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    diff --git a/www/_sources/tutorial/ja/graph_coloring.md b/www/_sources/tutorial/ja/graph_coloring.md new file mode 100644 index 0000000..52905a2 --- /dev/null +++ b/www/_sources/tutorial/ja/graph_coloring.md @@ -0,0 +1,189 @@ +# グラフ彩色問題 + +こちらでは、[Lucas, 2014, "Ising formulations of many NP problems"](https://doi.org/10.3389/fphy.2014.00005)の 6.1. Graph Coloring を OpenJij と [JijModeling](https://www.ref.documentation.jijzept.com/jijmodeling/)、そして[JijModeling transpiler](https://www.ref.documentation.jijzept.com/jijmodeling-transpiler/) を用いて解く方法をご紹介します。 + +## 概要: グラフ彩色問題とは + +グラフ彩色問題とは、あるグラフ上の辺で繋がれた頂点どうしを異なる色になるように彩色する問題です。NP完全な問題として知られています。 + +### 具体例 + +下図のように6個の頂点といくつかの辺からなる無向グラフが与えられたとしましょう。 + +![](../../assets/graph_coloring_01.png) + +これを3色で全ての頂点を塗り分けると以下のようになります。 + +![](../../assets/graph_coloring_02.png) + +全ての辺において、その両端に位置する頂点は異なる色で塗り分けられていることがわかります。 + +### 問題の一般化 + +それではこの問題を一般化し、数式で表現してみましょう。無向グラフ$G = (V, E)$を、辺で結ばれた頂点の色が重複しないように$N$色で塗り分けることを考えます。 +頂点の色分けをバイナリ変数$x_{v, n}$で表すことにします。$v$番目の頂点を$n$の色で塗り分けるとき、$x_{v, n} = 1$、それ以外では$x_{v, n} = 0$となります。 + +**制約: 頂点はどれか一色で塗り分けなければならない** + +例えば、青色と緑色の2色で1つの頂点を塗ることは許されません。これを数式で表現すると、以下のようになります。 + +$$ +\sum_{n=0}^{N-1} x_{v, n} = 1 \quad (\forall n \in \{ 0, 1, \dots, N-1 \}) \tag{1} +$$ + +**目的関数: 同じ色の頂点を両端に持つ辺の数を最小にする** + +グラフ彩色問題の問題設定から、全ての辺の両端の頂点が異なる色で塗り分けられる必要があります。これを数式で表現すると + +$$ +\min \quad \sum_{n=0}^{N-1} \sum_{(uv) \in E} x_{u, n} x_{v, n} \tag{2} +$$ + +もし、全ての辺の両端の頂点が異なる色で塗り分けられているなら、この目的関数値はゼロとなります。 + +## JijModelingによるモデル構築 + +### ナップサック問題で用いる変数を定義 + +式(1), (2)で用いられている変数を、以下のようにして定義しましょう。 + +```python +import jijmodeling as jm + + +# define variables +V = jm.Placeholder('V') +E = jm.Placeholder('E', dim=2) +N = jm.Placeholder('N') +x = jm.Binary('x', shape=(V, N)) +n = jm.Element('i', (0, N)) +v = jm.Element('v', (0, V)) +e = jm.Element('e', E) +``` + +`V=jm.Placeholder('V')`でグラフの頂点数、`E=jm.Placeholder('E', dim=2)`でグラフの辺集合を定義します。`N=jm.Placeholder('N')`でグラフを塗り分ける色数を定義し、その`V, N`を用いてバイナリ変数$x_{v, n}$を`x=jm.Binary('x', shape=(V, N))`のように定義します。`n, v`はバイナリ変数の添字に用いる変数です。最後の`e`は辺を表す変数です。`e[0], e[1]`が辺`e`の両端に位置する頂点となります。すなわち$(uv) = (e[0] e[1])$です。 + +### 制約の追加 + +式(1)を制約として実装します。 + +```python +# set problem +problem = jm.Problem('Graph Coloring') +# set one-hot constraint that each vertex has only one color +const = x[v, :] +problem += jm.Constraint('color', const==1, forall=v) +``` + +問題を作成し、そこに制約を追加しましょう。`x[v, :]`とすることで`Sum(n, x[v, n])`を簡潔に実装することができます。 + +### 目的関数の追加 + +式(2)の目的関数を実装しましょう。 + +```python +# set objective function: minimize edges whose vertices connected by edges are the same color +sum_list = [n, e] +problem += jm.Sum(sum_list, x[e[0], n]*x[e[1], n]) +``` + +`sum_list=[n, e], jm.Sum(sum_list, ...)`とすることで、$\sum_n \sum_e$を表現することができます。`x[e[0], n]`は$x_{e[0], n}$、`x[e[1], n]`は$x_{e[1], n}$を表していいます。 + +実際に実装された数式をJupyter Notebookで表示してみましょう。 + +![](../../assets/graph_coloring_03.png) + +### インスタンスの作成 + +実際にグラフ彩色を行うグラフを設定しましょう。 + +```python +# set the number of vertices +inst_V = 12 +# set the number of colors +inst_N = 4 +# create a random graph +inst_G = nx.gnp_random_graph(inst_V, 0.4) +# get information of edges +inst_E = [list(edge) for edge in inst_G.edges] +instance_data = {'V': inst_V, 'N': inst_N, 'E': inst_E, 'G': inst_G} +``` + +今回はグラフの頂点数を12個、グラフを塗り分ける色数を4つとします。 + +### 未定乗数の設定 + +グラフ彩色問題には制約が一つあります。よってその制約の重みを設定する必要があります。 +先程の`Constraint`部分で付けた名前と一致させるように、辞書型を用いて設定を行います。 + +```python +# set multipliers +lam1 = 1.0 +multipliers = {'color': lam1} +``` + +### JijModeling transpilerによるPyQUBOへの変換 + +ここまで行われてきた実装は、全てJijModelingによるものでした。 +これを[PyQUBO](https://pyqubo.readthedocs.io/en/latest/)に変換することで、OpenJijはもちろん、他のソルバーを用いた組合せ最適化計算を行うことが可能になります。 + +```python +from jijmodeling.transpiler.pyqubo import to_pyqubo + +# convert to pyqubo +pyq_model, pyq_chache = to_pyqubo(problem, instance_data, {}) +qubo, bias = pyq_model.compile().to_qubo(feed_dict=multipliers) +``` + +JijModelingで作成された`problem`、そして先ほど値を設定した`instance_data`を引数として、`to_pyqubo`によりPyQUBOモデルを作成します。次にそれをコンパイルすることで、OpenJijなどで計算が可能なQUBOモデルにします。 + +### OpenJijによる最適化計算の実行 + +今回はOpenJijのシミュレーテッド・アニーリングを用いて、最適化問題を解くことにします。 +それには以下のようにします。 + +```python +# set sampler +sampler = oj.SASampler() +# solve problem +response = sampler.sample_qubo(qubo) +``` + +`SASampler`を設定し、そのサンプラーに先程作成したQUBOモデルの`qubo`を入力することで、計算結果が得られます。 + +### デコードと解の表示 + +返された計算結果をデコードし、解析を行いやすくします。 + +```python +# decode solution +result = pyq_chache.decode(response) +``` + +このようにして得られた結果から、グラフ彩色された結果を見てみましょう。 + +```python +# get indices of x = 1 +indices, _, _ = result.record.solution['x'][0] +# get vertex number and color +vertices, colors = indices +# sort lists by vertex number +zip_lists = zip(vertices, colors) +zip_sort = sorted(zip_lists) +sorted_vertices, sorted_colors = zip(*zip_sort) +# initialize vertex color list +node_colors = [-1] * len(vertices) +# set color list for visualization +colorlist = ['gold', 'violet', 'limegreen', 'darkorange'] +# set vertex color list +for i, j in zip(sorted_vertices, sorted_colors): + node_colors[i] = colorlist[j] +# make figure +fig = plt.figure() +nx.draw_networkx(instance_data['G'], node_color=node_colors, with_labels=True) +fig.savefig('graph_coloring.png') +``` + +すると以下のような画像を得ます。 + +![](../../assets/graph_coloring_04.png) diff --git a/www/_sources/tutorial/ja/knapsack.md b/www/_sources/tutorial/ja/knapsack.md new file mode 100644 index 0000000..6513346 --- /dev/null +++ b/www/_sources/tutorial/ja/knapsack.md @@ -0,0 +1,187 @@ +# ナップサック問題 + +こちらでは、[Lucas, 2014, "Ising formulations of many NP problems"](https://doi.org/10.3389/fphy.2014.00005)の 5.2. Knapsack with Integer Weights を OpenJij と [JijModeling](https://www.ref.documentation.jijzept.com/jijmodeling/)、そして[JijModeling transpiler](https://www.ref.documentation.jijzept.com/jijmodeling-transpiler/) を用いて解く方法をご紹介します。 + +## 概要: ナップサック問題とは + +ナップサック問題は、具体的には以下のような状況で最適解を求める問題です。 +最も有名なNP困難な整数計画問題の一つとして知られています。まずは具体例を考えてみましょう。 + +### 具体例 + +この問題の具体例として、以下のような物語を考えます。 + +> ある探検家がある洞窟を探検していました。しばらく洞窟の中を歩いていると、思いがけなく複数の宝物を発見しました。 + +||宝物A|宝物B|宝物C|宝物D|宝物E|宝物F| +|---|---|---|---|---|---|---| +|値段|$5000|$7000|$2000|$1000|$4000|$3000| +|重さ|800g|1000g|600g|400g|500g|300g| + +> しかし探検家の手持ちの荷物の中で宝物を運べるような袋としては、残念ながら小さなナップサックしか持ち合わせていませんでした。 +> このナップサックには2kgの荷物しか入れることができません。探検家はこのナップサックに入れる宝物の価値をできるだけ高くしたいのですが、どの荷物を選べば最も効率的に宝物を持って帰ることができるでしょうか。 + +### 問題の一般化 + +この問題を一般化するには、ナップサックに入れる荷物$N$個の集合$\{ 0, 1, \dots, i, \dots, N-1\}$があり、各荷物が$i$をインデックスとして持っているものとして考えます。 +ナップサックに入れる各荷物$i$のコストのリスト$\bm{v}$と重さのリスト$\bm{w}$を作ることで、問題を表現することができます。 + +$$ + \bm{v} = \{v_0, v_1, \dots, v_i, \dots, v_{N-1}\} +$$ + +$$ + \bm{w} = \{w_0, w_1, \dots, w_i, \dots, w_{N-1}\} +$$ + +さらに$i$番目の荷物を選んだことを表すバイナリ変数を$x_i$としましょう。この変数は$i$をナップサックに入れるとき$x_i = 1$、入れないとき$x_i = 0$となるような変数です。最後にナップサックの最大容量を$W$とします。 +最大化したいのは、ナップサックに入れる荷物の合計です。よってこれを目的関数として表現しましょう。さらにナップサックの容量制限以下にしなければならない制約を考えると、ナップサック問題は以下のような数式で表現されます。 + +$$ + \max \ \sum_{i=0}^{N-1} v_i x_i \tag{1} +$$ + +$$ + \mathrm{s.t.} \quad \sum_{i=0}^{N-1} w_i x_i \leq W \tag{2} +$$ + +$$ + x_i \in \{0, 1\} \quad (\forall i \in \{0, 1, \dots, N-1\}) \tag{3} +$$ + +## JijModelingによるモデル構築 + +### ナップサック問題で用いる変数を定義 + +式(1), (2), (3)で用いられている変数$\bm{v}, \bm{w}, N, W, x_i, i$を、以下のようにして定義しましょう。 + +```python +import jijmodeling as jm + + +# define variables +v = jm.Placeholder('v', dim=1) +N = v.shape[0] +w = jm.Placeholder('w', shape=(N)) +W = jm.Placeholder('W') +x = jm.Binary('x', shape=(N)) +i = jm.Element('i', (0, N)) +``` + +`v = jm.Placeholder('v', dim=1)`でナップサックに入れる物の価値を表す一次元のリストを宣言し、その具体的な要素数を`N`としています。その`N`を用いて、ナップサックに入れる物の重さを表す一次元のリストを`w = jm.Placeholder('w', shape=(N))`のように定義することで、`v`と`w`が同じ長さであることを保証できます。`W = jm.Placeholder('W')`ではナップサックの容量制限を表す$W$を定義しています。続く`x = jm.Binary('x', shape=(N))`により、`v, w`と同じ長さのバイナリ変数リスト`x`を定義します。最後に`i = jm.Element('i', (0, N))`は$v_i, w_i, x_i$の添字を定義しており、これは$0\leq i < N$の範囲の整数であることを表しています。 + +### 目的関数の追加 + +式(1)を目的関数として実装します。 + +```python +# set problem +problem = jm.Problem('Knapsack') +# set objective function +obj = - jm.Sum(i, v[i]*x[i]) +problem += obj +``` + +問題を作成し、そこに目的関数を追加しましょう。`Sum(i, 数式)`とすることで、数式部分の総和を添字`i`に対して行うことができます。 + +### 制約の追加 + +式(2)の制約を実装しましょう。 + +```python +# set total weight constraint +const = jm.Sum(i, w[i]*x[i]) +problem += jm.Constraint('weight', const<=W) +``` + +`Constraint(制約名, 制約式)`とすることで、制約式に適当な制約名を付与することができます。 +実際に実装された数式をJupyter Notebookで表示してみましょう。 + +![](../../assets/knapsack_01.png) + +### インスタンスの作成 + +先程の冒険家の物語を、インスタンスとして設定しましょう。ただし宝物の価値は$1000で規格化、さらに宝物の重さも100gで規格化された値を用います。 + +```python +# set a list of values & weights +inst_v = [5, 7, 2, 1, 4, 3] +inst_w = [8, 10, 6, 4, 5, 3] +# set maximum weight +inst_W = 20 +instance_data = {'v': inst_v, 'w': inst_w, 'W': inst_W} +``` + +### 未定乗数の設定 + +このナップサック問題には制約が一つあります。よってその制約の重みを設定する必要があります。 +先程の`Constraint`部分で付けた名前と一致させるように、辞書型を用いて設定を行います。 + +```python +# set multipliers +lam1 = 1.0 +multipliers = {'weight': lam1} +``` + +### JijModeling transpilerによるPyQUBOへの変換 + +ここまで行われてきた実装は、全てJijModelingによるものでした。 +これを[PyQUBO](https://pyqubo.readthedocs.io/en/latest/)に変換することで、OpenJijはもちろん、他のソルバーを用いた組合せ最適化計算を行うことが可能になります。 + +```python +from jijmodeling.transpiler.pyqubo import to_pyqubo + +# convert to pyqubo +pyq_model, pyq_chache = to_pyqubo(problem, instance_data, {}) +qubo, bias = pyq_model.compile().to_qubo(feed_dict=multipliers) +``` + +JijModelingで作成された`problem`、そして先ほど値を設定した`instance_data`を引数として、`to_pyqubo`によりPyQUBOモデルを作成します。次にそれをコンパイルすることで、OpenJijなどで計算が可能なQUBOモデルにします。 + +### OpenJijによる最適化計算の実行 + +今回はOpenJijのシミュレーテッド・アニーリングを用いて、最適化問題を解くことにします。 +それには以下のようにします。 + +```python +# set sampler +sampler = oj.SASampler() +# solve problem +response = sampler.sample_qubo(qubo) +``` + +`SASampler`を設定し、そのサンプラーに先程作成したQUBOモデルの`qubo`を入力することで、計算結果が得られます。 + +### デコードと解の表示 + +返された計算結果をデコードし、解析を行いやすくします。 + +```python +# decode solution +result = pyq_chache.decode(response) +``` + +このようにして得られた結果から、実際にどの宝物をナップサックに入れたのかを見てみましょう。 + +```python +indices, _, _ = result.record.solution['x'][0] +inst_w = instance_data['w'] +sum_w = 0 +for i in indices[0]: + sum_w += inst_w[i] +print('Indices of x = 1: ', indices[0]) +print('Value of objective function: ', result.evaluation.objective) +print('Value of constraint term: ', result.evaluation.constraint_violations['weight']) +print('Total weight: ', sum_w) +``` + +すると以下のような出力を得ます。 + +```bash +Indices of x = 1: [0, 3, 4, 5] +Value of objective function: [-13.0] +Value of constraint term: [0.0] +Total weight: 20 +``` + +目的関数の値にマイナスをかけたものが、実際にナップサックに入れた宝物の価値の合計です。また`.evaluation.constarint_violations[制約名]`とすることで、その制約がどれだけ満たされていないかを取得することができます。 diff --git a/www/genindex-A.html b/www/genindex-A.html index 4fad9de..f5fdc79 100644 --- a/www/genindex-A.html +++ b/www/genindex-A.html @@ -369,9 +369,19 @@

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to OpenJij\u2019s documentation!","openjij","openjij.model.chimera_model","openjij.model","openjij.model.king_graph","openjij.model.model","openjij.sampler.chimera_gpu.base_gpu_chimera","openjij.sampler.chimera_gpu.gpu_sa_sampler","openjij.sampler.chimera_gpu.gpu_sqa_sampler","openjij.sampler.chimera_gpu","openjij.sampler.csqa_sampler","openjij.sampler","openjij.sampler.response","openjij.sampler.sa_sampler","openjij.sampler.sampler","openjij.sampler.sqa_sampler","openjij.utils.benchmark","openjij.utils.decorator","openjij.utils.graph_utils","openjij.utils","openjij.utils.res_convertor","openjij.utils.time_measure","openjij.variable_type","1. 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    OpenJij Book

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    1. OpenJij Tutorialsopenjij.variable_type

     - +

    next

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    1. OpenJij チュートリアル

    +

    1. グラフ彩色問題

     diff --git a/www/tutorial/ja/graph_coloring.html b/www/tutorial/ja/graph_coloring.html new file mode 100644 index 0000000..e1cdb93 --- /dev/null +++ b/www/tutorial/ja/graph_coloring.html @@ -0,0 +1,950 @@ + + + + + + + + + 1. グラフ彩色問題 — OpenJij Book + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    グラフ彩色問題

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    1. グラフ彩色問題#

    +

    こちらでは、Lucas, 2014, “Ising formulations of many NP problems”の 6.1. Graph Coloring を OpenJij と JijModeling、そしてJijModeling transpiler を用いて解く方法をご紹介します。

    +
    +

    1.1. 概要: グラフ彩色問題とは#

    +

    グラフ彩色問題とは、あるグラフ上の辺で繋がれた頂点どうしを異なる色になるように彩色する問題です。NP完全な問題として知られています。

    +
    +

    1.1.1. 具体例#

    +

    下図のように6個の頂点といくつかの辺からなる無向グラフが与えられたとしましょう。

    +

    +

    これを3色で全ての頂点を塗り分けると以下のようになります。

    +

    +

    全ての辺において、その両端に位置する頂点は異なる色で塗り分けられていることがわかります。

    +
    +
    +

    1.1.2. 問題の一般化#

    +

    それではこの問題を一般化し、数式で表現してみましょう。無向グラフG=(V,E)G = (V, E)を、辺で結ばれた頂点の色が重複しないようにNN色で塗り分けることを考えます。 +頂点の色分けをバイナリ変数xv,nx_{v, n}で表すことにします。vv番目の頂点をnnの色で塗り分けるとき、xv,n=1x_{v, n} = 1、それ以外ではxv,n=0x_{v, n} = 0となります。

    +

    制約: 頂点はどれか一色で塗り分けなければならない

    +

    例えば、青色と緑色の2色で1つの頂点を塗ることは許されません。これを数式で表現すると、以下のようになります。

    +
    +∑n=0N−1xv,n=1(∀n∈{0,1,…,N−1})(1) +\sum_{n=0}^{N-1} x_{v, n} = 1 \quad (\forall n \in \{ 0, 1, \dots, N-1 \}) \tag{1} +

    目的関数: 同じ色の頂点を両端に持つ辺の数を最小にする

    +

    グラフ彩色問題の問題設定から、全ての辺の両端の頂点が異なる色で塗り分けられる必要があります。これを数式で表現すると

    +
    +min⁡∑n=0N−1∑(uv)∈Exu,nxv,n(2) +\min \quad \sum_{n=0}^{N-1} \sum_{(uv) \in E} x_{u, n} x_{v, n} \tag{2} +

    もし、全ての辺の両端の頂点が異なる色で塗り分けられているなら、この目的関数値はゼロとなります。

    +
    +
    +
    +

    1.2. JijModelingによるモデル構築#

    +
    +

    1.2.1. ナップサック問題で用いる変数を定義#

    +

    式(1), (2)で用いられている変数を、以下のようにして定義しましょう。

    +
    import jijmodeling as jm
+
+
+# define variables
+V = jm.Placeholder('V')
+E = jm.Placeholder('E', dim=2)
+N = jm.Placeholder('N')
+x = jm.Binary('x', shape=(V, N))
+n = jm.Element('i', (0, N))
+v = jm.Element('v', (0, V))
+e = jm.Element('e', E)
+
    +
    +

    V=jm.Placeholder('V')でグラフの頂点数、E=jm.Placeholder('E', dim=2)でグラフの辺集合を定義します。N=jm.Placeholder('N')でグラフを塗り分ける色数を定義し、そのV, Nを用いてバイナリ変数xv,nx_{v, n}をx=jm.Binary('x', shape=(V, N))のように定義します。n, vはバイナリ変数の添字に用いる変数です。最後のeは辺を表す変数です。e[0], e[1]が辺eの両端に位置する頂点となります。すなわち(uv)=(e[0]e[1])(uv) = (e[0] e[1])です。

    +
    +
    +

    1.2.2. 制約の追加#

    +

    式(1)を制約として実装します。

    +
    # set problem
+problem = jm.Problem('Graph Coloring')
+# set one-hot constraint that each vertex has only one color
+const = x[v, :]
+problem += jm.Constraint('color', const==1, forall=v)
+
    +
    +

    問題を作成し、そこに制約を追加しましょう。x[v, :]とすることでSum(n, x[v, n])を簡潔に実装することができます。

    +
    +
    +

    1.2.3. 目的関数の追加#

    +

    式(2)の目的関数を実装しましょう。

    +
    # set objective function: minimize edges whose vertices connected by edges are the same color
+sum_list = [n, e]
+problem += jm.Sum(sum_list, x[e[0], n]*x[e[1], n])
+
    +
    +

    sum_list=[n, e], jm.Sum(sum_list, ...)とすることで、∑n∑e\sum_n \sum_eを表現することができます。x[e[0], n]はxe[0],nx_{e[0], n}、x[e[1], n]はxe[1],nx_{e[1], n}を表していいます。

    +

    実際に実装された数式をJupyter Notebookで表示してみましょう。

    +

    +
    +
    +

    1.2.4. インスタンスの作成#

    +

    実際にグラフ彩色を行うグラフを設定しましょう。

    +
    # set the number of vertices
+inst_V = 12
+# set the number of colors
+inst_N = 4
+# create a random graph
+inst_G = nx.gnp_random_graph(inst_V, 0.4)
+# get information of edges
+inst_E = [list(edge) for edge in inst_G.edges]
+instance_data = {'V': inst_V, 'N': inst_N, 'E': inst_E, 'G': inst_G}
+
    +
    +

    今回はグラフの頂点数を12個、グラフを塗り分ける色数を4つとします。

    +
    +
    +

    1.2.5. 未定乗数の設定#

    +

    グラフ彩色問題には制約が一つあります。よってその制約の重みを設定する必要があります。 +先程のConstraint部分で付けた名前と一致させるように、辞書型を用いて設定を行います。

    +
    # set multipliers
+lam1 = 1.0
+multipliers = {'color': lam1}    
+
    +
    +
    +
    +

    1.2.6. JijModeling transpilerによるPyQUBOへの変換#

    +

    ここまで行われてきた実装は、全てJijModelingによるものでした。 +これをPyQUBOに変換することで、OpenJijはもちろん、他のソルバーを用いた組合せ最適化計算を行うことが可能になります。

    +
    from jijmodeling.transpiler.pyqubo import to_pyqubo
+
+# convert to pyqubo
+pyq_model, pyq_chache = to_pyqubo(problem, instance_data, {})
+qubo, bias = pyq_model.compile().to_qubo(feed_dict=multipliers)
+
    +
    +

    JijModelingで作成されたproblem、そして先ほど値を設定したinstance_dataを引数として、to_pyquboによりPyQUBOモデルを作成します。次にそれをコンパイルすることで、OpenJijなどで計算が可能なQUBOモデルにします。

    +
    +
    +

    1.2.7. OpenJijによる最適化計算の実行#

    +

    今回はOpenJijのシミュレーテッド・アニーリングを用いて、最適化問題を解くことにします。 +それには以下のようにします。

    +
    # set sampler
+sampler = oj.SASampler()
+# solve problem
+response = sampler.sample_qubo(qubo)
+
    +
    +

    SASamplerを設定し、そのサンプラーに先程作成したQUBOモデルのquboを入力することで、計算結果が得られます。

    +
    +
    +

    1.2.8. デコードと解の表示#

    +

    返された計算結果をデコードし、解析を行いやすくします。

    +
    # decode solution
+result = pyq_chache.decode(response)
+
    +
    +

    このようにして得られた結果から、グラフ彩色された結果を見てみましょう。

    +
    # get indices of x = 1
+indices, _, _ = result.record.solution['x'][0]
+# get vertex number and color
+vertices, colors = indices
+# sort lists by vertex number
+zip_lists = zip(vertices, colors)
+zip_sort = sorted(zip_lists)
+sorted_vertices, sorted_colors = zip(*zip_sort)
+# initialize vertex color list
+node_colors = [-1] * len(vertices)
+# set color list for visualization
+colorlist = ['gold', 'violet', 'limegreen', 'darkorange']
+# set vertex color list
+for i, j in zip(sorted_vertices, sorted_colors):
+    node_colors[i] = colorlist[j]
+# make figure
+fig = plt.figure()
+nx.draw_networkx(instance_data['G'], node_color=node_colors, with_labels=True)
+fig.savefig('graph_coloring.png')
+
    +
    +

    すると以下のような画像を得ます。

    +

    +
    +
    +
    + + + + +
    + +
    + +
    +
    +
    +

    + + By Jij Inc.
    + + © Copyright 2022.
    +

    +
    +
    + +
    + + +
    +
    + + + + + + + \ No newline at end of file diff --git a/www/tutorial/ja/index.html b/www/tutorial/ja/index.html index f867961..21bffd9 100644 --- a/www/tutorial/ja/index.html +++ b/www/tutorial/ja/index.html @@ -6,7 +6,7 @@ - 1. OpenJij チュートリアル — OpenJij Book + 2. OpenJij チュートリアル — OpenJij Book @@ -60,7 +60,8 @@ - + + @@ -370,9 +371,19 @@

    OpenJij Book

    @@ -555,7 +566,7 @@

    OpenJij チュートリアル

    -

    1. OpenJij チュートリアル#

    +

    2. OpenJij チュートリアル#

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + + + + + +
    +
    + + + + + + + + + + +
    + +
    + +
    + + + + +
    +
    + + + + +
    +
    + + +
    + +
    + +
    +
    + +
    + +
    + +
    +
    + + +
    +
    + + +
    +
    + Contents +
    + +
    +
    +
    +
    + +
    +

    ナップサック問題

    + +
    + +
    +
    +
    + +
    + +
    +

    3. ナップサック問題#

    +

    こちらでは、Lucas, 2014, “Ising formulations of many NP problems”の 5.2. Knapsack with Integer Weights を OpenJij と JijModeling、そしてJijModeling transpiler を用いて解く方法をご紹介します。

    +
    +

    3.1. 概要: ナップサック問題とは#

    +

    ナップサック問題は、具体的には以下のような状況で最適解を求める問題です。 +最も有名なNP困難な整数計画問題の一つとして知られています。まずは具体例を考えてみましょう。

    +
    +

    3.1.1. 具体例#

    +

    この問題の具体例として、以下のような物語を考えます。

    +
    +

    ある探検家がある洞窟を探検していました。しばらく洞窟の中を歩いていると、思いがけなく複数の宝物を発見しました。

    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

    宝物A

    宝物B

    宝物C

    宝物D

    宝物E

    宝物F

    値段

    $5000

    $7000

    $2000

    $1000

    $4000

    $3000

    重さ

    800g

    1000g

    600g

    400g

    500g

    300g

    +
    +

    しかし探検家の手持ちの荷物の中で宝物を運べるような袋としては、残念ながら小さなナップサックしか持ち合わせていませんでした。 +このナップサックには2kgの荷物しか入れることができません。探検家はこのナップサックに入れる宝物の価値をできるだけ高くしたいのですが、どの荷物を選べば最も効率的に宝物を持って帰ることができるでしょうか。

    +
    +
    +
    +

    3.1.2. 問題の一般化#

    +

    この問題を一般化するには、ナップサックに入れる荷物NN個の集合{0,1,…,i,…,N−1}\{ 0, 1, \dots, i, \dots, N-1\}があり、各荷物がiiをインデックスとして持っているものとして考えます。
    +ナップサックに入れる各荷物iiのコストのリストv\bm{v}と重さのリストw\bm{w}を作ることで、問題を表現することができます。

    +
    +v={v0,v1,…,vi,…,vN−1} + \bm{v} = \{v_0, v_1, \dots, v_i, \dots, v_{N-1}\} +
    +w={w0,w1,…,wi,…,wN−1} + \bm{w} = \{w_0, w_1, \dots, w_i, \dots, w_{N-1}\} +

    さらにii番目の荷物を選んだことを表すバイナリ変数をxix_iとしましょう。この変数はiiをナップサックに入れるときxi=1x_i = 1、入れないときxi=0x_i = 0となるような変数です。最後にナップサックの最大容量をWWとします。
    +最大化したいのは、ナップサックに入れる荷物の合計です。よってこれを目的関数として表現しましょう。さらにナップサックの容量制限以下にしなければならない制約を考えると、ナップサック問題は以下のような数式で表現されます。

    +
    +max⁡ ∑i=0N−1vixi(1) + \max \ \sum_{i=0}^{N-1} v_i x_i \tag{1} +
    +s.t.∑i=0N−1wixi≤W(2) + \mathrm{s.t.} \quad \sum_{i=0}^{N-1} w_i x_i \leq W \tag{2} +
    +xi∈{0,1}(∀i∈{0,1,…,N−1})(3) + x_i \in \{0, 1\} \quad (\forall i \in \{0, 1, \dots, N-1\}) \tag{3} +
    +
    +
    +

    3.2. JijModelingによるモデル構築#

    +
    +

    3.2.1. ナップサック問題で用いる変数を定義#

    +

    式(1), (2), (3)で用いられている変数v,w,N,W,xi,i\bm{v}, \bm{w}, N, W, x_i, iを、以下のようにして定義しましょう。

    +
    import jijmodeling as jm
+
+
+# define variables
+v = jm.Placeholder('v', dim=1)
+N = v.shape[0]
+w = jm.Placeholder('w', shape=(N))
+W = jm.Placeholder('W')
+x = jm.Binary('x', shape=(N))
+i = jm.Element('i', (0, N))
+
    +
    +

    v = jm.Placeholder('v', dim=1)でナップサックに入れる物の価値を表す一次元のリストを宣言し、その具体的な要素数をNとしています。そのNを用いて、ナップサックに入れる物の重さを表す一次元のリストをw = jm.Placeholder('w', shape=(N))のように定義することで、vとwが同じ長さであることを保証できます。W = jm.Placeholder('W')ではナップサックの容量制限を表すWWを定義しています。続くx = jm.Binary('x', shape=(N))により、v, wと同じ長さのバイナリ変数リストxを定義します。最後にi = jm.Element('i', (0, N))はvi,wi,xiv_i, w_i, x_iの添字を定義しており、これは0≤i<N0\leq i < Nの範囲の整数であることを表しています。

    +
    +
    +

    3.2.2. 目的関数の追加#

    +

    式(1)を目的関数として実装します。

    +
    # set problem
+problem = jm.Problem('Knapsack')    
+# set objective function
+obj = - jm.Sum(i, v[i]*x[i])
+problem += obj
+
    +
    +

    問題を作成し、そこに目的関数を追加しましょう。Sum(i, 数式)とすることで、数式部分の総和を添字iに対して行うことができます。

    +
    +
    +

    3.2.3. 制約の追加#

    +

    式(2)の制約を実装しましょう。

    +
    # set total weight constraint
+const = jm.Sum(i, w[i]*x[i])
+problem += jm.Constraint('weight', const<=W)
+
    +
    +

    Constraint(制約名, 制約式)とすることで、制約式に適当な制約名を付与することができます。
    +実際に実装された数式をJupyter Notebookで表示してみましょう。

    +

    +
    +
    +

    3.2.4. インスタンスの作成#

    +

    先程の冒険家の物語を、インスタンスとして設定しましょう。ただし宝物の価値は$1000で規格化、さらに宝物の重さも100gで規格化された値を用います。

    +
    # set a list of values & weights 
+inst_v = [5, 7, 2, 1, 4, 3]
+inst_w = [8, 10, 6, 4, 5, 3]
+# set maximum weight
+inst_W = 20
+instance_data = {'v': inst_v, 'w': inst_w, 'W': inst_W}    
+
    +
    +
    +
    +

    3.2.5. 未定乗数の設定#

    +

    このナップサック問題には制約が一つあります。よってその制約の重みを設定する必要があります。 +先程のConstraint部分で付けた名前と一致させるように、辞書型を用いて設定を行います。

    +
    # set multipliers
+lam1 = 1.0
+multipliers = {'weight': lam1}    
+
    +
    +
    +
    +

    3.2.6. JijModeling transpilerによるPyQUBOへの変換#

    +

    ここまで行われてきた実装は、全てJijModelingによるものでした。 +これをPyQUBOに変換することで、OpenJijはもちろん、他のソルバーを用いた組合せ最適化計算を行うことが可能になります。

    +
    from jijmodeling.transpiler.pyqubo import to_pyqubo
+
+# convert to pyqubo
+pyq_model, pyq_chache = to_pyqubo(problem, instance_data, {})
+qubo, bias = pyq_model.compile().to_qubo(feed_dict=multipliers)
+
    +
    +

    JijModelingで作成されたproblem、そして先ほど値を設定したinstance_dataを引数として、to_pyquboによりPyQUBOモデルを作成します。次にそれをコンパイルすることで、OpenJijなどで計算が可能なQUBOモデルにします。

    +
    +
    +

    3.2.7. OpenJijによる最適化計算の実行#

    +

    今回はOpenJijのシミュレーテッド・アニーリングを用いて、最適化問題を解くことにします。 +それには以下のようにします。

    +
    # set sampler
+sampler = oj.SASampler()
+# solve problem
+response = sampler.sample_qubo(qubo)
+
    +
    +

    SASamplerを設定し、そのサンプラーに先程作成したQUBOモデルのquboを入力することで、計算結果が得られます。

    +
    +
    +

    3.2.8. デコードと解の表示#

    +

    返された計算結果をデコードし、解析を行いやすくします。

    +
    # decode solution
+result = pyq_chache.decode(response)
+
    +
    +

    このようにして得られた結果から、実際にどの宝物をナップサックに入れたのかを見てみましょう。

    +
    indices, _, _ = result.record.solution['x'][0]
+inst_w = instance_data['w']
+sum_w = 0
+for i in indices[0]:
+    sum_w += inst_w[i]
+print('Indices of x = 1: ', indices[0])
+print('Value of objective function: ', result.evaluation.objective)
+print('Value of constraint term: ', result.evaluation.constraint_violations['weight'])
+print('Total weight: ', sum_w)
+
    +
    +

    すると以下のような出力を得ます。

    +
    Indices of x = 1:  [0, 3, 4, 5]
+Value of objective function:  [-13.0]
+Value of constraint term:  [0.0]
+Total weight:  20
+
    +
    +

    目的関数の値にマイナスをかけたものが、実際にナップサックに入れた宝物の価値の合計です。また.evaluation.constarint_violations[制約名]とすることで、その制約がどれだけ満たされていないかを取得することができます。

    +
    +
    +
    + + + + +
    + +
    + +
    +
    +
    +

    + + By Jij Inc.
    + + © Copyright 2022.
    +

    +
    +
    + +
    + + +
    +
    + + + + + + + \ No newline at end of file