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Exam 3 Solutions: Complexity Classes, NP-Completeness, and Reductions, Exams of Insurance law

North Carolina Central University (NCCU)Insurance law

Solutions and explanations for exam questions related to complexity classes, np-completeness, and reductions. It covers concepts like p, np, np-hard, np-complete, and reduction techniques. Useful for students studying computer science or related fields.

Typology: Exams

2024/2025

Available from 02/11/2025

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C S6515 Exam 3 with verified solutions
The |!||!|Class |!||!|P |!||!|- |!||!|correct |!||!|answer |!||!|✔A |!||!|solution |!||!|may |!||!|be |!||!|found |!||!|in |!||!|polynomial |!||!|time
The |!||!|Class |!||!|NP |!||!|- |!||!|correct |!||!|answer |!||!|✔A |!||!|solution |!||!|may |!||!|be |!||!|verified |!||!|in |!||!|polynomial |!||!|time
NP-Hard |!||!|- |!||!|correct |!||!|answer |!||!|✔We |!||!|are |!||!|yet |!||!|to |!||!|find |!||!|a |!||!|polynomial |!||!|time |!||!|solution
NP-Complete |!||!|- |!||!|correct |!||!|answer |!||!|✔Both |!||!|in |!||!|NP |!||!|and |!||!|at |!||!|least |!||!|as |!||!|hard |!||!|as |!||!|NP-Hard |!||!|
problems
NP |!||!|Reduction |!||!|- |!||!|Steps |!||!|- |!||!|correct |!||!|answer |!||!|✔1. |!||!|Demonstrate |!||!|that |!||!|problem |!||!|B |!||!|is |!||!|in
|!||!|the |!||!|class |!||!|of |!||!|NP |!||!|Problems
- |!||!|validate |!||!|a |!||!|solution |!||!|in |!||!|polynomial |!||!|time
2. |!||!|Demonstrate |!||!|that |!||!|problem |!||!|B |!||!|is |!||!|at |!||!|least |!||!|as |!||!|hard |!||!|as |!||!|a |!||!|problem |!||!|believed |!||!|to
|!||!|be |!||!|NP-Complete. |!||!|This |!||!|is |!||!|done |!||!|via |!||!|a |!||!|reduction |!||!|from |!||!|a |!||!|known |!||!|problem |!||!|A |!||!|(A-
>B)
a) |!||!|Show |!||!|how |!||!|an |!||!|instance |!||!|of |!||!|A |!||!|is |!||!|converted |!||!|to |!||!|B |!||!|in |!||!|polynomial |!||!|time
b) |!||!|Show |!||!|how |!||!|a |!||!|solution |!||!|to |!||!|B |!||!|can |!||!|be |!||!|converted |!||!|to |!||!|a |!||!|solution |!||!|for |!||!|A, |!||!|again |!||!|
in |!||!|polynomial |!||!|time
c) |!||!|Show |!||!|that |!||!|a |!||!|solution |!||!|for |!||!|B |!||!|exists |!||!|IF |!||!|AND |!||!|ONLY |!||!|IF |!||!|a |!||!|solution |!||!|to |!||!|A |!||!|exists
- |!||!|most |!||!|prove |!||!|both |!||!|parts: |!||!|if |!||!|you |!||!|you |!||!|have |!||!|a |!||!|solution |!||!|to |!||!|B, |!||!|you |!||!|have |!||!|a |!||!|
solution |!||!|to |!||!|A
- |!||!|If |!||!|there |!||!|is |!||!|no |!||!|solution |!||!|to |!||!|B, |!||!|then |!||!|no |!||!|solution |!||!|exists |!||!|to |!||!|A
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Download Exam 3 Solutions: Complexity Classes, NP-Completeness, and Reductions and more Exams Insurance law in PDF only on Docsity!

CS6515 Exam 3 with verified solutions

The |!||!|Class |!||!|P |!||!|- |!||!|correct |!||!|answer |!||!| ✔ A |!||!|solution |!||!|may |!||!|be |!||!|found |!||!|in |!||!|polynomial |!||!|time The |!||!|Class |!||!|NP |!||!|- |!||!|correct |!||!|answer |!||!| ✔ A |!||!|solution |!||!|may |!||!|be |!||!|verified |!||!|in |!||!|polynomial |!||!|time NP-Hard |!||!|- |!||!|correct |!||!|answer |!||!| ✔ We |!||!|are |!||!|yet |!||!|to |!||!|find |!||!|a |!||!|polynomial |!||!|time |!||!|solution NP-Complete |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Both |!||!|in |!||!|NP |!||!|and |!||!|at |!||!|least |!||!|as |!||!|hard |!||!|as |!||!|NP-Hard |!||!| problems NP |!||!|Reduction |!||!|- |!||!|Steps |!||!|- |!||!|correct |!||!|answer |!||!| ✔ 1. |!||!|Demonstrate |!||!|that |!||!|problem |!||!|B |!||!|is |!||!|in |!||!|the |!||!|class |!||!|of |!||!|NP |!||!|Problems

  • |!||!|validate |!||!|a |!||!|solution |!||!|in |!||!|polynomial |!||!|time
  1. |!||!|Demonstrate |!||!|that |!||!|problem |!||!|B |!||!|is |!||!|at |!||!|least |!||!|as |!||!|hard |!||!|as |!||!|a |!||!|problem |!||!|believed |!||!|to |!||!|be |!||!|NP-Complete. |!||!|This |!||!|is |!||!|done |!||!|via |!||!|a |!||!|reduction |!||!|from |!||!|a |!||!|known |!||!|problem |!||!|A |!||!|(A-

B) a) |!||!|Show |!||!|how |!||!|an |!||!|instance |!||!|of |!||!|A |!||!|is |!||!|converted |!||!|to |!||!|B |!||!|in |!||!|polynomial |!||!|time b) |!||!|Show |!||!|how |!||!|a |!||!|solution |!||!|to |!||!|B |!||!|can |!||!|be |!||!|converted |!||!|to |!||!|a |!||!|solution |!||!|for |!||!|A, |!||!|again |!||!|

in |!||!|polynomial |!||!|time c) |!||!|Show |!||!|that |!||!|a |!||!|solution |!||!|for |!||!|B |!||!|exists |!||!|IF |!||!|AND |!||!|ONLY |!||!|IF |!||!|a |!||!|solution |!||!|to |!||!|A |!||!|exists

  • |!||!|most |!||!|prove |!||!|both |!||!|parts: |!||!|if |!||!|you |!||!|you |!||!|have |!||!|a |!||!|solution |!||!|to |!||!|B, |!||!|you |!||!|have |!||!|a |!||!| solution |!||!|to |!||!|A
  • |!||!|If |!||!|there |!||!|is |!||!|no |!||!|solution |!||!|to |!||!|B, |!||!|then |!||!|no |!||!|solution |!||!|exists |!||!|to |!||!|A

LP: |!||!|Why |!||!|optimum |!||!|occurs |!||!|at |!||!|a |!||!|vertex |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Feasible |!||!|region |!||!|is |!||!| convex LP: |!||!|Optimum |!||!|achieved |!||!|at |!||!|a |!||!|vertex |!||!|except: |!||!|- |!||!|correct |!||!|answer |!||!| ✔ 1. |!||!|Infeasible |!||!|- |!||!| feasible |!||!|region |!||!|is |!||!|empty

  1. |!||!|Unbounded |!||!|- |!||!|optimal |!||!|value |!||!|of |!||!|objective |!||!|function |!||!|is |!||!|arbitrarily |!||!|large Independent |!||!|Set |!||!|-> |!||!|Vertex |!||!|Cover |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Lemma: |!||!|I |!||!|is |!||!|an |!||!|independent |!||!|set |!||!|of |!||!|G(V, |!||!|E) |!||!|iff |!||!|V |!||!|- |!||!|I |!||!|is |!||!|a |!||!|vertex |!||!|cover |!||!|of |!||!|G Simply |!||!|check |!||!|if |!||!|there |!||!|is |!||!|a |!||!|vertex |!||!|cover |!||!|of |!||!|size |!||!|V |!||!|- |!||!|b |!||!|in |!||!|G(V, |!||!|E). |!||!|If |!||!|there |!||!| is, |!||!|output |!||!|V |!||!|- |!||!|S 3SAT |!||!|-> |!||!|Independent |!||!|Set |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Lemma: |!||!|f |!||!|is |!||!|satisfiable |!||!|iff |!||!|the |!||!| resulting |!||!|set |!||!|has |!||!|an |!||!|independent |!||!|set |!||!|of |!||!|size |!||!|m |!||!|(# |!||!|of |!||!|clauses |!||!|in |!||!|f) |!||!|in |!||!|G(V, |!||!| E). To |!||!|construct |!||!|G(V, |!||!|E), |!||!|create |!||!|a |!||!|node |!||!|for |!||!|each |!||!|literal |!||!|in |!||!|each |!||!|clause |!||!|and |!||!| connect |!||!|them |!||!|by |!||!|an |!||!|edge. |!||!|Also |!||!|connect |!||!|any |!||!|literal |!||!|with |!||!|it's |!||!|negation. Independent |!||!|Set |!||!|-> |!||!|Clique |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Lemma: |!||!|G(V, |!||!|E) |!||!|has |!||!|an |!||!| independent |!||!|set |!||!|of |!||!|size |!||!|g |!||!|iff |!||!|G'(V, |!||!|E) |!||!|has |!||!|a |!||!|clique |!||!|of |!||!|size |!||!|g. To |!||!|construct |!||!|G'(V, |!||!|E), |!||!|add |!||!|all |!||!|the |!||!|vertices |!||!|in |!||!|G(V, |!||!|E) |!||!|to |!||!|G'(V, |!||!|E) |!||!|and |!||!|add |!||!| edges |!||!|to |!||!|G'(V, |!||!|E) |!||!|if |!||!|there |!||!|is |!||!|no |!||!|edge |!||!|in |!||!|G(V, |!||!|E) SAT |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Input: |!||!|C |!||!|is |!||!|a |!||!|CNF |!||!|with |!||!|any |!||!|# |!||!|of |!||!|variables |!||!|(n) |!||!|and |!||!| clauses |!||!|(m) Output: |!||!|assignment |!||!|of |!||!|each |!||!|variable |!||!|s.t. |!||!|the |!||!|CNF |!||!|is |!||!|True 3SAT |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Input: |!||!|C |!||!|is |!||!|a |!||!|CNF |!||!|whose |!||!|clauses |!||!|have |!||!|at |!||!|most |!||!| 3 |!||!| literals

Output: |!||!|An |!||!|array |!||!|of |!||!|integers |!||!|that |!||!|is |!||!|a |!||!|subset |!||!|of |!||!|A |!||!|and |!||!|sums |!||!|to |!||!|t KnapsackSearch |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Input: |!||!|W |!||!|is |!||!|an |!||!|array |!||!|of |!||!|weights, |!||!|V |!||!|is |!||!|an |!||!| array |!||!|of |!||!|values, |!||!|B |!||!|is |!||!|the |!||!|capacity |!||!|of |!||!|the |!||!|knapsack, |!||!|and |!||!|g |!||!|is |!||!|the |!||!|goal. Output: |!||!|Array |!||!|of |!||!|items |!||!|of |!||!|value |!||!|at |!||!|least |!||!|g |!||!|with |!||!|weight |!||!|less |!||!|than |!||!|or |!||!|equal |!||!|to |!||!|B TSP |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Input: |!||!|G |!||!|is |!||!|a |!||!|weighted, |!||!|fully |!||!|connected |!||!|graph |!||!|with |!||!| weights |!||!|for |!||!|each |!||!|n(n-1)/2 |!||!|edges; |!||!|b |!||!|is |!||!|a |!||!|budget Output: |!||!|A |!||!|path |!||!|that |!||!|visits |!||!|every |!||!|vertex |!||!|in |!||!|the |!||!|graph |!||!|exactly |!||!|once |!||!|and |!||!|has |!||!|a |!||!|total |!||!|cost |!||!|of |!||!|b |!||!|or |!||!|less 3DMatching |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Input: |!||!|disjoint |!||!|sets |!||!|X, |!||!|Y, |!||!|Z |!||!|of |!||!|items |!||!|to |!||!|be |!||!| matched. |!||!|Collection |!||!|of |!||!|compatibility |!||!|triples |!||!|(X, |!||!|Y, |!||!|Z) Output: |!||!|A |!||!|disjoint |!||!|set |!||!|(no |!||!|elements |!||!|in |!||!|common) |!||!|of |!||!|n |!||!|compatible |!||!|triples ZOE |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Input: |!||!|an |!||!|mXn |!||!|matrix |!||!|A, |!||!|all |!||!|of |!||!|whose |!||!|entries |!||!|are |!||!| 0 |!||!| or |!||!| 1 Output: |!||!|An |!||!|n-vector |!||!|x, |!||!|all |!||!|of |!||!|whose |!||!|entries |!||!|are |!||!| 0 |!||!|or |!||!|1, |!||!|such |!||!|that |!||!|AX |!||!|= |!||!| 1 Clique, |!||!|Independent |!||!|Set |!||!|and |!||!|Vertex |!||!|Cover |!||!|Relation |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Lemma: |!||!| Given |!||!|an |!||!|undirected |!||!|graph |!||!|G |!||!|= |!||!|(V, |!||!|E) |!||!|with |!||!|n |!||!|vertices |!||!|and |!||!|a |!||!|subset |!||!|V |!||!| ′ |!||!|⊆ |!||!|V |!||!|of |!||!|size |!||!|k. |!||!|The |!||!|following |!||!|are |!||!|equivalent: (i) |!||!|V |!||!| ′ |!||!|is |!||!|a |!||!|clique |!||!|of |!||!|size |!||!|k |!||!|for |!||!|the |!||!|complement |!||!|Graph (ii) |!||!|V |!||!| ′ |!||!|is |!||!|an |!||!|independent |!||!|set |!||!|of |!||!|size |!||!|k |!||!|for |!||!|Graph (iii) |!||!|V |!||!|- |!||!|V ′ |!||!|is |!||!|a |!||!|vertex |!||!|cover |!||!|of |!||!|size |!||!|n−k |!||!|for |!||!|G, |!||!|(where |!||!|n=|V|)

NP-Complete |!||!|relationship |!||!|to |!||!|NP-Hard |!||!|- |!||!|correct |!||!|answer |!||!| ✔ All |!||!|NP-Complete |!||!|Problems |!||!|are |!||!|NP-Hard, |!||!|but |!||!|not |!||!|all |!||!|NP-Hard |!||!|Problems |!||!|are |!||!|NP-Complete You |!||!|are |!||!|given |!||!|two |!||!|problems |!||!|A |!||!|and |!||!|B |!||!|such |!||!|that |!||!|A |!||!|is |!||!|NP-complete, |!||!|B |!||!|does |!||!| not |!||!|belong |!||!|to |!||!|the |!||!|class |!||!|NP |!||!|and |!||!|A |!||!|-> |!||!|B |!||!|- |!||!|correct |!||!|answer |!||!| ✔ 1. |!||!|B |!||!|is |!||!|NP-Hard

  1. |!||!|If |!||!|S |!||!|is |!||!|a |!||!|candidate |!||!|solution |!||!|to |!||!|an |!||!|instance |!||!|I |!||!|for |!||!|problem |!||!|A, |!||!|we |!||!|can |!||!| check |!||!|in |!||!|polynomial |!||!|time |!||!|if |!||!|S |!||!|is |!||!|indeed |!||!|a |!||!|solution
  2. |!||!|If |!||!|B |!||!|-> |!||!|MST, |!||!|then |!||!|P |!||!|= |!||!|NP |!||!|(Because |!||!|A |!||!|-> |!||!|B |!||!|-> |!||!|MST)
  3. |!||!|A |!||!|is |!||!|NP-Hard What |!||!|is |!||!|a |!||!|subset |!||!|of |!||!|what? |!||!|(NP, |!||!|P, |!||!|NP-Complete, |!||!|NP-Hard) |!||!|- |!||!|correct |!||!|answer |!||!| ✔ 1. |!||!|NP-complete |!||!|is |!||!|a |!||!|subset |!||!|of |!||!|NP-Hard
  4. |!||!|P |!||!|is |!||!|a |!||!|subset |!||!|of |!||!|NP
  5. |!||!|NP-complete |!||!|is |!||!|a |!||!|subset |!||!|of |!||!|NP A |!||!|problem |!||!|in |!||!|NP |!||!|is |!||!|NP-Complete |!||!|if: |!||!|- |!||!|correct |!||!|answer |!||!| ✔ all |!||!|NP |!||!|problems |!||!|can |!||!| be |!||!|reduced |!||!|to |!||!|it |!||!|in |!||!|polynomial |!||!|time. |!||!|This |!||!|is |!||!|same |!||!|as |!||!|reducing |!||!|any |!||!|of |!||!|the |!||!| NPC |!||!|problem |!||!|to |!||!|it NP-Complete |!||!|Quiz |!||!|- |!||!|correct |!||!|answer |!||!| ✔ https://www.geeksforgeeks.org/algorithms-gq/np- complete-gq/ Undecidable |!||!|Problems |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Computationally |!||!|Impossible |!||!|-- |!||!|No |!||!| algorithm |!||!|solves |!||!|the |!||!|problem |!||!|on |!||!|every |!||!|input |!||!|even |!||!|with |!||!|unlimited |!||!|time |!||!|and |!||!| space. |!||!|The |!||!|key |!||!|is |!||!|every |!||!|input. |!||!|Even |!||!|if |!||!|some |!||!|inputs |!||!|or |!||!|most |!||!|inputs |!||!|can |!||!|be |!||!| solved, |!||!|as |!||!|long |!||!|as |!||!|it |!||!|can't |!||!|solve |!||!|all, |!||!|then |!||!|it |!||!|is |!||!|considered |!||!|undecidable. Halting |!||!|Problem |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Input: |!||!|Program |!||!|P |!||!|with |!||!|an |!||!|Input |!||!|I Output: |!||!|True |!||!|if |!||!|P(I) |!||!|terminates, |!||!|False |!||!|if |!||!|P(I) |!||!|Never |!||!|terminates.
  1. |!||!|Optimum |!||!|always |!||!|lies |!||!|at |!||!|a |!||!|vertex |!||!|(but |!||!|there |!||!|may |!||!|be |!||!|other |!||!|equivalent |!||!| optimum |!||!|that |!||!|don't |!||!|lie |!||!|at |!||!|vertex)
  2. |!||!|Feasible |!||!|region |!||!|is |!||!|convex |!||!|(because |!||!|it |!||!|is |!||!|constructed |!||!|by |!||!|intersection |!||!|of |!||!|half |!||!| planes) LP |!||!|Standard |!||!|Form |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Max |!||!|C^T |!||!|* |!||!|x Subject |!||!|to:
  3. |!||!|A*X |!||!|<= |!||!|b
  4. |!||!|x |!||!|>= x |!||!|= |!||!|n-vector |!||!|of |!||!|variables c |!||!|= |!||!|n-vector b |!||!|= |!||!|m-vector |!||!|of |!||!|constraints A: |!||!|m |!||!|X |!||!|n |!||!|matrix LP |!||!|Convert |!||!|to |!||!|Standard |!||!|Form |!||!|- |!||!|correct |!||!|answer |!||!| ✔ 1. |!||!|min |!||!|C^T |!||!|* |!||!|X |!||!|<=> |!||!|max |!||!|- C^T |!||!|* |!||!|X
  5. |!||!|a_1x_1 |!||!|+ |!||!|... |!||!|+ |!||!|a_nx_n |!||!|>= |!||!|b: |!||!|<=> |!||!|-a_1x_1 |!||!|- |!||!|... |!||!|- |!||!|a_nx_n |!||!|<= |!||!|-b
  6. |!||!|a_1x_1 |!||!|+ |!||!|... |!||!|+ |!||!|a_nx_n |!||!|= |!||!|b: |!||!|<=> |!||!|
  • |!||!|a_1x_1 |!||!|+ |!||!|... |!||!|+ |!||!|a_nx_n |!||!|>= |!||!|b
  • |!||!|-a_1x_1 |!||!|- |!||!|... |!||!|- |!||!|a_nx_n |!||!|<= |!||!|-b
  1. |!||!|Unconstrained |!||!|x |!||!|(that |!||!|can |!||!|be |!||!|pos |!||!|or |!||!|neg) |!||!|<=>
  • |!||!|add |!||!|x+ |!||!|and |!||!|x- |!||!|where |!||!|x+ |!||!|>= |!||!| 0 |!||!|and |!||!|x- |!||!|>=
  • |!||!|replace |!||!|x |!||!|= |!||!|x+ |!||!|- |!||!|x- LP |!||!|Strict |!||!|Inequalities |!||!|- |!||!|correct |!||!|answer |!||!| ✔ - |!||!|Aren't |!||!|allowed
  • |!||!|the |!||!|points |!||!|on |!||!|the |!||!|boundary |!||!|do |!||!|not |!||!|fall |!||!|in |!||!|polyhedron
  • |!||!|open |!||!|set |!||!|where |!||!|we |!||!|don't |!||!|know |!||!|optimum |!||!|-> |!||!|ill |!||!|defined LP |!||!|Feasible |!||!|Region |!||!|- |!||!|correct |!||!|answer |!||!| ✔ - |!||!|the |!||!|intersection |!||!|of |!||!|n |!||!|+ |!||!|m |!||!|half |!||!|spaces |!||!|which |!||!|forms |!||!|a |!||!|convex |!||!|polyhedron
  • |!||!|vertices |!||!|are |!||!|the |!||!|points |!||!|satisfying |!||!|n |!||!|constraints |!||!|with |!||!|equality |!||!|and |!||!|m |!||!|constraints |!||!|with |!||!|<=
  • |!||!|vertices |!||!|are |!||!|upper |!||!|bounded |!||!|by |!||!|(n+m |!||!|/ |!||!|n)
  • |!||!|# |!||!|of |!||!|neighboring |!||!|vertices |!||!|is |!||!|bounded |!||!|by |!||!|n*m LP |!||!|Simplex |!||!|Algorithm |!||!|- |!||!|correct |!||!|answer |!||!| ✔ - |!||!|worst |!||!|case |!||!|runtime |!||!|is |!||!|exponential
  • |!||!|guaranteed |!||!|to |!||!|give |!||!|an |!||!|optimum
  • |!||!|widely |!||!|used |!||!|on |!||!|HUGE |!||!|LPs LP |!||!|Optimum |!||!|is |!||!|always |!||!|achieved |!||!|at |!||!|a |!||!|vertex |!||!|on |!||!|the |!||!|feasible |!||!|region |!||!|except |!||!| when: |!||!|- |!||!|correct |!||!|answer |!||!| ✔ 1. |!||!|Infeasible: |!||!|feasible |!||!|region |!||!|is |!||!|empty
  • |!||!|this |!||!|depends |!||!|only |!||!|on |!||!|constraints |!||!|(not |!||!|objective)
  1. |!||!|Unbounded: |!||!|optimal |!||!|value |!||!|is |!||!|arbitrarily |!||!|large
  • |!||!|this |!||!|depends |!||!|only |!||!|on |!||!|objective |!||!|(not |!||!|constraints) LP |!||!|Determine |!||!|Infeasible |!||!|- |!||!|correct |!||!|answer |!||!| ✔ max |!||!|z Subject |!||!|to:
  1. |!||!|AX |!||!|+ |!||!|z |!||!|<= |!||!|b
  2. |!||!|x |!||!|>= Then |!||!|simply |!||!|check |!||!|if |!||!|z |!||!|> |!||!| 0

LP |!||!|Check |!||!|Unbounded |!||!|- |!||!|correct |!||!|answer |!||!| ✔ - |!||!|check |!||!|if |!||!|primal |!||!|and |!||!|dual |!||!|are |!||!| infeasible LP |!||!|Strong |!||!|Duality |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Theorem: |!||!|Primal |!||!|LP |!||!|is |!||!|feasible |!||!|and |!||!| bounded |!||!|iff |!||!|Dual |!||!|LP |!||!|is |!||!|feasible |!||!|and |!||!|bounded --> |!||!|Primal |!||!|has |!||!|optimal |!||!|x* |!||!|iff |!||!|dual |!||!|has |!||!|optimal |!||!|y* --> |!||!|c^T |!||!|* |!||!|x |!||!|= |!||!|b^T |!||!|* |!||!|y LP |!||!|Max |!||!|Flow |!||!|- |!||!|Min |!||!|Cut |!||!|- |!||!|correct |!||!|answer |!||!| ✔ c^T |!||!|* |!||!|x |!||!|= |!||!|max |!||!|flow b^T |!||!|* |!||!|y |!||!|= |!||!|min |!||!|cut LP |!||!|Ek-SAT |!||!|approximation: |!||!|Simple |!||!|and |!||!|LP-Based |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Simple: |!||!| 1 |!||!|- |!||!| 2^-k LP: |!||!| 1 |!||!|- |!||!|(1 |!||!|- |!||!|1/k)^k

  • |!||!|simple |!||!|scheme |!||!|is |!||!|better |!||!|for |!||!|large |!||!|clauses, |!||!|LP |!||!|is |!||!|better |!||!|for |!||!|small |!||!|clauses
  • |!||!|better |!||!|of |!||!|two |!||!|schemes |!||!|always |!||!|better |!||!|than |!||!|3/4 |!||!|m* LP |!||!|If |!||!|Dual |!||!|is |!||!|Infeasible |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Then |!||!|Primal |!||!|is |!||!|either |!||!|infeasible |!||!|or |!||!| unbounded LP |!||!|If |!||!|Dual |!||!|is |!||!|feasible |!||!|- |!||!|correct |!||!|answer |!||!| ✔ Then |!||!|Primal |!||!|is |!||!|bounded Halting |!||!|Problem |!||!|Implication: |!||!|If |!||!|P |!||!|ne |!||!|NP |!||!|- |!||!|correct |!||!|answer |!||!| ✔ - |!||!|No |!||!|algorithm |!||!|can |!||!|run |!||!|in |!||!|polynomial |!||!|time |!||!|on |!||!|every |!||!|input