can represent a forbidden assignment.
lapjvc treats NaN and either infinity as forbidden assignments.
lapmod checks its sparse arrays. Stored costs must be finite, non-negative, and less than lap.LARGE.
With return_cost=True (the default), the solvers sum total costs in float64 from the original costs, even when they solve in float32.
Set return_cost=False when you need only assignments. This skips the total-cost calculation.
Cost limits and filtering after solving
lapjv, lapjvx, lapjvxa, lapjvr, and lapjvh accept cost_limit.
The JVX and R/H batch solvers support it too.
The value must be finite or positive infinity. The default np.inf applies no finite cost limit.
With a finite limit, the solver considers leaving items unmatched during optimization.
Unmatched rows and columns each incur half the limit during optimization;
returned totals exclude those penalties.
JV/JVX solvers expand each nonempty (N, M) problem to shape (N+M, N+M),
which requires more memory and can slow the solve. R/H handles finite limits without square padding.
Some workflows, such as object tracking, apply a cost threshold after solving.
For this approach, leave cost_limit=np.inf. Then remove assignment pairs whose costs exceed the threshold.
Filtering after the solve can produce different assignments from using cost_limit.
It removes pairs without solving again, so it cannot replace discarded matches.
Terms used in the examples
- A cost matrix contains the cost of each possible assignment from a row to a column.
- A mapping array gives the assigned index for each row or column.
- An assignment pair contains a row index and its assigned column index.
Bis the batch size.NandMare the matrix dimensions.Kis the number of assignment pairs.- A dtype is an array's data type, such as float32 or float64.
- The global interpreter lock (GIL) limits concurrent execution of Python code within one interpreter.
🅰️ Single-matrix Solvers 📄
1. The original function lapjv
lapjv supports square and rectangular cost matrices.
It returns optimal assignments as mapping arrays x (size N) and y (size M).
If return_cost=True, it also returns the total cost. See the API documentation for details.
import lap
import numpy as np
cost = np.random.rand(100, 150)
total, x, y = lap.lapjv(cost, extend_cost=True)
valid = x >= 0
pairs = np.column_stack((np.arange(len(x))[valid], x[valid]))
lapjv returns (total, x, y) by default, or (x, y) with return_cost=False.
For a cost matrix cost with shape (N, M), x has N entries and y has M entries.
x[i] gives the assigned column for row i. y[j] gives the assigned row for column j.
Unmatched entries are -1.
For example, x = [1, 0] means row 0 uses column 1 and row 1 uses column 0.
The example above converts mapping arrays to pairs; use
lapjvx
for aligned row and column indices, as in SciPy.
See the original lapjv documentation at gatagat/lap.
Show lapjvx
2. The new function lapjvx
lapjvx uses the same algorithm as lapjv. It returns aligned row and column indices, as in SciPy's linear_sum_assignment.
See the API documentation for details.
import lap
import numpy as np
cost = np.random.rand(100, 150)
total, rows, cols = lap.lapjvx(cost, extend_cost=True)
pairs = np.column_stack((rows, cols))
See the object tracking benchmark to compare lapjvx with other solvers.
Show lapjvxa
3. The new function lapjvxa
lapjvxa provides the same solver behavior as lapjvx. It returns assignment pairs with shape (K, 2).
The caller does not need to convert the output.
The pair format also suits tracking workflows that apply a cost threshold after solving, as described in Cost values.
See the API documentation for details.
import lap
import numpy as np
cost = np.random.rand(100, 150)
total, pairs = lap.lapjvxa(cost, extend_cost=True)
Show lapjvc
4. The new function lapjvc
lapjvc extends Christoph Heindl's py-lapsolver.
It returns the same output format as lapjvx and SciPy's linear_sum_assignment.
See the API documentation for details.
import lap
import numpy as np
cost = np.random.rand(100, 150)
total, rows, cols = lap.lapjvc(cost)
pairs = np.column_stack((rows, cols))
Show lapjvs
5. The new function lapjvs
lapjvs extends Vadim Markovtsev's lapjv.
Its scalar kernel does not require AVX2 instructions.
It supports square and rectangular cost matrices. It can return either of these output formats:
- Aligned index arrays, as in SciPy's
linear_sum_assignment. - Mapping arrays
xandy, as inlapjv.
See the wrapper documentation for details.
import lap
import numpy as np
cost = np.random.rand(100, 150)
total, rows, cols = lap.lapjvs(cost, jvx_like=True)
pairs = np.column_stack((rows, cols))
Show lapjvsa
6. The new function lapjvsa
lapjvsa provides the same solver behavior as lapjvs. It returns assignment pairs with shape (K, 2).
The caller does not need to convert the output.
See the wrapper documentation for details.
import lap
import numpy as np
cost = np.random.rand(100, 150)
total, pairs = lap.lapjvsa(cost)
Show lapmod
7. The original function lapmod
See the wrapper documentation for details.
import lap
import numpy as np
n = 3
cc = np.array([3.0, 1.0, 1.0, 4.0, 2.0, 1.0]) # Costs grouped by row
ii = np.array([0, 2, 4, 6]) # Row offsets into cc
kk = np.array([0, 1, 0, 2, 1, 2]) # Column indices for each cost
total, x, y = lap.lapmod(n, cc, ii, kk)
8. Rust and hybrid functions lapjvr and lapjvh
R/H were about 3–10× faster than SciPy across benchmarks; see benchmark.md.
lapjvr and lapjvh share a Rust Jonker-Volgenant implementation. For square
problems with infinite cost_limit, lapjvh uses C++ completion. Both return
aligned row and column indices, like lapjvx; use output="pairs" for the
pair layout of lapjvxa.
import lap
import numpy as np
cost = np.random.rand(100, 150)
total, rows, cols = lap.lapjvr(cost, extend_cost=True)
total, pairs = lap.lapjvh(cost, extend_cost=True, output="pairs")
Both support rectangular inputs (extend_cost=True), finite cost_limit, and
precision="native", "float32", or "float64". Rectangular and finite-limit
problems are solved without square padding. See the
API guide
for full signatures, return types, and precision rules.
🅱️ Batch Solvers 🗂️
All batch solvers use the same n_threads rules:
0(the default) orNoneselects the CPU count fromos.cpu_count(), capped by the batch size. If the CPU count is unavailable, the solver uses one worker.1or a negative value runs sequentially.- Positive values select the worker count. The number of workers cannot exceed the batch size.
- Explicit positive values are not capped by the detected CPU count.
- Empty and single-item batches do not create a thread pool.
1. The new function lapjvx_batch
lapjvx_batch runs lapjvx on a batch of cost matrices with shape (B, N, M).
See the wrapper documentation for details.
import lap
import numpy as np
batch_costs = np.random.rand(500, 100, 150) # (B, N, M)
totals, rows, cols = lap.lapjvx_batch(batch_costs, extend_cost=True, n_threads=2)
print(f"total costs = {totals.sum()}")
pairs_7 = np.column_stack((rows[7], cols[7])) # (K_b, 2)
print(f"pairs_7.shape = {pairs_7.shape}")
Show lapjvxa_batch
2. The new function lapjvxa_batch
lapjvxa_batch runs lapjvxa on a batch of cost matrices with shape (B, N, M).
See the wrapper documentation for details.
import lap
import numpy as np
batch_costs = np.random.rand(500, 100, 150) # (B, N, M)
totals, pairs = lap.lapjvxa_batch(batch_costs, extend_cost=True, n_threads=2)
print(f"total costs = {totals.sum()}")
print(f"pairs[7].shape = {pairs[7].shape}")
Show lapjvs_batch
3. The new function lapjvs_batch
lapjvs_batch runs lapjvs on a batch of cost matrices with shape (B, N, M).
See the wrapper documentation for details.
import lap
import numpy as np
batch_costs = np.random.rand(500, 100, 150) # (B, N, M)
totals, rows, cols = lap.lapjvs_batch(batch_costs, extend_cost=True, n_threads=2)
print(f"total costs = {totals.sum()}")
pairs_7 = np.column_stack((rows[7], cols[7])) # (K_b, 2)
print(f"pairs_7.shape = {pairs_7.shape}")
Show lapjvsa_batch
4. The new function lapjvsa_batch
lapjvsa_batch runs lapjvsa on a batch of cost matrices with shape (B, N, M).
See the wrapper documentation for details.
import lap
import numpy as np
batch_costs = np.random.rand(500, 100, 150) # (B, N, M)
totals, pairs = lap.lapjvsa_batch(batch_costs, extend_cost=True, n_threads=2)
print(f"total costs = {totals.sum()}")
print(f"pairs[7].shape = {pairs[7].shape}")
Show lapmod_batch
5. The new function lapmod_batch
lapmod_batch solves a sequence of sparse (n, cc, ii, kk) problems.
It uses the same inputs and checks as lapmod. Each problem must be square.
Problem sizes and numbers of stored entries can differ.
Results keep the input order. For problem b, x_list[b] maps rows to columns and y_list[b] maps columns to rows.
Both lists contain the same int32 arrays as the single solver.
import lap
import numpy as np
problems = [
(2, np.array([1.0, 4.0, 3.0, 2.0]),
np.array([0, 2, 4]), np.array([0, 1, 0, 1])),
(3, np.array([4.0, 1.0, 3.0]),
np.array([0, 1, 2, 3]), np.array([2, 0, 1])),
]
totals, x_list, y_list = lap.lapmod_batch(problems, n_threads=2)
print(totals) # [3. 8.]
print(x_list[1]) # [2 0 1]
# Skip total-cost calculation when only the mappings are needed.
x_list, y_list = lap.lapmod_batch(problems, return_cost=False, n_threads=2)
With the default fast=True, the native solver releases the GIL. Independent problems can then run concurrently.
Python code that checks inputs and calculates total costs can limit speedup.
With fast=False, the Python solver's performance depends largely on the GIL.
fp_version selects the native path-search version, as in lapmod.
Do not change input arrays while the solver runs. If a problem raises an exception, the caller receives it after the thread pool stops.
See the wrapper documentation for the full signature and return types.
6. lapjvr_batch and lapjvh_batch
lapjvr_batch and lapjvh_batch run their single-matrix solvers on a batch of
cost matrices with shape (B, N, M). Results keep the input order and use the
same output and precision options as the single-matrix calls.
import lap
import numpy as np
batch_costs = np.random.rand(500, 100, 150) # (B, N, M)
totals, rows, cols = lap.lapjvr_batch(batch_costs, extend_cost=True, n_threads=2)
totals, pairs = lap.lapjvh_batch(batch_costs, extend_cost=True, output="pairs", n_threads=2)
The n_threads rules above apply; automatic mode may run small workloads
serially. See the batch API guide
for return types and native worker details.
🏆 Benchmark and Test
See the benchmark guide for commands, repeat controls, timing methods, and historical results.
See the test guide to run the full test suite.
📝 License
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