Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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152 views

Partitioning the vertices of a complete graph with weights on both vertices and edges with constraints

Given the complete graph on n vertices. Each vertex and each edge has a positive weight associated with it. What is desired is to partition the vertices into parts so that the sum of the weights of ...
3
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0answers
216 views

Ordered routing problem which is NP-hard

All the np-hard routing problems I know are of the form, minimize some quantity while visiting the verticies in an unordered way. Are there problems which are still np-hard, if one has to visit the ...
2
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3answers
972 views

Complexity of Portal 2

I am studying the complexity of Portal 2 and I would like to know if the problem has been studied before. In particular, I would be interested any reference discussing the complexity of Portal 2 and ...
2
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2answers
872 views

$k$-clique in $k$-partite graph

Is the decision whether a $k$-clique exists in a $k$-partite graph NP-hard? I have found only a very limited number of references on this problem, and they seem to be concerned with heuristics to ...
2
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2answers
475 views

Is the following optimization problem NP-hard?

Set S, which is an non-empty finite subset of $\{ (i,j) : i, j \in N \land i \neq j \}$, is given. E.g. $S=\{(1,3), (2,3), (1,4), (2,4), (3,1), (3,4)\}$ . For each element $(i,j)$, we have weight $w_{...
2
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2answers
465 views

Complexity of a weighted subset selection problem.

I'm currently thinking about a problem I'd like to manage in one of my applications. There is a set of objects $\{A, B, C, D, \ldots\}$. Every object has the same attributes with different values. ...
2
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2answers
359 views

For which values of $k$ is the $k$-disjoint paths problem in $\mathcal{P}$?

The $k$-Vertex-Disjoint Paths Problem ($k$-$\text{DPP}$) is defined as follows: Input: A graph $G=(V,E)$ and $k$ pairs of vertices $(s_1,t_1),\ldots,(s_k,t_k)$. Question: Does there exist $k$-...
2
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2answers
399 views

NP-hard problem on planar unit disk graph

I am curious to know whether there are problems which are np-hard even on planar unit disk graphs. A unit disk graph is the intersection graph of a collection of unit disks in the plane, where we ...
2
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1answer
291 views

What are some example problems for integer programming that are *not binary*

I'm interested in NP-hard problems that have a "nice" integer-programming formulation (quadratic or linear, with quadratic or linear constraints) that is not binary. Of course it is always possible ...
2
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3answers
181 views

How hard is it to find a “well-distributed” subset of models of a propositional formula?

We consider the propositional language $\mathcal{L}_{\mathit{PS}}$ defined over a finite alphabet $\mathit{PS}$ and the usual logical connectives. An interpretation is an assignment $\mathit{PS} \...
2
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2answers
94 views

complexity of CSPs with 2-variable constraints and multi-valued variables

(CSP = "constraint satisfaction problem") CSPs for which either the variables are restricted to less than two values or the constraints take less than two inputs are obviously trivial, and Schaefer's ...
2
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1answer
188 views

Hardness of computing circles with max number of lattice points

You're given an $n\times n$ lattice $\mathcal{L}$, and you're asked to compute the maximum number of points in $\mathcal{L}$ that can belong to the same circle (the circle has to be enclosed by the ...
2
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1answer
237 views

Does there exist polytime algorithm for this partitioning problem?

I would like to know if there exists a polytime probablistic algorithm for the problem described below. It is relevant for construction of a crossvalidation-partitioning in statistics, fulfilling ...
2
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2answers
243 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 ...
2
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1answer
215 views

Set cover with budget on sets

I am wondering if this is a studied variant of the Set Cover problem. We are given a universe $X$, a collection of sets $S = \{S_1, ..., S_m\}$ and integers $c_i$. We want to cover all elements in $...
2
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1answer
87 views

Diameter vs. Tractability

I would like to find examples of (unweighted) optimization problems $\Pi$ that satisfy 1) or 2): 1) $\Pi$ is NP-hard but polytime solvable in the class of graphs with diameter $\le k$, for all $k\ge ...
2
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1answer
140 views

A least sized partition of a set under a distance metric

What is the worst case complexity of an algorithm to find a least partition of a set under a distance metric, described as follows: Input: A set $S=\{s_1,\ldots,s_n\}$, where the elements $s_i$ are ...
2
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1answer
74 views

Constant Width Max Sum Product Multi-objective Shortest path problem

This question is a follow-up on the question I asked three days ago here. For convenience I restate it here. I am given a graph. Each edge is labelled by a vector of numbers, called weights. They ...
2
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1answer
168 views

When can a convex function induce submodularity?

Say I have a real valued convex function $f$ on the hypercube $[-1,1]^n$. Let $f'$ be the induced function on the discrete hypercube $\{-1,1\}^n$. Now I want to find a vertex on $\{-1,1\}^n$ on which ...
2
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1answer
184 views

What is the known complexity of this game? (similar to PushPush-1)

I've been looking at a few entries in http://library.msri.org/books/Book56/files/10demaine.pdf (on combinatorial algorithmic game theory). I didn't see the following game listed there, and I want to ...
2
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1answer
132 views

Is partitioning a multiset into two multisets with equal averages NP-complete?

The subset sum problem of partitioning a multiset of integers into two multisets with equal sums is NP-complete. Is the seemingly related problem of partitioning a multiset of integers into two ...
2
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1answer
146 views

Complexity of Haemers' minimum rank

In 1978 Willem H. Haemmers published "An upper bound on the Shannon capacity of a graph". Tims has a survey of more recent results his thesis. What is the computational complexity of computing ...
2
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1answer
132 views

What Is the Complexity of This Two-to-One Matching Problem?

Given a graph $G=(V,E)$ and a function $c:V\mapsto\{1,2\}$. The function $c(\cdot)$ divides the vertices into two disjoint sets $V_1$ and $V_2$, where for all $v_1\in V_1$, we have $c(v_1)=1$ and for ...
2
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1answer
154 views

Maximum stable matching/allocation

I checked some papers on two-side stable allocation/matching (marriage, worker/company, doctor/hospital), but has not found any literature on the following problem. Can someone point out if I missed ...
2
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1answer
265 views

Design of exact algorithms for non-local hard problems

In the connected dominating set problem (CDS) we are given an $n$-vertex undirected graph, and asked to find the smallest connected subset $S$ of vertices such that each vertex not in $S$ is adjacent ...
2
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1answer
157 views

Computational complexity of finding a (1D deterministic) CA that achieves a desired spacetime history “patch” after $t$ iterations

Question description: Consider the problem of finding a minimum $n$-color $k$-state one-dimensional cellular automata (minimizing $k$ for some fixed value of $n$ or vice versa), with communication ...
2
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1answer
4k views

Difference between NP-Hard and NP-Complete [closed]

Can someone please summarize the exact difference between NP-Complete and NP-Hard problems in simple language? Wiki and my standard books aren't exactly helping.
2
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1answer
130 views

Solving linear program with 1 quadratic constraint complexity

Consider the following linear program, $$\min y \\ xc_1 \leq c_2 + yz,\\ x = x_1 + \dots + x_n,\\ z \leq x_1 + x_2, \\ z \leq x_2 + x_3, \\ \vdots\\ z \leq x_{n-1} + x_n, \\ x,x_1, \dots, x_n,y,z \...
2
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1answer
772 views

Is “meeting/room planning” NP-complete?

Short summary: The problem is to assign people to meetings at different days while respecting the capacities and (inter-day) constraints of the meetings. Each person can only attend at most one ...
2
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0answers
68 views

How hard is it to approximate distance of linear code

I'm trying to figure out what is the current knowledge about how hard it is, given a generating matrix of a linear code over a field $F_{q}$, approximate it's distance. I of course found that ...
2
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0answers
52 views

Complexity of Block Design?

What is known about the complexity of creating Block Designs (https://en.wikipedia.org/wiki/Block_design)? I've found one paper that creates approximately solutions using Metaheuristics that claims ...
2
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0answers
88 views

How hard is it to determine ex(n,G)?

Define the extremal Turán function $ex(n,G)$ of a graph $G$ as the most edges a graph on $n$ vertices can have without having a subgraph that is isomorphic to $G$. This function is known ...
2
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0answers
51 views

Is minimal cover under symmetric 3-deduction NP-complete?

Forgive me if this problem is known by another name, I do not know any references for it. Symmetric deduction. An equation $e \in E$ is a subset of variables $V$ such that knowing $|e| - 1$ of the ...
2
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0answers
82 views

Finding 3SUM witness when promised a solution

Suppose we have a 3SUM instance given with the promise that there exists at least one solution. Is the trivial $O(n^2)$ (modulo logarithmic improvements) solution still the best algorithm or is there ...
2
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0answers
162 views

Best polynomial-time approximation factor for NP-optimization problems

Let us say that a function $f(n)$ is the best approximation factor for an NP-optimization problem, if both of the following hold: There exist a polynomial-time algorithm $A,$ and an integer $n_0$, ...
2
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0answers
104 views

Counting the maximum number of paths of length $n$ that differ in at least $k$ edges

What is known about the complexity of solving (or approximately solving) the following problem? INPUT: Graph $G=(V,E)$ and constants $L$ and $K$. OUTPUT: The maximum size of any set $S$ of simple ...
2
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0answers
171 views

Is Non-linear Constrained Optimal Exact Cover NP-Hard?

Playing around I ran into a problem which looks like a Exact Set Covering / Partition Problem, but I am unable to find a reduction to categorize the complexity of the problem, despite it looks NP-Hard....
2
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0answers
132 views

Reduction Unbounded Knapsack < k-Exact Unbounded Knapsack

I'd like to have an explicit reduction among these two problems: (1) Unbounded Knapsack: Given a set of $n$ item types with weight $w_i$ and quality $q_i$ solve: $$maximize \sum_{i=1}^n q_ix_i $$ ...
2
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0answers
150 views

Computational Complexity of cycle double cover

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 $G$...
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0answers
52 views

Examples of “Sandpile” TSP Instances

This question is closely related to this MO question. I would like to know, whether any (planar Euclidean) TSP instances are known, that exhibit avalanche effects similar to those ecountered in ...
2
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0answers
202 views

Graph optimization problem with multiple objectives/constraints

Let's assume that we have a directed acyclic graph $G = (V, E)$, non-negative vertex weight functions $w_a(v)$ and $w_b(v)$, and a non-negative edge weight function $t(u,v)$. We can divide vertices in ...
2
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0answers
65 views

Is there a relationship between the probabilistic interepretation of Sherali-Adams SDP hierarchy and the Lasserre SDP hierarchy?

Firstly note this paper http://ttic.uchicago.edu/~madhurt/Papers/reductions.pdf where a Lasserre SDP is being setup for the independent set probblem at the bottom of page 4 where the author says says, ...
2
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0answers
84 views

Max common sub forest on $k$ graphs

Not sure how to phrase this really, but here goes. Suppose you are given $k$ simple graphs, each having exactly $m$ edges. The edges in each graph are labeled from 1 to $m$. The problem is to find ...
2
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0answers
92 views

About representability of optimization versions of NP-complete questions as polynomial optimization over the hypercube

Naively, in my very limited awareness, it feels that the Max-CUT is a very "special" NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a $n$...
2
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0answers
125 views

Empty sudoku and NP-completeness [closed]

My question is straightforward: Is an empty sudoku grid (not partially completed) still NP-complete?
2
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0answers
210 views

Convex hull of polytopes

Consider a set of polytopes $P_i : i=1,2,...,k$ each of which has a structure as $P_i:= \{(x_{i1},x_{i2},..., x_{in})\; |\; x_{ij} \in [a_{ij}, b_{ij}] \subseteq [0,1]\}\;\; \text{for all}\;\; j=1,......
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0answers
150 views

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 ...
2
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0answers
126 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 ...
2
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0answers
120 views

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 ...
2
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0answers
284 views

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 ...