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
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
, 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 |!||!| |!||!| |!||!|
2. |!||!| 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
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
, 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 |!||!| |!||!| |!||!|
2. |!||!| 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
