diff --git a/examples/rule_discovery.jl b/examples/rule_discovery.jl
index 49e423b..732622d 100644
--- a/examples/rule_discovery.jl
+++ b/examples/rule_discovery.jl
@@ -82,4 +82,31 @@ branching_region = SimpleGraph(Graphs.SimpleEdge.(edges))
 # Generate the tree-like N3 neighborhood of R
 graph = tree_like_N3_neighborhood(branching_region)
 
-solve_opt_rule(branching_region, graph, vs)
\ No newline at end of file
+solve_opt_rule(branching_region, graph, vs)
+
+
+# ## Generating rules for large scale problems
+# For large scale problems, we can use the greedy merge rule to generate rules, which avoids generating all candidate clauses.
+function solve_greedy_rule(branching_region, graph, vs)
+    ## Use default solver and measure
+    m = D3Measure()
+    table_solver = TensorNetworkSolver(; prune_by_env=true)
+
+    ## Pruning irrelevant entries
+    ovs = OptimalBranchingMIS.open_vertices(graph, vs)
+    subg, vmap = induced_subgraph(graph, vs)
+    @info "solving the branching table..."
+    tbl = OptimalBranchingMIS.reduced_alpha_configs(table_solver, subg, Int[findfirst(==(v), vs) for v in ovs])
+    @info "the length of the truth_table after pruning irrelevant entries: $(length(tbl.table))"
+
+    @info "generating the optimal branching rule via greedy merge..."
+    candidates = OptimalBranchingCore.bit_clauses(tbl)
+    result = OptimalBranchingMIS.OptimalBranchingCore.greedymerge(candidates, MISProblem(graph), vs, m)
+    return result
+    @info "the greedily minimized gamma: $(result.γ)"
+
+    @info "the branching rule on R:"
+    viz_dnf(result.optimal_rule, vs)
+end
+
+result = solve_greedy_rule(branching_region, graph, vs)
diff --git a/lib/OptimalBranchingCore/src/OptimalBranchingCore.jl b/lib/OptimalBranchingCore/src/OptimalBranchingCore.jl
index 30ac60c..8631f23 100644
--- a/lib/OptimalBranchingCore/src/OptimalBranchingCore.jl
+++ b/lib/OptimalBranchingCore/src/OptimalBranchingCore.jl
@@ -30,5 +30,5 @@ include("interfaces.jl")
 include("branching_table.jl")
 include("setcovering.jl")
 include("branch.jl")
-
+include("greedymerge.jl")
 end
diff --git a/lib/OptimalBranchingCore/src/branch.jl b/lib/OptimalBranchingCore/src/branch.jl
index e69b9e8..9c2d8b4 100644
--- a/lib/OptimalBranchingCore/src/branch.jl
+++ b/lib/OptimalBranchingCore/src/branch.jl
@@ -15,10 +15,15 @@ A [`OptimalBranchingResult`](@ref) object representing the optimal branching rul
 """
 function optimal_branching_rule(table::BranchingTable, variables::Vector, problem::AbstractProblem, m::AbstractMeasure, solver::AbstractSetCoverSolver)
     candidates = candidate_clauses(table)
-    size_reductions = [measure(problem, m) - measure(first(apply_branch(problem, candidate, variables)), m) for candidate in candidates]
-    return minimize_γ(table, candidates, size_reductions, solver; γ0=2.0)
+    size_reductions = [size_reduction(problem, m, candidate, variables) for candidate in candidates]
+    return minimize_γ(table, candidates, size_reductions, solver; γ0 = 2.0)
 end
 
+function size_reduction(p::AbstractProblem, m::AbstractMeasure, cl::Clause{INT}, variables::Vector) where {INT}
+    return measure(p, m) - measure(first(apply_branch(p, cl, variables)), m)
+end
+
+
 """
     BranchingStrategy
     BranchingStrategy(; kwargs...)
@@ -31,20 +36,20 @@ A struct representing the configuration for a solver, including the reducer and
 - `selector::AbstractSelector`: The selector to select the next branching variable or decision.
 - `m::AbstractMeasure`: The measure to evaluate the performance of the branching strategy.
 """
-@kwdef struct BranchingStrategy{TS<:AbstractTableSolver, SCS<:AbstractSetCoverSolver, SL<:AbstractSelector, M<:AbstractMeasure}
+@kwdef struct BranchingStrategy{TS <: AbstractTableSolver, SCS <: AbstractSetCoverSolver, SL <: AbstractSelector, M <: AbstractMeasure}
     set_cover_solver::SCS = IPSolver()
     table_solver::TS
     selector::SL
     measure::M
 end
-Base.show(io::IO, config::BranchingStrategy) = print(io, 
-"""
-BranchingStrategy
-├── table_solver - $(config.table_solver)
-├── set_cover_solver - $(config.set_cover_solver)
-├── selector - $(config.selector)
-└── measure - $(config.measure)
-""")
+Base.show(io::IO, config::BranchingStrategy) = print(io,
+    """
+    BranchingStrategy
+    ├── table_solver - $(config.table_solver)
+    ├── set_cover_solver - $(config.set_cover_solver)
+    ├── selector - $(config.selector)
+    └── measure - $(config.measure)
+    """)
 
 """
     branch_and_reduce(problem::AbstractProblem, config::BranchingStrategy; reducer::AbstractReducer=NoReducer(), result_type=Int)
@@ -73,7 +78,7 @@ function branch_and_reduce(problem::AbstractProblem, config::BranchingStrategy,
     variables = select_variables(rp, config.measure, config.selector)  # select a subset of variables
     tbl = branching_table(rp, config.table_solver, variables)      # compute the BranchingTable
     result = optimal_branching_rule(tbl, variables, rp, config.measure, config.set_cover_solver)  # compute the optimal branching rule
-    return sum(result.optimal_rule.clauses) do branch  # branch and recurse
+    return sum(get_clauses(result)) do branch  # branch and recurse
         subproblem, localvalue = apply_branch(rp, branch, variables)
         branch_and_reduce(subproblem, config, reducer, result_type) * result_type(localvalue) * reducedvalue
     end
diff --git a/lib/OptimalBranchingCore/src/greedymerge.jl b/lib/OptimalBranchingCore/src/greedymerge.jl
new file mode 100644
index 0000000..26df3dc
--- /dev/null
+++ b/lib/OptimalBranchingCore/src/greedymerge.jl
@@ -0,0 +1,43 @@
+struct GreedyMerge <: AbstractSetCoverSolver end
+function optimal_branching_rule(table::BranchingTable, variables::Vector, problem::AbstractProblem, m::AbstractMeasure, solver::GreedyMerge)
+    candidates = bit_clauses(table)
+    return greedymerge(candidates, problem, variables, m)
+end
+
+function bit_clauses(tbl::BranchingTable{INT}) where {INT}
+    n, bss = tbl.bit_length, tbl.table
+    temp_clauses = [[Clause(bmask(INT, 1:n), bs) for bs in bss1] for bss1 in bss]
+    return temp_clauses
+end
+
+function greedymerge(cls::Vector{Vector{Clause{INT}}}, problem::AbstractProblem, variables::Vector, m::AbstractMeasure) where {INT}
+    cls = copy(cls)
+    size_reductions = [size_reduction(problem, m, first(candidate), variables) for candidate in cls]
+    local γ
+    while true
+        γ = complexity_bv(size_reductions)
+        minval = zero(γ)
+        minidx = (-1, -1, -1, -1)
+        local minclause
+        local minred
+        for i ∈ 1:length(cls), j ∈ i+1:length(cls)
+            for ii in 1:length(cls[i]), jj in 1:length(cls[j])
+                cl12 = gather2(length(variables), cls[i][ii], cls[j][jj])
+                if cl12.mask == 0
+                    continue
+                end
+                reduction = size_reduction(problem, m, cl12, variables)
+                val = γ^(-reduction) - γ^(-size_reductions[i]) - γ^(-size_reductions[j])
+                if val < minval
+                    minval, minidx, minclause, minred = val, (i, j, ii, jj), cl12, reduction
+                end
+            end
+        end
+        minidx == (-1, -1, -1, -1) && break  # no more merging
+        deleteat!(cls, minidx[1:2])
+        deleteat!(size_reductions, minidx[1:2])
+        push!(cls, [minclause])
+        push!(size_reductions, minred)
+    end
+    return OptimalBranchingResult(DNF([cl[1] for cl in cls]), size_reductions, γ)
+end
\ No newline at end of file
diff --git a/lib/OptimalBranchingCore/src/setcovering.jl b/lib/OptimalBranchingCore/src/setcovering.jl
index bc58b82..5d1f582 100644
--- a/lib/OptimalBranchingCore/src/setcovering.jl
+++ b/lib/OptimalBranchingCore/src/setcovering.jl
@@ -101,18 +101,18 @@ end
 The result type for the optimal branching rule.
 
 ### Fields
-- `selected_ids::Vector{Int}`: The indices of the selected rows in the branching table.
 - `optimal_rule::DNF{INT}`: The optimal branching rule.
 - `branching_vector::Vector{T<:Real}`: The branching vector that records the size reduction in each subproblem.
 - `γ::Float64`: The optimal γ value (the complexity of the branching rule).
 """
 struct OptimalBranchingResult{INT <: Integer, T <: Real}
-    selected_ids::Vector{Int}
     optimal_rule::DNF{INT}
     branching_vector::Vector{T}
     γ::Float64
 end
-Base.show(io::IO, results::OptimalBranchingResult{INT, T}) where {INT, T} = print(io, "OptimalBranchingResult{$INT, $T}:\n selected_ids: $(results.selected_ids)\n optimal_rule: $(results.optimal_rule)\n branching_vector: $(results.branching_vector)\n γ: $(results.γ)")
+Base.show(io::IO, results::OptimalBranchingResult{INT, T}) where {INT, T} = print(io, "OptimalBranchingResult{$INT, $T}:\n optimal_rule: $(results.optimal_rule)\n branching_vector: $(results.branching_vector)\n γ: $(results.γ)")
+get_clauses(results::OptimalBranchingResult) = results.optimal_rule.clauses
+get_clauses(res::AbstractArray) = res
 
 """
     minimize_γ(table::BranchingTable, candidates::Vector{Clause}, Δρ::Vector, solver)
@@ -140,7 +140,7 @@ function minimize_γ(table::BranchingTable, candidates::Vector{Clause{INT}}, Δ
 
     # Note: the following instance is captured for time saving, and also for it may cause IP solver to fail
     for (k, subset) in enumerate(subsets)
-        (length(subset) == num_items) && return OptimalBranchingResult([k], DNF([candidates[k]]), [Δρ[k]], 1.0)
+        (length(subset) == num_items) && return OptimalBranchingResult(DNF([candidates[k]]), [Δρ[k]], 1.0)
     end
 
     cx_old = cx = γ0
@@ -153,7 +153,7 @@ function minimize_γ(table::BranchingTable, candidates::Vector{Clause{INT}}, Δ
         cx ≈ cx_old && break  # convergence
         cx_old = cx
     end
-    return OptimalBranchingResult(picked_scs, DNF([candidates[i] for i in picked_scs]), Δρ[picked_scs], cx)
+    return OptimalBranchingResult(DNF([candidates[i] for i in picked_scs]), Δρ[picked_scs], cx)
 end
 
 # TODO: we need to extend this function to trim the candidate clauses
diff --git a/lib/OptimalBranchingCore/test/branching_table.jl b/lib/OptimalBranchingCore/test/branching_table.jl
index 7c2e0b5..468eeaf 100644
--- a/lib/OptimalBranchingCore/test/branching_table.jl
+++ b/lib/OptimalBranchingCore/test/branching_table.jl
@@ -1,5 +1,5 @@
 using OptimalBranchingCore, GenericTensorNetworks
-using BitBasis
+using OptimalBranchingCore.BitBasis
 using Test
 
 @testset "branching table" begin
diff --git a/lib/OptimalBranchingCore/test/greedymerge.jl b/lib/OptimalBranchingCore/test/greedymerge.jl
new file mode 100644
index 0000000..4de7b09
--- /dev/null
+++ b/lib/OptimalBranchingCore/test/greedymerge.jl
@@ -0,0 +1,15 @@
+using Test
+using OptimalBranchingCore
+using OptimalBranchingCore: bit_clauses
+using OptimalBranchingCore.BitBasis
+using GenericTensorNetworks
+
+@testset "bit_clauses" begin
+	tbl = BranchingTable(5, [
+		[StaticElementVector(2, [0, 0, 1, 0, 0]), StaticElementVector(2, [0, 1, 0, 0, 0])],
+		[StaticElementVector(2, [1, 0, 0, 1, 0])],
+		[StaticElementVector(2, [0, 0, 1, 0, 1])],
+	])
+
+	bc = bit_clauses(tbl)
+end
\ No newline at end of file
diff --git a/lib/OptimalBranchingCore/test/setcovering.jl b/lib/OptimalBranchingCore/test/setcovering.jl
index 57d723a..3209b47 100644
--- a/lib/OptimalBranchingCore/test/setcovering.jl
+++ b/lib/OptimalBranchingCore/test/setcovering.jl
@@ -16,7 +16,6 @@ end
     Δρ = [count_ones(c.mask) for c in clauses]
     result_ip = OptimalBranchingCore.minimize_γ(tbl, clauses, Δρ, IPSolver(; max_itr = 10, verbose = false))
     result_lp = OptimalBranchingCore.minimize_γ(tbl, clauses, Δρ, LPSolver(; max_itr = 10, verbose = false))
-    @test result_ip.selected_ids == result_lp.selected_ids
     @test result_ip.branching_vector ≈ result_lp.branching_vector
     @test result_ip.γ ≈ result_lp.γ ≈ 1.0
 
@@ -29,7 +28,6 @@ end
     Δρ = [count_ones(c.mask) for c in clauses]
     result_ip = OptimalBranchingCore.minimize_γ(tbl, clauses, Δρ, IPSolver(; max_itr = 10, verbose = false))
     result_lp = OptimalBranchingCore.minimize_γ(tbl, clauses, Δρ, LPSolver(; max_itr = 10, verbose = false))
-    @test result_ip.selected_ids == result_lp.selected_ids
     @test result_ip.branching_vector ≈ result_lp.branching_vector
     @test result_ip.γ ≈ result_lp.γ ≈ 1.1673039782614185
 
@@ -47,7 +45,6 @@ end
     Δρ = [count_ones(c.mask) for c in clauses]
     result_ip = OptimalBranchingCore.minimize_γ(tbl, clauses, Δρ, IPSolver(max_itr = 10, verbose = false))
     result_lp = OptimalBranchingCore.minimize_γ(tbl, clauses, Δρ, LPSolver(max_itr = 10, verbose = false))
-    @test result_ip.selected_ids == result_lp.selected_ids
     @test result_ip.branching_vector ≈ result_lp.branching_vector
     @test OptimalBranchingCore.covered_by(tbl, result_ip.optimal_rule)
     @test OptimalBranchingCore.covered_by(tbl, result_lp.optimal_rule)
diff --git a/lib/OptimalBranchingMIS/src/interfaces.jl b/lib/OptimalBranchingMIS/src/interfaces.jl
index de29ab0..83e397a 100644
--- a/lib/OptimalBranchingMIS/src/interfaces.jl
+++ b/lib/OptimalBranchingMIS/src/interfaces.jl
@@ -1,37 +1,37 @@
 """
-    mis_size(g::AbstractGraph; bs::BranchingStrategy = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure=D3Measure()), reducer::AbstractReducer = MISReducer())
+    mis_size(g::AbstractGraph; branching_strategy::BranchingStrategy = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure=D3Measure()), reducer::AbstractReducer = MISReducer())
 
 Calculate the size of the Maximum Independent Set (MIS) for a given graph.
 
 ### Arguments
 - `g::AbstractGraph`: The graph for which the MIS size is to be calculated.
-- `bs::BranchingStrategy`: (optional) The branching strategy to be used. Defaults to a strategy using `table_solver=TensorNetworkSolver`, `selector=MinBoundaryHighDegreeSelector(2, 6, 0)`, and `measure=D3Measure`.
+- `branching_strategy::BranchingStrategy`: (optional) The branching strategy to be used. Defaults to a strategy using `table_solver=TensorNetworkSolver`, `selector=MinBoundaryHighDegreeSelector(2, 6, 0)`, and `measure=D3Measure`.
 - `reducer::AbstractReducer`: (optional) The reducer to be applied. Defaults to `MISReducer`.
 
 ### Returns
 - An integer representing the size of the Maximum Independent Set for the given graph.
 """
-function mis_size(g::AbstractGraph; bs::BranchingStrategy = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure=D3Measure()), reducer=MISReducer())
+function mis_size(g::AbstractGraph; branching_strategy::BranchingStrategy = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure = D3Measure()), reducer = MISReducer())
     p = MISProblem(g)
-    res = branch_and_reduce(p, bs, reducer, MaxSize)
+    res = branch_and_reduce(p, branching_strategy, reducer, MaxSize)
     return res.size
 end
 
 """
-    mis_branch_count(g::AbstractGraph; bs::BranchingStrategy = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure=D3Measure()), reducer=MISReducer())
+    mis_branch_count(g::AbstractGraph; branching_strategy::BranchingStrategy = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure=D3Measure()), reducer=MISReducer())
 
 Calculate the size and the number of branches of the Maximum Independent Set (MIS) for a given graph.
 
 ### Arguments
 - `g::AbstractGraph`: The graph for which the MIS size and the number of branches are to be calculated.
-- `bs::BranchingStrategy`: (optional) The branching strategy to be used. Defaults to a strategy using `table_solver=TensorNetworkSolver`, `selector=MinBoundaryHighDegreeSelector(2, 6, 0)`, and `measure=D3Measure`.
+- `branching_strategy::BranchingStrategy`: (optional) The branching strategy to be used. Defaults to a strategy using `table_solver=TensorNetworkSolver`, `selector=MinBoundaryHighDegreeSelector(2, 6, 0)`, and `measure=D3Measure`.
 - `reducer::AbstractReducer`: (optional) The reducer to be applied. Defaults to `MISReducer`.
 
 ### Returns
 - A tuple `(size, count)` where `size` is the size of the Maximum Independent Set and `count` is the number of branches.
 """
-function mis_branch_count(g::AbstractGraph; branching_strategy::BranchingStrategy = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure=D3Measure()), reducer=MISReducer())
+function mis_branch_count(g::AbstractGraph; branching_strategy::BranchingStrategy = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure = D3Measure()), reducer = MISReducer())
     p = MISProblem(g)
     res = branch_and_reduce(p, branching_strategy, reducer, MaxSizeBranchCount)
     return (res.size, res.count)
-end
\ No newline at end of file
+end
diff --git a/lib/OptimalBranchingMIS/src/types.jl b/lib/OptimalBranchingMIS/src/types.jl
index 7a1f44f..28b9594 100644
--- a/lib/OptimalBranchingMIS/src/types.jl
+++ b/lib/OptimalBranchingMIS/src/types.jl
@@ -12,7 +12,7 @@ Represents a Maximum Independent Set (MIS) problem.
 
 """
 mutable struct MISProblem <: AbstractProblem
-    g::SimpleGraph
+    g::SimpleGraph{Int}
 end
 Base.copy(p::MISProblem) = MISProblem(copy(p.g))
 Base.show(io::IO, p::MISProblem) = print(io, "MISProblem($(nv(p.g)))")
@@ -83,4 +83,18 @@ function OptimalBranchingCore.measure(p::MISProblem, ::D3Measure)
         dg = degree(g)
         return Int(sum(max(d - 2, 0) for d in dg))
     end
-end
\ No newline at end of file
+end
+
+function OptimalBranchingCore.size_reduction(p::MISProblem, m::D3Measure, cl::Clause{INT}, variables::Vector) where {INT}
+    vertices_removed = removed_vertices(variables, p.g, cl)
+    isempty(vertices_removed) && return 0
+    sum = 0
+    for v in vertices_removed
+        sum += max(degree(p.g, v) - 2, 0)
+    end
+    vertices_removed_neighbors = setdiff(mapreduce(v -> neighbors(p.g, v), ∪, vertices_removed), vertices_removed)
+    for v in vertices_removed_neighbors
+        sum += max(degree(p.g, v) - 2,0) - max(degree(p.g, v) - 2 - count(vx -> vx ∈ vertices_removed, neighbors(p.g, v)), 0)
+    end
+    return sum
+end
diff --git a/lib/OptimalBranchingMIS/test/greedymerge.jl b/lib/OptimalBranchingMIS/test/greedymerge.jl
new file mode 100644
index 0000000..1799841
--- /dev/null
+++ b/lib/OptimalBranchingMIS/test/greedymerge.jl
@@ -0,0 +1,43 @@
+using OptimalBranchingMIS
+using OptimalBranchingMIS.EliminateGraphs.Graphs
+using Test
+using Random
+using OptimalBranchingCore
+using OptimalBranchingCore.BitBasis
+using GenericTensorNetworks
+using OptimalBranchingCore: bit_clauses
+Random.seed!(1234)
+
+# Example from arXiv:2412.07685 Fig. 1
+@testset "GreedyMerge" begin
+    edges = [(1, 4), (1, 5), (3, 4), (2, 5), (4, 5), (1, 6), (2, 7), (3, 8)]
+    example_g = SimpleGraph(Graphs.SimpleEdge.(edges))
+    p = MISProblem(example_g)
+    tbl = BranchingTable(5, [
+        [StaticElementVector(2, [0, 0, 0, 0, 1]), StaticElementVector(2, [0, 0, 0, 1, 0])],
+        [StaticElementVector(2, [0, 0, 1, 0, 1])],
+        [StaticElementVector(2, [0, 1, 0, 1, 0])],
+        [StaticElementVector(2, [1, 1, 1, 0, 0])],
+    ])
+    cls = bit_clauses(tbl)
+    res = OptimalBranchingCore.greedymerge(cls, p, [1, 2, 3, 4, 5], NumOfVertices())
+    clsf = res.optimal_rule.clauses
+    @test clsf[1].mask == cls[3][1].mask
+    @test clsf[1].val == cls[3][1].val
+    @test clsf[2].mask == cls[4][1].mask
+    @test clsf[2].val == cls[4][1].val
+    @test clsf[3].mask == 27
+    @test clsf[3].val == 16
+end
+
+@testset "GreedyMerge" begin
+    g = random_regular_graph(20, 3)
+    mis_num, count2 = mis_branch_count(g)
+    for reducer in [NoReducer(), MISReducer()]
+        for measure in [D3Measure(), NumOfVertices()]
+            bs = BranchingStrategy(table_solver = TensorNetworkSolver(), selector = MinBoundaryHighDegreeSelector(2, 6, 0), measure = measure, set_cover_solver = OptimalBranchingCore.GreedyMerge())
+            mis1, count1 = mis_branch_count(g; branching_strategy = bs, reducer)
+            @test mis1 == mis_num
+        end
+    end
+end
diff --git a/lib/OptimalBranchingMIS/test/types.jl b/lib/OptimalBranchingMIS/test/types.jl
new file mode 100644
index 0000000..f683507
--- /dev/null
+++ b/lib/OptimalBranchingMIS/test/types.jl
@@ -0,0 +1,59 @@
+using OptimalBranchingMIS
+using OptimalBranchingMIS.EliminateGraphs.Graphs
+using Test
+using Random
+using OptimalBranchingCore
+using OptimalBranchingCore.BitBasis
+using GenericTensorNetworks
+using OptimalBranchingCore: size_reduction, apply_branch
+
+@testset "size_reduction" begin
+    g = random_regular_graph(60, 3)
+    vs = collect(1:20)
+    cl = Clause(bit"1111111111", bit"1011010111")
+    p = MISProblem(g)
+    m = D3Measure()
+    @test size_reduction(p, m, cl, vs) == measure(p, m) - measure(first(apply_branch(p, cl, vs)), m)
+
+    edges = [(1, 4), (1, 5), (3, 4), (2, 5), (4, 5), (1, 6), (2, 7), (3, 8)]
+    example_g = SimpleGraph(Graphs.SimpleEdge.(edges))
+    p = MISProblem(example_g)
+    cl = Clause(bit"11111", bit"10000")
+    vs = collect(1:5)
+    m = D3Measure()
+    @test size_reduction(p, m, cl, vs) == measure(p, m) - measure(first(apply_branch(p, cl, vs)), m)
+end
+
+
+@testset "covered_by" begin
+    tbl = BranchingTable(9, [
+        [[0,0,0,0,0,1,1,0,0], [0,0,0,0,0,0,1,1,0]],
+        [[0,0,0,0,1,1,1,0,0]],
+        [[0,0,1,1,0,0,0,0,1], [0,0,1,1,0,1,0,0,0], [0,0,1,1,0,0,0,1,0]],
+        [[0,0,1,1,1,0,0,0,1], [0,0,1,1,1,1,0,0,0]],
+        [[0,1,0,0,0,0,1,1,0]],
+        [[0,1,0,1,1,0,0,0,1]],
+        [[0,1,1,0,1,0,0,0,1]],
+        [[0,1,1,1,0,0,0,0,1], [0,1,1,1,0,0,0,1,0]],
+        [[0,1,1,1,1,0,0,0,1]],
+        [[1,0,0,0,0,0,1,1,0]],
+        [[1,0,0,1,1,0,0,0,1]],
+        [[1,0,1,0,1,0,0,0,1]],
+        [[1,0,1,1,0,0,0,0,1], [1,0,1,1,0,0,0,1,0]],
+        [[1,0,1,1,1,0,0,0,1]],
+        [[1,1,0,0,0,0,1,1,0]],
+        [[1,1,0,1,1,0,0,0,1]],
+        [[1,1,1,0,1,0,0,0,1]],
+        [[1,1,1,1,0,0,0,0,1], [1,1,1,1,0,0,0,1,0]],
+        [[1,1,1,1,1,0,0,0,1]]
+    ])
+    clauses = OptimalBranchingCore.candidate_clauses(tbl)
+    Δρ = [count_ones(c.mask) for c in clauses]
+    result_ip = OptimalBranchingCore.minimize_γ(tbl, clauses, Δρ, IPSolver(max_itr = 10, verbose = false))
+    @test OptimalBranchingCore.covered_by(tbl, result_ip.optimal_rule)
+
+    p = MISProblem(random_regular_graph(20, 3))
+    cls = OptimalBranchingCore.bit_clauses(tbl)
+    res = OptimalBranchingCore.greedymerge(cls, p, [1, 2, 3, 4, 5], D3Measure())
+    @test OptimalBranchingCore.covered_by(tbl, res.optimal_rule)
+end
\ No newline at end of file