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A reduction is the transformation of one problem into another problem. A example of using a reduction would be to be to show if a problem P is undecidable. This would be achieved by transforming or performing a reduction of a decision problem $P$ into an undecidable problem. If this can be achieved then we have shown that this problem P is undecidable.

22 votes
1 answer
1k views

How to prove that USTCONN requires logarithmic space?

However, USTCONN being complete for L under logspace reductions does not seem to imply this. … This might then perhaps be adapted to show that USTCONN is complete for L under FO-reductions. …
12 votes
1 answer
215 views

Reductions between languages of different densities?

The density of a language $X$ is a function $d_X \colon \mathbb{N} \to \mathbb{N}$ defined as $$d_X(n) = |\{x\in X \mid |x| \le n\}|.$$ Suppose $A$ and $B$ are languages over some finite alphabet, $A$ …
13 votes
1 answer
3k views

Improving Cook's generic reduction for Clique to SAT?

I am interested in reducing $k$-Clique to SAT without making the instance much larger. Clique is in NP so it can be reduced to SAT using logarithmic space. The straightforward Garey/Johnson textbook …
1 vote

Reduction from independent set in hypergraphs to independent set in graphs

I am not as convinced as @Andrew Morgan is that this is "fair standard fare", and would also welcome pointers to a citable reduction. In particular, I do not see how to maintain a linear blowup if $k$ …
András Salamon's user avatar
13 votes

Nontrivial membership in NP

My favourite example is a classic 1977 result of Ashok Chandra and Philip Merlin. They showed that the query containment problem was decidable for conjunctive queries. The conjunctive query containm …
András Salamon's user avatar
4 votes

Nontrivial membership in NP

Deciding reachability for various kinds of infinite-state systems is sometimes decidable, often not. For some interesting special cases a small enough and efficiently checkable certificate always exi …
András Salamon's user avatar
13 votes
3 answers
345 views

Can limit of hard languages be easy?

Can the following all hold simultaneously? $L_s$ is contained in $L_{s+1}$ for all positive integers $s$. $L = \bigcup_s L_s$ is the language of all finite words over $\{0,1\}$. There is some comple …