Talk:NP-hardness

message: This page is almost incomprehensible due to the style of writing and the lack of attention to explaining the diagrams.  The Euler diagrams essentially looks similar with no clear description.  A diagram is supposed to support the text and make the concept clearer.  This is clearly not the case here.
Can someone help the author fix this PLEASE!
I'm not a stupid person. Granted, I haven't taken a math course since my high school years, but I am not utterly incapable of understanding math. I even have had flashes of understanding the basic implications of P=NP. This said, this article is absolute gibberish to my eyes - polynomial time? Truth value assignments? I can stop and think and squirt a little bit of sense out from this article, but pity be to the poor chump just looking for a quick, easily digestible explanation. I'm not saying that the technical explanation isn't valid (I don't think I'm qualified to even read it), but this article really needs a coherant opening, with the algebraic greek gibberish moved to "section 1 - technical description" or some such. I have thusly tagged this page as  and . --Action Jackson IV 02:15, 4 February 2007 (UTC)
thread_title:  Impossible to understand 

message: What has emergence to do with this article? Probably I should edit this straight away, but I'm curious.
--134.58.253.57 17:38, 5 June 2007 (UTC)
thread_title:  Links 

message: For a more familiar NP-Hard problem, how about Tetris? See here: http://www.arxiv.org/abs/cs.CC/0210020 
It will be easier for people to see what NP-hard means if you can read about something like a popular game and see how it relates.
Sci.Am., "Scientists agree: Tetris is hard."
24.218.46.235 03:28, 24 September 2007 (UTC)
replies: text: Maybe I am overly optimistic, but I would assume that the number of readers who are familiar with integers is larger than the number of readers who are familiar with Tetris. Further, "Tetris is hard" is clearly misleading, since "Tetris" is not a well-defined computational problem. In fact, the decision problems which are NP-hart are very much different from the problems an actual Tetris player needs to solve. --Mellum 19:39, 24 September 2007 (UTC)
replies: text:  LMAO. Well, actually, I don't think that assumption's all that obvious to be making. Maybe if we changed the actual integers in the subset problem to all positive, and check if they summed to fifteen (for instance) - instead of confusing readers not familiar with negative numbers. 220.224.246.97 (talk) 22:18, 5 June 2013 (UTC)
thread_title:  Tetris 

message: I've just removed this from the top of the page:
TO SOLVE NP PROBLEMS, FIRST EXPRESS THEM IN LEAST SPACE PICTORIAL FORM.  THEN PLANE THE nth dimensional picture you've created normal to its axis of radial symmetry.  Keep planing until you have a bunch of simple 2d solutions to 3d problems.  Then do a weighted average to arrive at certainty.
Signed, a math genius PS: please don't revert without thinking about it first. 
It could be useful to the article but would need writing into it rather than sitting above the main heading as it did. Cheez talk 18:24, 8 November 2007 (UTC)
thread_title:  Comment 

message: But not all NP-Hard problems that aren't NP-Complete are undecidable.
Maybe there is a clearer way of saying this.
--128.32.45.173 (talk) 01:35, 9 April 2009 (UTC)
replies: text: Thank you for the praise.  I've reworded it. LWizard @ 07:22, 9 April 2009 (UTC)
thread_title:  Quintuple negative, impressive! 

message: I find the two Euler diagrams – or at least, the one on the left – impossible to understand. Does the label "NP" refer to the whole diagram, or the the region outside the three convex curves? Does the label "NP-Hard" refer to the whole large ellipse, or just the part of it outside the circle? Does the label "NP-Complete" refer to the intersection of the circle and the large ellipse, or to more than this? Maproom (talk) 20:52, 19 March 2010 (UTC)
thread_title:  Those Euler diagrams 

message: IMO, it is VERY important to make clear that for many NP-hard or NP-complete problems, heuristic algorithms exist that will either give a "good" solution or the exact solution quickly in many cases - but which will fail in some cases. The KEY POINT from a "general understanding" perspective is that any algorithm that is guaranteed to find the exact solution in all cases will be slow. Without this information being in the article, a high proportion of people who read it will be mislead. Does anybody disagree? New Thought (talk) 21:18, 29 July 2010 (UTC)
replies: text: You are right, that's why I just added a few words on FPTAS, PTAS and APX in the Consequences section (perhaps even more words should be said though).EXPTIME-complete (talk) 00:32, 6 March 2017 (UTC)
thread_title:  VERY Important point: Heuristics Exist To Resolve Many NP-hard Problems 

message: Given if and only if is used in the very beginning of the introduction and doesn't mean exactly what a "casual reader" would think, maybe there's a way to rewrite or complement the sentence ("A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H") to make it more readable to us who only have a vague understanding of the concepts of P, NP, NP-complete and NP-hard? 
I navigated from P versus NP problem to NP-complete to NP-hard to quickly refresh my understanding of the relationships between the complexity classes and while I found the the first statements of the introduction of the NP-complete page to be very clear and sufficient, this introduction was easily misunderstood if I hadn't noticed the link behind if and only if. While these are pretty complex topics and requires background knowledge I think it's generally a good thing to try to have the introduction and especially the first few statements as easy to grasp as possible (while still accurate and precisely defined), don't you?
The usage of if and only if requires knowledge of a term I don't think is too commonly used to denote this kind of logical relationship when adressing a broader audience..? Even as a programmer having studied quite a lot of mathematics it was news to me at least. Anyway, long story for a small suggestion. This is not my expertise so I don't wan't to touch a well crafted article so I'm just making a suggestion instead: If you want to keep the strict language used, maybe you could complement with a clarification, along the lines with "if and only if (which in the field of mathematics means ...)" for example?
83.227.152.68 (talk) 16:51, 9 August 2010 (UTC)
thread_title:  "if and only if" - easily misunderstood? 

message: It says on the article : "Another example of an NP-hard problem is the optimization problem of finding the least-cost cyclic route through all nodes of a weighted graph. This is commonly known as the Traveling Salesman Problem. There are also decision problems that are NP-hard but not NP-complete" - but the TSP problem is not NP-Complete, it's just NP-Hard. Maybe you could remove the "also" from the text, or reorder it. The way it is, it gives the impression that TSP is NP-Complete. --Wladston (talk) 21:51, 15 November 2010 (UTC)
replies: text: Done. LWizard @ 04:28, 16 November 2010 (UTC)
thread_title: Examples and the TSP

message: I'm removing the Euler diagram for reasons I've spelled out on .  Twin Bird (talk) 23:09, 23 July 2011 (UTC)
replies: text: I re-removed it because it came back. To summarize the problem the trivial languages ∅ and Σ* are in both P and NP. However, they are not NP-complete even if P=NP. The right-hand diagram incorrectly states that, if P=NP, then P=NP=NP-complete but the two trivial languages are a counterexample to this claim. Dricherby (talk) 17:24, 31 January 2016 (UTC)

text: Not sure if useful or worth mentioning, but I have seen some complexity theory sources use a formal definition of language that explicitly excludes the two trivial languages from consideration. That is, the trivial languages are never contained in any complexity class and statements like "all languages" and "some language" always refer to non-trivial languages automatically. This is done precisely to allow "more elegant" statements like "if P=NP, then P=NP=NP-complete, and all languages are NP-hard" instead of "if P=NP, then P=NP="NP-complete plus the trivial languages", and all non-trivial languages are NP-hard".  — Preceding unsigned comment added by 190.229.37.164 (talk) 16:09, 6 July 2017 (UTC)
thread_title:  Euler diagram wrong. 

message: The definition given here says that  "A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H (i.e., L ≤ TH). In other words, L can be solved in polynomial time by an oracle machine with an oracle for H." This defines NP-hard in terms of NP-complete (essentially saying NP-hard includes anything that effectively reduces some NP-complete problem to a P one)
But the intro to the NP-complete article says "A decision problem L is NP-complete if it is in the set of NP problems and also in the set of NP-hard problems". This may be true but it is pretty useless as either a definition or an explanation. My uninformed guess is that what you want to say is that NPH includes *any* problem whose solution can be used to solve any NP problem in poly time and that NPC is the subset of NPH problems that are actually themselves in NP.
If that's right then what I'd say in the intro here is something like "A problem H is NP-hard if and only if every NP problem is polynomial time Turing-reducible to H." And on the other page I'd say "A decision problem L is NP-complete if it is in the set of NP problems and can be used to resolve any other NP problem in only polynomial additional time. (Problems with the latter property are called NP-Hard so NPC is just the intersection of NPH with NP itself)"
Of course it follows that H2 is guaranteed to be NPH if there is some H1 in NPH which is reducible to H2 and in particular if there is an NPC problem which is so reducible (but I'm not sure about the "only if" unless we know that NPC is nonempty)
 
P.S. This complaint appears to have been made by someone else (user Gaenciso)almost 5 years ago (in Dec 2007), so I won't hold my breath waiting for a fix!
alQpr (talk) 02:22, 5 July 2013 (UTC)
replies: text: I agree. Let me try to fix this. -- Pintoch (talk) 15:36, 31 July 2014 (UTC)
thread_title:  Circular Definitions 

message: From the article:
:If P ≠ NP, then NP-hard problems cannot be solved in polynomial time, while P = NP does not resolve whether the NP-hard problems can be solved in polynomial time;
The second part is misleading because it is known that there exist NP-hard problems that are not in P, namely those in EXPTIME or harder. The only part P=NP wouldn't resolve is whether there exist NP-hard problems that are not in PSPACE or in EXPTIME.
It wouldn't even make sense to say "P = NP wouldn't resolve whether any NP-hard problems can be solved in polynomial time", because in fact it would resolve it for NP-complete problems, which is exactly the class of NP-hard problems which can be solved in polynomial time. 
I'll remove the part of sentence.
-- 80.152.204.232 (talk) 13:55, 5 June 2015 (UTC)
thread_title:  Misleading statement 

message: From the article:
:;NP: Class of computational problems for which a given solution can be verified as a solution in polynomial time by a non-deterministic Turing machine.:
As far as I understand, a problem is in NP if it can either be verified in polynomial time by a deterministic machine or, equivalently, solved in polynomial time by a non-determistic one. I think this sentence is confusing the two definitions, maybe it can be clarified. Ksarnek (talk) 10:11, 6 April 2016 (UTC)
thread_title:  Naming convention issue 

message: Hi,
The article mentions cryptography in the application areas. However, based on this Reddit thread and this Stackoverflow thread, it seems like cryptography is a widely different topic: NP-hardness discusses worst-case scenarios (e.g. how hard it is to solve this particular problem, in the worst case), where cryptography tackles average-case scenarios (e.g. how hard it is to break the system for any case).
I hence suggest removing cryptography from the list, or at least adding a footnote discussing this. What do you think? #!/bin/DokReggar -talk 09:47, 2 February 2017 (UTC)
thread_title:  Cryptography and NP-harness 

message: The article speaks of a problem H that is NP-hard, saying "assuming a solution for H takes 1 unit time, we can use H‎'s solution to solve L in polynomial time" Since H takes less time than L, wouldn't that mean H is less hard than L? In what sense, then, is H "at least as hard as the hardest problems in NP"? Qwertie (talk) 13:04, 16 January 2018 (UTC)
replies: text: In traditional complexity theory two problem are considered to be equivalently hard if the time to solve one is a polynomial function of the time to solve the other. So in that sense H and L are equivalently hard.  Of course polynomials can be very large (e.g. x100), so whether that notion of equivalence has any practical meaning, is open to question. The language of the sentence you quote could be improved; what "unit time" means here is not clear.--agr (talk) 15:14, 16 January 2018 (UTC)
thread_title: How is NP-hard hard again?

message: NP-completeness
compare it with this text - it differs in an important aspect. I dont understand it sadly so I cant correct — Preceding unsigned comment added by LasagneAlForno (talk • contribs) 01:56, 15 July 2021 (UTC)
thread_title:  Different text for picture 

message: Doesn’t NP hard means the gamer has to use Nintendo Power magazine tips in order to progress through the game?
As in Nintendo Power hard?
2600:8801:0:2B10:4011:67BD:24CD:EB4D (talk) 20:05, 26 February 2023 (UTC)
thread_title:  NP?