If a solution is found, the answer to the overall problem is yes, and if no solution is found, the answer to the overall problem must be no. The number of sets $B$ with $|B| \le l$ is $|B| \choose l$, which is polynomial in the size of the input since $l$ is a fixed constant.įor each candidate value of $B$, we simply apply the lemma to determine whether there exists a corresponding value of $A$ such that $A \cup B$ solves the problem. If $l$ is a fixed constant then the problem is polynomial time solvable The number of sets $A$ with $|A| \le k$ is $|A| \choose kl$, which is polynomial in the size of the input since $k$ is a fixed constant.įurthermore, these sets are easily enumerable.įor each candidate value of $A$, we simply apply the lemma to determine whether there exists a corresponding value of $B$ such that $A \cup B$ solves the problem. If $k$ is a fixed constant then the problem is polynomial time solvable Since the input problem (clique) is NP-hard, the output problem is NP-hard as well.
We showed above that the reduction is polynomial time and answer preserving. We see then that in the case that the input instance is a no instance, the answer to the output instance is no as well. Then the edge $(e, v)$ (which is in $E$ by the definition of $E$) is not covered, and so $A \cup B$ is not a vertex cover. Let $e = (w, v)$ be an edge in $E'$ with $e \not\in A$ and $v \not\in B$. Let $B' = \$ elements, each of which is an edge of $G'$, we conclude that at least one edge in $E' \backslash A$ has an endpoint not in $B$. Ubi is also where you can find automobile-related businesses like car showrooms (Toyota Motor, Kah Motor, Star Auto etc), car garages and driving centres. Therefore, the situation is symmetric in the two parts of the bipartite graph, and we can restrict our attention to just one part of this lemma: we wish to show that if $A \subseteq V_1$ with $|A| \le l$ then we can decide in polynomial time whether there exists $B \subseteq V_2$ with $|B| \le k$ such that $A \cup B$ is a vertex cover. Vertex is located at Ubi, a district near Paya Lebar and Eunos.Ubi is better known as an industrial estate due to the myriad of industrial businesses available here i.e JTC, garment, plumbing, etc. lemma: if we know $A \subseteq V_1$ with $|A| \le l$ (or $B \subseteq V_2$ with $|B| \le k$), we can decide in polynomial time whether there exists $B \subseteq V_2$ with $|B| \le k$ (resp $A \subseteq V_1$ with $|A| \le l$) such that $A \cup B$ is a vertex coverįor the proof of this lemma, we do not use the constraint that vertices in $V_1$ have degree at most $d$. The first two bullet points address question number 1 and the last two address question number 2.
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