# Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

131 questions with no upvoted or accepted answers
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### What is consequence of $PH\subseteq NSPACE((\log n)^2)$?

What is consequences of $PH\subseteq NSPACE((\log n)^2)$? We don't even know PH is equals to L or not. I am wondering what will be happened when $PH\subseteq NSPACE((\log n)^2)$?
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### PTIME or NP-Hardness of stochastic objective function

I will begin by linking a previous post where I asked a general question for a stochastic setting which I describe below. It turns out that my "proof" for a restricted case had a mistake and there is ...
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### Showing hardness of maximizing stochastic objective function over graph

Consider a graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each vertex $v_i$ can take positive value $a_i$ with probability $p_i$ and value $0$ with probability $1-p_i$. The challenge is to ...
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### Random self reducibility and NP

I was reading the Wikipedia page Random self-reducibility and it states: If an NP-complete problem is non-adaptively random self-reducible the polynomial hierarchy collapses to $\Sigma_3$. I am ...
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### Some questions about the Ryan O'Donnel and Yuan Zhou's paper “Approximability and proof complexity”

My question is particularly about the set-up in section $8$ (Analysis of the KV Max-Cut instances") of the paper, https://arxiv.org/pdf/1211.1958.pdf. What they call the Khot-Vishnoi UG instance ...
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### Can the Lasserre relaxation be defined over the reals?

If one wants to say minimize a function $f : \{-1,1\}^n \rightarrow \mathbb{R}$ on its domain then a degree$-d$ Lasserre relaxation of it would be to solve the problem of $\min \mathbb{E}_\mu [f(x)]$ ...
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### SOS and the small set expansion property

For what graphs do we know that their small set expansion property has a low degree SOS proof? Is this known to be true for say the complete graphs? A terminology issue about what is low degree" :...
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### Given a matrix $A$ maximize the number of positive elements in $Ax$ under specific constraints for $x$

Let $A = [a_{ij}]$ be a symmetric matrix with nonnegative values and $k << n/2$ a given constant. We want to rearrange the columns of the matrix such that the number of rows with the following ...
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### Min dominating set software

I need a fast min-dominating-set code for some complexity lower bounds research I am doing. I could transform to SAT and use an off the shelf SAT solver; but I was hoping min-dom-set had something ...
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### What's the hardest problem with a non-trivial exact algorithm?

Exact algorithms for NP-complete problems are sometimes feasible, if the input is small enough. I’ve also came across some algorithms which are not practical even for very small inputs, and their ...
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### A version of bipartite graph turnpike problem

Given a set of "required" weights for edges of a bipartite graph, I am looking for assignments to the nodes so that there is at least an edge carrying a weight from that set. Each edge can have ...
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### Reduction from OR-SAT to Exact CNF-SAT, keeping the number of variables polynomially bounded?

Let me define both the problems first: $OR$-$SAT$: $m$ Boolean formulae are given in $CNF$, $\phi_1$,$\phi_2$, $\ldots$, $\phi_m$, each on variable set, $x_1, x_2, \ldots, x_n$. ($m$ $<$ $2^n$, ...
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### Is the nonnegativeness of a polynomial hard for $\mathsf{NP}_\mathbb{R}$?

It is clear that the following problem is in $\mathsf{NP}_\mathbb{R}$. Input: a list $P$ of triplets $(a,s,t)$ where $s$ and $t$ are nonnegative integers. Output: is there an $x\in \mathbb{R}$ such ...
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### 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 ...
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### Calculating the ground state of an Ising model with $\sigma_i = (0,1)$ spin state assignments (do Barahona & Istrail's NP-hardness results hold?)

In a typical Ising model, one has possible spin assignments of $\sigma_i = \pm 1$. However, one can also imagine a $q = 2$ Potts model generalization with spin assignments $\sigma_i = (0,1)$. Is ...
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### Which complexity information of Ising model is more important?

In 1982, Barahona proved that finding the ground state of an Ising model is NP-hard. Later, in 2000, Istrail proved that it is NP-complete. When I look up the citations of these two papers using ...
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### 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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### Different hardness proofs w.r.t different classes

Consider a language $L$ which is hard for some class $C$ (e.g. PSPACE-hard). Trivially, $L$ is also $D$-hard for every class $D\subseteq C$ under the same type of reduction (e.g. NP-hard). Is there a ...
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### NP-hardness of a winner determination auction

We would need some suggestions for the proof of NP-hardness of an optimization problem. The problem  \max_{x_{a,s}} \sum_s \sum_a x_{a,s} q_a \lambda_s \prod_{s' < s} \prod_{a' \neq a} (1 + ...
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### Scheduling with Separation Constraint

There are $N$ types of jobs. For each $i$, we have to schedule $T/D_i$ jobs of type $i$ in $T$ timeslots. We know that $\sum_{i=1}^N 1/(D_i+1) = 1$. For each type $i$, the distance between two ...
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### Minimum cut with nonlinear objective function

Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum. Let us generalize it the following way: let $f$ be a ...
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### Reasoning about NP hardness of optimization problems with closed form functions as input

(This may not be a research level question per se. I can delete this question if the community thinks this way too) I am trying to understand how to reason about hardness of optimization problems ...
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### Directed NP Hard Problem on DAG

There are problems that are NP-Hard on undirected graphs(maximum weight independent set and graph coloring) but are polynomial time solvable on trees. Tree decomposition is a good tool to talk about ...
What is the best way to represent a NP-Hardness result for a decision problem with two decision parameters? Suppose we have a problem $P$ which asks to minimize two parameters $x$ and $y$ and we show ...
Let $G=(V,E)$ be a directed acyclic graph. Two vertices is transitive if there is a directed path between them. A Path Cover for a Set of Transitive Pairs (PCSTP) is a set of directed paths such that ...