# Questions tagged [reductions]

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.

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### Is it possible to boost the error probability of a Consensus protocol over dynamic network?

Consider the binary consensus problem in a synchronous setting over dynamic network (thus, there are $n$ nodes, and some of them are connected by edges that may change round to round). Given a ...
422 views

### Which complexity class does this number theory problem belong to?

'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by=c$' is $\mathsf{NP}$-complete. Which complexity class does 'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by^2=c$' belong to? 2k views

### Limited number of variable occurrences in 1-in-3 SAT

Is there a known result on complexity class of 1-in-3-SAT with restricted number of variable occurrences? I've come up with the following parsimonious reduction with Peter Nightingale, but I want ...
179 views

### Will a non-linear lower bound on some NP complete problem prove non-linear lower bound on 3SAT?

A problem $\Pi$ is $\mathsf{NP}$ complete if there is a polynomial time reduction from an $\mathsf{NP}$ complete problem $\Pi^\circ$ to $\Pi$ with polynomial blow up on number of variables and ...
1k views

### How to prove that USTCONN requires logarithmic space?

USTCONN is the problem that requires deciding whether there is a path from the source vertex $s$ to the target vertex $t$ in a graph $G$, where these are all given as part of the input. Omer Reingold ...
421 views

### PSPACE completeness, with different kinds of reductions

PSPACE-complete$_{FP}$ problems are the PSPACE problems such that every other PSPACE problem can be transformed to it with a polynomial time reduction, i.e. the reduction is an algorithm $\in$ FP. ...
298 views

### Reducing sorting to max-flow

Is there a linear-time reduction from the sorting problem to the max-flow problem? If so, what would such a reduction look like?
3k views

### Proof for Kolmogorov complexity is uncomputable using reductions

I am looking for a proof that Kolmogorov complexity is uncomputable using a reduction from another uncomputable problem. The common proof is a formalization of Berry's paradox rather than a reduction, ...
93 views

### Is this some variant of the Knapsack Problem?

We are a set of items $I = \{I_1, I_2,.. I_n\}$, which need to be placed in a certain number of knapsacks $K$. We can use as many knapsacks as we want and each knapsack has an infinite capacity but ...
133 views

### Complexity of Knapsack-type problem with applications to computational workflows

Consider the following problem: Let there be a set A of $n$ items $A=\{z_1, ..., z_n\}$, and let $W$ be a strictly positive integer. Each item $z_i$ has a value $v_i$ and a weight $w_i$. Finding a ...
607 views

### #P- vs PP-Completeness

Suppose $A$ is any #P-complete problem. Now, $A$ is modified to obtain a decision problem $A'$ not by asking whether there is a solution but whether at least half of the potential solutions are ...
1 vote
1k views

### Is there a reduction from a 0-1 knapsack problem to the unbounded problem?

As we know, an unbounded knapsack problem could be described as: $\max \sum_{i=1}^nc_1x_i$ s.t. $\sum_{i=1}^na_ix_i\le b$ $x_i\ge0,x_i\in\mathbb Z,i=1,\cdots,n$ And for an 0-1 knapsack problem, we ...
158 views

### Does learning conjunctions with malicious noise reduce to learning conjunctions with random noise?

In Feldman-Gopalan-Khot-Ponnuswami 06 the authors show that agnostically learning parities reduces to learning parities with random classification noise. They also remark (among other things) that ...
881 views

### Are there subexponential algorithms for PLANAR SAT known?

Some NP-hard problems which are exponential on general graphs are subexponential on planar graphs because the treewidth is at most $4.9 \sqrt{|V(G)|}$ and they are exponential in the treewidth. ...
349 views

### Reduction SAT to a problem on a planar graph with as few vertices as possible

Let $\phi$ be CNF formula with $n$ variables and $m$ clauses. I am looking for a reduction is $\phi$ satisfiable to a problem on a planar graph $G$ with as few vertices as possible. The majority of ...
187 views

### Introduction to Black-Box Separations in Cryptography

Are there any textbook-style material on black-box separations in cryptography? I tried to read the paper of Impagliazzo and Rudich but couldn't get much of it. A previous StackExchange entry gives a ...
1 vote
276 views

### Many-one reduction from inequality problem to equality problem

Let the k-inequality-MIS problem be the decision problem whether an arbitrary graph $G=(V, E)$ contains a maximal independent set of at least size $k$, that is the corresponding language is: \...
918 views

### Curious about computer-assisted NP-completeness proofs

In the paper "THE COMPLEXITY OF SATISFIABILITY PROBLEMS" by Thomas J. Schaefer, the author has mentioned that ...
170 views

### The relationship between completeness and strength of reductions

Ladner theorem can be stated as: $P \ne NP$ if and only if there exists an incomplete set in $NP-P$. Here an incomplete set is a set that is not complete for $NP$ under many-one polynomial time ...
249 views

### Balls & Bins: A punishment and reward game

Consider a game where one has a set of bins $(b_1, b_2, ...)$, and each bin has an associated initial count of balls $(c_1, c_2, ...)$. The rules of the game are as follows: (1) Once a bin has a ...
736 views

### Minimal polynomial reduction of dominating set to max clique

Let $G$ be a simple undirected graph. Recall that $S \subseteq V(G)$ is a dominating set of $G$ if every vertex of $v \in V(G) \setminus S$ has a neighbour in $S.$ It is well known that it is NP ...
1 vote
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### Implication of lower bounds in Boolean circuit to other models of computation

Suppose that one can prove that some hard function $f$ with $n$ bit input does not admit any Boolean circuit of size at most $n^t$. Then, how strong can we say about how hard $f$ is in other models? ...
720 views

### Efficient Reduction from Min Cut to st-Min Cut

I am aware that many known algorithms for min cut problem is not by reducing the problem to $st$-min cut. But the question of efficient reduction from min cut to $st$-min cut is still interesting to ...
2k views

### Is there simple reduction Dominating Set to Vertex Cover?

Is there simple reduction Dominating Set to Vertex Cover? In the other direction the reduction is simple. Searching the web returned blog. It warns This is not finished yet and experiments suggest ...
277 views

### Proof of an Ising model representation of graph isomorphism problem

I am going to through Ising formulations of many NP problems by Andrew Lucas. In section $9$ on page 22, the author introduced an exact Ising formulation of the graph isomorphism problem. Given two ...
260 views

### Solving problems by deciding a logic

I am curious to know when open problems have been solved by expressing them in a specific logic, and then showing that this logic is decidable. I have two distinct cases in mind: The problem is ...
171 views

### What is known about reduction by "$P_1$ interprets $P_2$" for generalized programming languages?

Inspired by this answer, let's say that a programming language is given by the data $L=(P,ev)$ where $P$ (the set of "valid programs") is a computable subset of $\Sigma^*$ and $ev$ (the "evaluator") ...
183 views

### Root finding in [0,1]

I am interested in the problem of finding a real root of a polynomial equation $f(x)=0$ where $f(x)=\sum_{i=0}^n a_ix^i$. Is it possible to give a reduction, i.e, to compute a different polynomial $g$ ...
178 views

### Graph partition with objective over intra-partition weights

I have a problem in which I need to find an optimal graph cut that maximizes an objective over weights not on the cut. I have looked at the literature but have not been able to find any similar ...
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### On Equivalence of "gen-CONF" problem and "DH-exp-inv" problem

Let $g$ be a generator of a group of prime order $p$; $a,b ∈_R Z_p^+$ Consider an Algorithm $\mathcal{A}$ which on input $g,g^a,g^{ab}$ outputs $g^{br},r$, for some non zero $r ∈ Z_p$ And an ...
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### NP-hardness proof: looking for some good restricted np-hard problems

To show the NP-hardness of a problem, one need to choose a known NP-hard problem and find a polynomial reduction from the known problem to his problem in hand. Theoretically, any NP-hard problem can ...
737 views

### Should reductions make us more or less optimistic for the tractability of a problem?

It seems to me that most complexity theorists generally believe the following philosophical rule: If we can't figure out an efficient algorithm for problem $A$, and we can reduce problem $A$ to ...
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### On proving it is hard to compute $g^{rb}$ with knowledge of $r$, given $g, g^a, g^{ab}$

I am trying to prove the following Given $g, g^a, g^{ab}$ it is hard to compute $r, g^r, g^{rb}$, for some arbitrarily chosen value of $r$ where $g ∈ \mathbb{G}, \mathbb{G}$ is a cyclic group of ...
969 views

### Max-weight connected subgraph problem in planar graphs

The maximum-weight connected subgraph problem is as follows: Input: a graph $G=(V,E)$ and a weight $w_i$ (possibly negative) for each vertex $i \in V$. Output: a maximum-weight subset $S$ of vertices ...
2k views

### Nontrivial membership in NP

Is there an example of a language which is in $NP$, but where we cannot prove this fact directly by showing that there exists a polynomial witness for membership in this language? Instead, the fact ...
343 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 ...
1k views

### Transitive feedback arc set (TFAS): NP-complete?

Some time ago, I posted a reference request for graph problems where we want to find a 2-partition of the edges where both sets fulfill a property not related to their cardinality. I was trying to ...
261 views

### A reduction between small cliques problems

I'm looking for a reduction that gets a graph $G=(V_G,E_G)$ and outputs a graph $H$ that satisfies the following requirements. If $G$ contains a triangle, then $H$ contains a clique of size 9. If $G$ ...
246 views

### Maximize the expected number of "losers" - Is it NP-hard?

I am trying to find a reduction for a problem that seems NP-hard: Let me start from a toy example. Consider 3 elements, $a$, $b$, and $c$. You want to choose two pairs out of the three pairs and ...
617 views

### On Random Self-reducible properties

Permanent is random self-reducible. $\mathsf{SAT}$ is not random self-reducible since otherwise the polynomial hierarchy collapses to $\mathsf{\Sigma_3}$. 1) Is $k$-sum random self-reducible? That ...
270 views

### Permanent Approximation - Why can the JSV algorithm not handle matrices with negative entries?

Going through the literature, it seems that what it comes down to is that if one could efficiently approximate permanents of matrices with negative entries, then that would imply an efficient ...
297 views

### Graph Isomorphism: Polynomial time reduction from GI for disconnected graphs to GI for connected graphs? [closed]

Let the Graph Isomorphism Problem be the problem to decide whether there is a one-to-one mapping between the vertices of two graphs that preserves the edge relations. Let the Graph Isomorphism ...
342 views

### Is there a reduction to "door and pressure plate" games that doesn't explode solution length?

This paper gives a proof that in a game with doors and pressure plates, it is PSPACE-hard to determine whether or not the (player's) avatar can reach a given location. This is proven by a reduction ... 2k views

### Advanced techniques for determining complexity lower bounds

Some of you may have been following this question, which was closed due to not being research level. So, I'm extracting the part of the question which is at a research level. Beyond the "simpler" ...
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### $\mathsf{TC^0}$-completeness and reductions

AFAIU, we don't know any problem which is complete for $\mathsf{TC^0}$ w.r.t. many-one $\mathsf{AC^0}$ reductions ($\leq^\mathsf{AC^0}_m$). On the other hand, proving that they don't exist would ...
265 views

### What is the minimum required depth of reductions for NP-hardness of SAT?

As everyone knows, SAT is complete for $\mathsf{NP}$ w.r.t. polynomial-time many-one reductions. It is still complete w.r.t. $\mathsf{AC^0}$ many-one reductions. My questions is what is the minimum ...
250 views

### Complexity reductions to Hamiltonian Path?

I am looking for a NP-hardness reduction from an arbitrary problem to the Hamiltonian Path problem such that the reduced no-instances of Hamiltonian path are "far" from having a Hamiltonian path. Do ...
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### Is there a list of canonical problems in distributed systems?

Last week, I was reading again Leslie's Lamport's 1982 trasncript of a conference he gave about Solved Problems, Unsolved Problems and Non-Problems in Concurrency. The paper is easily readable, but ...