intractibility 1 (8)
numerical problems
my hobby
subset sum
\(\SubsetSum\): Given natural numbers \(w_1, \ldots, w_n\) and an integer \(W\), is there a subset that adds up to exactly \(W\)?
Ex: \(\{ 215, 215, 275, 275, 355, 355, 420, 420, 580, 655 \}\), \(W=1505\)
Yes: \(215+355+355=1505\)
Remark: With arithmetic problems, input integers are encoded in binary. Poly-time reducion must be polynomial in binary encoding.
subset sum
Theorem: \(\ThreeSat \leq_P \SubsetSum\)
Pf: Given an instance \(\Phi\) of \(\ThreeSat\), we construct an instance of \(\SubsetSum\) that has a solution iff \(\Phi\) is satisfied.
3-sat reduces to subset sum
Construction: Given \(\ThreeSat\) instance \(\Phi\) with \(n\) variables and \(k\) clauses, form \(2n + 2k\) decimal integers, each having \(n+k\) digits:
- Include one digit for each variable \(x_i\) and one digit for each clause \(C_j\)
- Include two numbers for each variable \(x_i\)
- Include two numbers for each clause \(C_j\)
- Sum of each \(x_i\) digit is \(1\); sum of each \(C_j\) is \(4\)
Key property: No carries possible ⇒ each digit yields one equation.
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\[\begin{eqnarray} C_1 & = & \overline{x_1} \vee x_2 \vee x_3 \\ C_2 & = & x_1 \vee \overline{x_2} \vee x_3 \\ C_3 & = & \overline{x_1} \vee \overline{x_2} \vee \overline{x_3} \end{eqnarray}\] |
x1 x2 x3 C1 C2 C3
x1 1 0 0 0 1 0 100010
-x1 1 0 0 1 0 1 100101
x2 0 1 0 1 0 0 10100
-x2 0 1 0 0 1 1 10011
x3 0 0 1 1 1 0 1110
-x3 0 0 1 0 0 1 1001
0 0 0 1 0 0 100
0 0 0 2 0 0 200
0 0 0 0 1 0 10
0 0 0 0 2 0 20
0 0 0 0 0 1 1
0 0 0 0 0 2 2
--------------------------
W 1 1 1 4 4 4 111444
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3-sat reduces to subset sum
Lemma: \(\Phi\) is satisfiable iff there exists a subset that sums to \(W\)
Pf ⇒:
- Suppose \(\ThreeSat\) instance \(\Phi\) has satisfying assignment \(x^*\)
- If \(x^*_i = \text{true}\), select integer in row \(x_i\); otherwise, select integer in row \(\overline{x_i}\)
- Each \(x_i\) digit sums to \(1\)
- Since \(\Phi\) is satisfiable, each \(C_j\) digit sums to at least \(1\) from \(x_i\) and \(\overline{x_i}\) rows
- Select dummy integers to make \(C_j\) digits sum to \(4\)
3-sat reduces to subset sum
Lemma: \(\Phi\) is satisfiable iff there exists a subset that sums to \(W\)
Pf ⇐:
- Suppose there exists a subset \(S^*\) that sums to \(W\)
- Digit \(x_i\) forces subset \(S^*\) to select either row \(x_i\) or row \(\overline{x_i}\) (but not both)
- If row \(x_i\) selected, assign \(x^*_i = \text{true}\); otherwise, assign \(x^*_i = \text{false}\)
- Digit \(C_j\) forces subset \(S^*\) to select at least one liter in clause ∎
Group: Subset Sum reduces to Knapsack
\(\SubsetSum\): Given a set \(X\), values \(u_i \geq 0\), and an integer \(U\), is there a subset \(S \subseteq X\) whose elements sum to exactly \(U\)?
\(\Knapsack\): Given a set \(X\), weights \(u_i \geq 0\), values \(v_i \geq 0\), a weight limit \(U\), and a target value \(V\), is there a subset \(S \subseteq X\) such that
\[ \sum_{i \in S} u_i \leq U,\quad \sum_{i \in S} v_i \geq V \]
Theorem: \(\SubsetSum \leq_P \Knapsack\)
Pf: Given instance \((w_1,\ldots,w_n,W)\) of \(\SubsetSum\), create \(\Knapsack\) instance...
poly-time reductions
Karp's 21 poly-time reductions from sat
quote
“If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in "creative leaps", no fundamental gap between solving a problem and recognizing the solution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss.
”
—Scott Aaronson