Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

18
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497 views

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 ...
16
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644 views

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 ...
16
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352 views

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 ...
14
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550 views

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, ...
14
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951 views

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 ...
11
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316 views

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 ...
9
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297 views

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 ...
9
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304 views

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 (...
9
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116 views

Reconstructing labeled poset from linear extensions

Let $(P, <, \mu)$ be a labeled poset, that is, a partial order $(P, <)$ with a labeling function $\mu$ that maps the elements of $P$ to labels in an alphabet $\Sigma$. A label list (or word) is ...
9
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237 views

Advances towards proving the Held-Karp conjecture for TSP

I've only began my research into the Held-Karp conjecture and I was wondering about recent progress in proving the conjecture. The Held-Karp relaxation is conjectured to have an integrality gap of $\...
8
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222 views

Subset sum problem with at most one solution for any target

This question was originally asked on CS.se. A little bit of initial discussion can be found in the comments there. We first consider the search version of the subset sum problem: Given a set $S$ of ...
8
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119 views

Complexity of fractional SAT

Let $(a, k)$-SAT be $k$-SAT with the promise that if there is there is a satisfying assignment, then there is such an assignment that satisfies at least $a$ literals of every clause. Can 3-SAT with $...
8
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171 views

NP-hardness of approximation for unconstrained submodular maximization

The problem of unconstrained submodular maximization can be phrased as follows: Given a non-negative submodular function $f$ on a domain $D$ find a set $S \subseteq D$ maximizing $f(S)$. Here a ...
8
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701 views

Is this minimization problem NP-Complete?

We are given an $n \times (n + k)$ matrix $A$, with entries in GF(2), of the form $A =[I_n\ B]$, where $I_n$ is the $n \times n$ identity matrix, and $B$ has no "zero" rows or columns. The problem is ...
8
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962 views

Is Exact Cover by Equally-Sized Sets reducible to Multi-Dimensional Matching in a certain nice way?

This question is motivated by my other question “Gap hardness of Multi-Dimensional Cover,” which is in turn motivated by the question “Set Cover for Permutation Matrices” by Brayden Ware. Informal ...
7
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133 views

NP-hardness proof of selecting the ''fittest'' topological sequence of a DAG

Given a directed acyclic graph (DAG) with $n$ vertexes $V=\{v_1, v_2,...,v_n\}$ and a given permutation of those $n$ vertexes $P=[p_1, p_2,..., p_n]$ that $\forall i, p_i\in V$. Note that $P$ could ...
7
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214 views

How hard is PromiseFlowFree?

Playing more Flow Free, I think I've realized why I'm so amazingly brilliant at this game: The objective is to connect all pairs while covering the entire board, but in every puzzle there is always ...
7
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163 views

What relations are there between a problem hardness and the hardness of verifying a witness?

I had some hard times trying to formulate the question, so I'll start with some examples: Suppose you are given a Dominating Set instance, $<G,k>$. Now suppose I give you a set of vertices $D$ ...
7
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499 views

Another Solution Problem (ASP) of integer multi-commodity flow: is it NP-complete?

I know that integer multicommodity flow is NP-complete. It was proven in On the complexity of time table and multi-commodity flow problems that any SAT problem can be reduced to an integer ...
7
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160 views

Universal tractable problem solver

Consider $X$ an $\mathsf{NP}$-complete language e.g. $3-SAT$. I'm looking for an algorithm $A$ for solving $X$ with the following property. Given $M \subset \lbrace 0, 1 \rbrace^*$ any set of words s....
7
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193 views

What is the complexity of pallet loading for identical non-rectangular objects?

In the pallet loading problem, we are asked to place a set of small identical 2-D rigid objects into a large bounding rectangle such that no two objects overlap. This problem is a special case of the ...
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158 views
+50

Planar Exact Cover by even-size sets

Major edit on June 6, 2019: Replaced the target problem with a simpler (but equivalent) one. Is the following problem NP-complete? Planar Exact Cover by even-size sets Input: A set $U$, a ...
6
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581 views

Is this permutation-sum problem NP-complete?

A new, tighter tardiness bound has been found for global Earliest-Deadline-First scheduling of jobs on symmetric multiprocessors. But this bound seems to be particularly hard to compute. In particular,...
6
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246 views

NP-hardness of tasks graph assignment to two heterogenous servers

I have a problem with determinig if the following assignment problem is NP-hard. Any comments and suggestions would be appreciated. Problem definition Given is a directed acyclic graph $G=(V,E)$ ...
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139 views

Complexity of DTMC subsystems

A discrete-time Markov chain (DTMC) is a tuple $M=(S,s_{init},P)$ where $S$ is a finite set of states, $s_{init}\in S$ the initial state, and $P:S\times S\to[0,1]$ the one-step transition probability ...
6
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208 views

Further question on hardness of node partitioning under shortest path constraint

This question first appeared at Hardness of node partitioning under shortest path constraint and I restate it here Given a direct graph $G=(V,E)$. $\forall (i,j) \in E$, there is a weight $w(i,j) \...
5
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66 views

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 ...
5
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79 views

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 ...
5
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107 views

An optimal subspace projection problem

Suppose we have a $k$-dimensional subspace $V$ in $\mathbb{R}^n$ given by a basis $\{v_1,\cdots,v_k\in \mathbb{R}^n\}$, find an index set $I\subset [n]$ with $|I|=m$ where $k\le m\le n$, such that $$\...
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274 views

On the shortest vector problem (is it $NP$-complete?)

Ajtai has shown that shortest vector problem is $NP$-hard by using randomized reduction from subset sum. Has this been derandomized?
5
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119 views

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 ...
5
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542 views

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 ...
5
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0answers
114 views

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 ...
5
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971 views

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 ...
5
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180 views

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,...
4
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0answers
82 views

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 ...
4
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112 views

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 $H \subseteq U$ ...
4
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0answers
168 views

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 ...
4
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0answers
137 views

What is the current “state-of-the-art” solver for quadratic knapsack problems?

New to this forum, so please let me know if my question format is incorrect. For linear KP with $n$ items and $c$ capacity, dynamic programming can find exact solutions in $\mathcal{O}(nc)$. I have ...
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225 views

How does part 5 of the paper “computational complexity of Minesweeper” work?

Marzio De Biasi mentioned this paper in this answer, and that paper's claim in part 5 would appear to resolve my question. However, there seems to be a significant gap in the proof. Even though one ...
4
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0answers
224 views

The weakly NP-complete problems and their associated counting problem

Are there weakly NP-complete problems whose associated counting problem can be computed in pseudo-polynomial time? And if one were to be found (and assuming it is #P-complete), what would be the ...
3
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0answers
145 views

Take a NEXP-complete problem and then have the input in unary. Why is this not NP-complete?

It is known that if any unary language is NP-complete, then P=NP. Suppose we take a NEXP-complete language with input $x$ in binary and witness $y\in\{0,1\}^{2^{poly(|x|)}}$ such that the verifying ...
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67 views

Rearranging angles of a convex polyline to make it closed

Let {$\alpha_1, \alpha_2, ... ,\alpha_n$} be a string of n positive reals summing up to 2$\pi$. We inductively construct the following 2D polyline, denoting with $R[\alpha]$ the clockwise rotation by ...
3
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52 views

Hardness of ancilla free quantum circuit extraction from circuit with ancillas

Is there any known result regarding the hardness of the following problem: Given a quantum circuit with ancillae implementing a unitary, find a quantum circuit that does not use any ancillae that ...
3
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0answers
190 views

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 $PH\subseteq NSPACE((\log n)^2)$?
3
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80 views

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 ...
3
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0answers
87 views

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 ...
3
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0answers
84 views

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 $k$ ...
3
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127 views

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 ...
3
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89 views

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 ...