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# Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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### Natural CLIQUE to k-Color reduction

There is clearly a reduction from CLIQUE to k-Color because they're both NP-Complete. In fact, I can construct one by composing a reduction from CLIQUE to 3-SAT with a reduction from 3-SAT to k-Color. ...
2answers
492 views

### Testing whether letters can be scheduled to achieve a word in a regular language

I fix a regular language $L$ on an alphabet $\Sigma$, and I consider the following problem that I call letter scheduling for $L$. Informally, the input gives me $n$ letters and an interval for each ...
3answers
641 views

### Is it hard to find optimal addition chains?

An addition chain is a sequence of positive integers $(x_1, x_2, \dots, x_n)$ where $x_1 = 1$ and each index $i\ge 2$, we have $x_i = x_j + x_k$ for some indices $1\le j,k < i$. The length of the ...
2answers
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### Are there known NP-complete problems, neither NP-hard in the strong sense nor having pseudopolynomial algorithm?

In their paper (p. 503) Garey and Johnson remark: ... there could exist an NP-complete problem which is neither NP-complete in the strong sense nor solvable by a pseudo-polynomial time algorithm ......
3answers
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### Complexity of edge coloring in planar graphs

3-edge coloring of cubic graphs is $NP$-complete. Four Color Theorem is equivalent to "Every cubic planar bridgeless graphs is 3-edge colorable". What is the complexity of 3-edge coloring of cubic ...
2answers
350 views

### H-free partition

This is a question inspired by the H-free cut problem. Given a graph, a partition of its vertex set $V$ into $r$ parts $V_1, V_2, \ldots, V_r$ is $H$-free if $G[V_i]$ does not induce a copy of $H$ for ...
1answer
739 views

### Intermediate $\mathsf{NP}$-complete problems?

Partition problem is weakly NP-complete since it has polynomial (pseudo-polynomial) time algorithm if input integers are bounded by some polynomial. However, 3-Partition is strongly NP-complete ...
3answers
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### 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 $D$ ...
3answers
437 views

### Partitioning a segmented stick

Problem : We are given a stick partitioned into n - equal parts. Each of these parts has a weight, let's say x. Number of times x appears as weight of some part is guaranteed to be even. For ...
1answer
207 views

### Graph class with easy chromatic number, but NP-hard coloring

Is there a graph class for which the chromatic number can be computed in polynomial time, but finding an actual $k$-coloring with $k=\chi(G)$ is NP-hard? Without any further restriction the answer ...
1answer
21k views

### Is integer factorization an NP-complete problem? [duplicate]

Possible Duplicate: What are the consequences of factoring being NP-complete? What notable reference works have covered this?
1answer
146 views

### Limits of variants of Independent Set?

Independent Set (IS) is the $\mathsf{NP}$-complete decision problem Input: graph $G$ with $v=|V(G)|$, integer $k$ Question: is there an independent set $S \subseteq V(G)$ with at least $k$ vertices? ...
1answer
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### Example of a function problem which is $\mathrm{FP}^{\mathrm{NP}}[wit, log]$-hard?

The usage of an $\mathrm{NP}$-oracles which delivers a witness has been proposed for example in [Buss1995]. I would like to see an example of an $\mathrm{FP}^{\mathrm{NP}}[wit, log]$-hard problem. Can ...
3answers
479 views

### bin packing with overlapping objects

I have $N$ bins with capacity $M$ and $k$ objects with size $s_i$. The goal is to pack these objects in the bins. Until now it is similar to the bin-packing problem. But the twist is that each object ...
1answer
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### Exact exponential-time algorithms for 0-1 programming

Are there known algorithms for the following problem that beat the naive algorithm? Input: A system $Ax \le b$ of $m$ linear inequalities. Output: A feasible solution $x^*\in \{0,1 \}^n$ if ...