✨ feat(knapsackGreedy): First draft.

Fixes #3.

src/index.js

@@ -4,6 +4,7 @@ import integerValuesKnapsackUnbounded from './integerValuesKnapsackUnbounded';
 import integerWeightsKnapsack from './integerWeightsKnapsack';
 import integerWeightsKnapsackUnbounded from './integerWeightsKnapsackUnbounded';
 import knapsackApprox from './knapsackApprox';
+import knapsackGreedy from './knapsackGreedy';
 import knapsackUnboundedGreedy from './knapsackUnboundedGreedy';
 import orderedByDecreasingUtility from './orderedByDecreasingUtility';
 
@@ -15,6 +16,7 @@ export default {
 	integerWeightsKnapsack,
 	integerWeightsKnapsackUnbounded,
 	knapsackApprox,
+	knapsackGreedy,
 	knapsackUnboundedGreedy,
 	orderedByDecreasingUtility,
 };
@@ -26,6 +28,7 @@ export {
 	integerWeightsKnapsack,
 	integerWeightsKnapsackUnbounded,
 	knapsackApprox,
+	knapsackGreedy,
 	knapsackUnboundedGreedy,
 	orderedByDecreasingUtility,
 };

src/knapsackGreedy.js

@@ -0,0 +1,52 @@
+import assert from 'assert';
+import {map, range, sorted, filter} from '@aureooms/js-itertools';
+
+import Item from './Item';
+import orderedByDecreasingUtility from './orderedByDecreasingUtility';
+
+/**
+ * 1/2-approximation to the 0-1 knapsack problem.
+ * Runs in O(n log n) time.
+ *
+ * Let OPT' be the value of the LP relaxation.
+ * Let v_i be the values of the items ordered by utility.
+ * Let k be the largest k such that sum(v[:k]) <= W.
+ * Let t = (W-sum(w[:k])) / w_{k+1},
+ * we have t < 1 and OPT' = sum(v[:k]) + t * v_{k+1}.
+ * Hence sum(v[:k+1]) > OPT' >= OPT.
+ * By partition one of { sum(v[:k]) , v_{k+1} } is at least OPT / 2.
+ * Assuming w_i <= W for all i in [N] each of these is feasible.
+ *
+ * @param {Array} v Values.
+ * @param {Array} w Weights.
+ * @param {Number} n Size of the problem.
+ * @param {Number} W Size of the knapsack.
+ */
+const knapsackGreedy = (v, w, n, W) => {
+	assert(v.length === n);
+	assert(w.length === n);
+	assert(W >= 0);
+
+	const items = sorted(
+		orderedByDecreasingUtility,
+		filter(
+			(item) => item.w <= W,
+			map((i) => new Item(w[i], v[i]), range(n)),
+		),
+	);
+	return subroutine(W, items);
+};
+
+const subroutine = (W, items) => {
+	let value = 0;
+	let capacity = W;
+	for (const {v, w} of items) {
+		if (capacity < w) return Math.max(v, value);
+		capacity -= w;
+		value += v;
+	}
+
+	return value;
+};
+
+export default knapsackGreedy;

test/src/0-1-knapsack-problem.js

@@ -5,6 +5,7 @@ import {all, map} from '@aureooms/js-itertools';
 import {
 	integerValuesKnapsack,
 	integerWeightsKnapsack,
+	knapsackGreedy,
 	knapsackApprox,
 } from '../../src';
 
@@ -52,6 +53,10 @@ const solvers = [
 		solve: integerWeightsKnapsack,
 		hypothesis: (_, w) => all(map((x) => Number.isInteger(x), w)),
 	},
+	{
+		solve: knapsackGreedy,
+		approx: 1 / 2,
+	},
 	approx(1 / 2),
 	approx(2 / 3),
 	approx(3 / 4),