Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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Complexity of the densest $k$-subgraph problem on planar graphs

In the densest $k$-subgraph problem, one is given an undirected graph $G$ and wants to find a set of vertices $N$ with $|N| = k$ such that the number of edges in the subgraph of $G$ induced by $N$ is ...
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Complexity of interval cover problem

Consider the following problem $Q$: We are given an integer $n$, and $k$ intervals $[l_i,r_i]$ with $1\leq l_i\leq r_i\leq 2n$. We are also given $2n$ integers $d_1,…,d_{2n}\geq 0$. The task is to ...
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Deeper look at Algorithmica?

Russell Impagliazzo published "A Personal View of Average-Case Complexity" (preprint) back in 1995. He presented five possible worlds we could be living in, depending on how P and NP were related. The ...
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a geometric variant of k-medians. NP-hard or in P?

The following problem is a special case of k-medians. Is it NP-hard? Is it in P? Input: $n$ points $(x_1,y_1), (x_2,y_2), \ldots, (x_n, y_n)$ with each $y_i \ge 0$, and an integer $k$. Output: a set ...
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Is Node Multiway Cut NP-complete on planar graphs when all terminals lie on the outer face?

I am interested in the following problem. Node Multiway Cut on Planar Graphs with terminals on the outer face Instance: A plane graph G, and integer k, and a set $S \subseteq V(G)$ of terminals ...
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Phase Transitions in NP Hard Problems

SAT Problems have a phase transition that depends on the ratio $r$ of variables to clauses. Below $r$, SAT problems are solvable quickly; above, they become difficult. The same is true of NP ...
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NP-Hardness of 4-cycle packing problem in complete bipartite digraph?

A directed complete bipartite graph is a bipartite graph where there is exactly one directed edge between any two vertices from its two different parts. In other words, it's an orientation of a ...
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Does solving matrix multiplication in quadratic time imply that SETH is false?

I have a little conjecture that if you could perform matrix multiplication (or solve 3-clique) in $O(n^2 \log(n))$ time, then you could solve CNF-SAT in $O(2^{(1-\epsilon)n})$ time. In other words, ...
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Which monotone DNFs are evasive?

A Boolean function $\phi$ on variables $X$ is evasive if every decision tree for $\phi$ has height $|X|$. In other words, for any strategy that picks variables of $X$ and asks for their value, an ...
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NP complete problem help

I'm currently trying to find a reduction to this problem: Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...
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What are the best known reductions from SAT to CNF-SAT?

Problems Let SAT denote the following problem: Given a boolean formula, does there exist a satisfying assignment? Let CNF-SAT denote the following problem: Given a ...
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Triangle arrangement problem

Suppose you are given an undirected graph $G$, with each vertex representing an equilateral triangle with sides of unit length. Does there exist an arrangement of these triangles in two dimensions (...
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Is this problem in P? Given a bipartite graph, find a minimum cardinality set of edges which intersect every vertex cover

This problem came up in my study of digraphs: Given a connected bipartite graph $G = (A \cup B, E)$, a vertex cover is a set $S$ of vertices such that every edge has some endpoint in $S$. Note that $A$...
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Complexity of a specific class of definite integrals

INTRODUCTION: From the answer to this question I learned that deciding whether a definite integral is $0$ or not can be NP-complete, as the following integral representation of the Number Partition ...
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Is monotone 1-in-3 MAXSAT known to be APX hard?

Monotone 1-in-3 SAT is the problem where each clause of the SAT problem contains exactly 3 positive variables. The goal is to find an assignment such that exactly one variable is true in each clause ...
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Complexity of bounded degree full contraction

This paper defines the problem $\mathrm{B{\scriptsize OUNDED} \ D{\scriptsize EGREE}\ C{\scriptsize ONTRACTION}}$ as follows: Instance: A graph $G$ and two integers $d$ and $k$. Question: Is there a ...
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NP-completeness of a specific topological sorting problem

Consider $(V, E)$ be a DAG, and $p_1, \dots, p_n$ be its topological sorting (i.e. such permutation $p$ of $V$ that $\forall(x, y) \in E.\ p^{-1}(x) < p^{-1}(y)$). Let's call the goodness of $p$ a ...
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Completing a matrix (over the reals) to be singular

Consider the following problem: you are given a matrix (say, with rational entries) some of whose entries are actually left blank; can these blanks be filled in with real numbers so that the resulting ...
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NP-completeness of the Dominating set problem for planar graphs of maximum degree 3?

I am trying to learn about some techniques that are used for proving the NP-completeness of domination related problems. One of the problems that is known to be NP-complete is the domination number of ...
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Variation on block design/set cover

Given 3 parameters $s, r$ and $t$, where $r \leq t$, I want to construct $t$ sets such that each integer $\{1, \ldots, s\}$ appears in exactly $r$ of these sets. The question is: Is it possible to ...
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Bounded Post Correspondence Problem NP-Complete Proof

I'm looking for a simple proof that shows that the Bounded-PCP problem belongs to NP-Complete as many text books say so. It is clear to me that the problem is decidable but I cannot find any reduction ...
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Reduction from planar bounded NCL to a static puzzle game

I call Fill3 the following simple game: the input is a $n \times n$ grid; every cell of the grid has a type: OR, AND, CHOICE, FANOUT and FIXED and can be rotated 0,...
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Analogue of Chow-Liu tree for $L_1$

Say $\Omega$ is a finite set and $f$ a probability mass function (pmf) over $\Omega^d$. Now let $T$ be a spanning tree on the set $V=\{1,2,\ldots,d\}$, and consider a collection of one- and two- ...
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An NP-hard Hidden Subgroup Problem

I've encountered a model which can be thought of as a version of the Hidden Subgroup Problem (https://en.wikipedia.org/wiki/Hidden_subgroup_problem), but that doesn't quite meet the standard problem's ...
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Inverting Kronecker product on vectors is in P?

Problem: Given a vector V of positive integers, find two vectors v1 and v2 such that the Kronecker product of v1 and v2 is equal to p(V) (where p(V) is a suitable permutation of V). Example: Input: V={...
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A variant of hitting set: finding a matching to hit all edge sets

In general, the hitting set problem is given a family $\cal S$ of sub-sets, $\{S_1, \cdots, S_h\}$, and a universal set $U = \bigcup_{i\in [1,h]} S_i$. It asks for a minimum set $H \subseteq U$ ...
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min weight k-closure on DAG

The problem Given a (connected) DAG $G(V,E)$ where each node is assigned an (non-negative) integer weight an integer k where $0\leq k\leq|V|$ Find a induced subgraph $H$ of $G$ consisting of $k$ ...
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 ...
Let $\mathcal{G}$ be the set of all finite simple graphs. Let graph $G\in \mathcal{G}$ and $C_G=\left <C_1,...,C_m \right >$ be a sequence of cycles of $G$ for some $m$. For every edge $e$ of $G$...