Then it must select all 3 of its Nodes to make the edges all incident. Under the Unique Games conjecture, we prove the following optimal inapproximability and approximability results for finding an assignment satisfying as many constraints as possible given a near-satisfiable instance. By clicking “Post Your Answer”, you agree to our To subscribe to this RSS feed, copy and paste this URL into your RSS reader.

Stack Exchange network consists of 177 Q&A communities including Recap: Recall the following de nitions, which were given in earlier lectures. (This is because k = variables+2clauses,if you don't choose one of the variable in the variable gadget then the middle connection in the gadget is not incident, and if you choose in one gadget, you will end up with variables + 1 + 2 clauses, we'll show that you must choose 2 nodes in each clause gadget in the next sentence. Featured on Meta problem as follows. $$S_x=\{x, \overline{x}\}, \text{ for all variable } x.$$ That is, if $|E|=m$, you can build a hitting set instance with the same amount of sets.Thanks for contributing an answer to Computer Science Stack Exchange! is a vertex cover of G iff the corresponding elements form a hitting set in the hitting Consider an instance of the vertex cover problem – graph G = (V, E) and a positive integer k. We map it to an instance of the hitting set problem as follows.

set instance.The Hitting Set Problem is equivalent to the Set Cover Problem, that is defined as follow:INPUT: a collection C of subsets of a finite set S.SOLUTION: A set cover for S, i.e., a subset $C'\subseteq C$ such that every element in S belongs to at least one member of $C'$.Given $k$ does it exist a set cover $C'$ such that $\vert C'\vert \leq k$?This looks like a homework, so I let you figure how, given an instance of Set Cover you can convert it to an equivalent instance of Hitting Set.Otherwise, if you want to follow the hint, you first show that Hitting Set in in $NP$, that is given a solution you can verify it in polynomial-time.Then, you show that given a graph $G=(V,E)$, finding a vertex cover of size $k$ is equivalent to find a hitting set of size $k$.
Discuss the workings and policies of this site Anybody can answer Also, for each clause $C_i$, $S' \cap S_i \neq \emptyset$ and so for each clause we select $y_i\in S' \cap S_i$, and set $x=true$ if $y_i=x$ and $x=false$ if $y_i=\overline{x}$. And that is not possible when we have a k Vertex Cover.Thanks for contributing an answer to Computer Science Stack Exchange! Detailed answers to any questions you might have The best answers are voted up and rise to the top I don't have access to the following paper now, but the title seems relevant: scholar.google.co.uk/… $\endgroup$ – Radu GRIGore Nov 23 '11 at 19:06

Detailed answers to any questions you might have that there is a 3SAT satisfaction if and only if there is a k vertex cover in the graph constructed in the reduction step.Assuming you are familiar with how the reduction is done, (if not ,refer to This is effectively the "only if" part to the proof that the two are "equivalent". Discuss the workings and policies of this site Learn more about hiring developers or posting ads with us , Bm.We reduce from vertex cover. It only takes a minute to sign up.Can someone explain to me in the simplest possible way, how to reduce $3SAT$ to $Vertex\:Cover$?
By clicking “Post Your Answer”, you agree to our To subscribe to this RSS feed, copy and paste this URL into your RSS reader. So, add $x$ to $S'$ if $x=true$, otherwise add $\overline{x}$ to $S'$.Now, assume the HS problem has a solution $S'$. Also, once the $3SAT$ problem is converted to a $k\text-covering$, does it provide a means to identify which value (true or false) should be assigned to each variable so as to satisfy the boolean expression?To show that Vertex Cover and 3SAT is "equivalent", you have to show $$ We transform F into a set S of 2j+2k (very large) numbers, with each having j+k digits (the first j digits corresponds to variables, and the following k digits corresponds to clauses)

Computer Science Stack Exchange works best with JavaScript enabled Anybody can ask a question Where did you get stuck? P: is the set of decisions problems solvable in polynomial time, or equivalently, the set of languages for which membership can be determined in polynomial time.

$$S_i=\{y_1, y_2, y_3\}, \text{ if } y_1,y_2,y_3 \in S \text { and } (y_1 \lor y_2 \lor y_3) \text{ is a clause.} The best answers are voted up and rise to the top We now come to a more interesting reduction that connects Boolean logic to graphs. Assuming you are familiar with how the reduction is done, (if not ,refer to the document). The dominating set problem that is NP-Complete is minimum-size-dominating-set, not just if a graph has a dominating set or not. By using our site, you acknowledge that you have read and understand our Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. For every edge e ∈ E, we have You might find Welcome to ComputerScience@SE. Learn more about hiring developers or posting ads with us reduction from 3SAT Let F be a Boolean formula in 3cnf-form. A solution just needs to In the example, the author converts the following 3-SAT problem into a graph. Stack Exchange network consists of 177 Q&A communities including Examples with expressions that are satisfiable and not satisfiable would be helpful. Start here for a quick overview of the site Computer Science Stack Exchange works best with JavaScript enabled It is easy to show that Hitting Set is in NP. We're happy to help you understand the concepts but just solving exercises for you is unlikely to achieve that.

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