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Questions tagged [polynomial-time]

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3
votes
0answers
71 views

Is Circuit Minimization $P$-hard under logspace reductions?

By Circuit Minimization, I am referring to the following decision problem. Circuit Minimization Input: A bit string $x$ and a number $k$. Question: Does there exist a Boolean Circuit $C$...
-2
votes
0answers
47 views

Is there and efficient procedure to test whether a clause is implied by a given CNF formula?

I have a CNF formula $F$ and a clause $c$ defined on the same set of variables. I would like to know if $c$ is implied by $F$ i.e. if $F \vDash c$. To achieve this I could just test if the formula $F\...
6
votes
1answer
90 views

Why is the “general notion of a reduction […] inherent to the notion of self-reducibility”?

While reading "Computational Complexity: A Conceptual Perspective" by Oded Goldreich, I have come across the following passage, which I simply cannot get my head around: Note that the general ...
5
votes
1answer
105 views

Complexity of counting integer roots of multivariate polynomials in a polyhedron?

Deciding integer roots of multivariate polyomials is undecidable. However what is known about counting integer roots of multivariate polynomials in $\mathbb Z[x_1,\dots,x_m]$ with both $m$ and total ...
4
votes
1answer
64 views

Complexity of finding Exact Size Cut-Sets in Bipartite Graphs

I am interested in the problem of deciding if a cut-set of a given size $k$ (i.e. the number of edges crossing the partitions is $k$) exists in a given bipartite graph (both the graph and $k$ are part ...
8
votes
2answers
252 views

Non-Orthogonal Vectors Problem

Consider the following problems: Orthogonal Vectors Problem Input: A set $S$ of $n$ Boolean vectors each of length $d$. Question: Do there exist distinct vectors $v_1$ and $v_2 \in S$ ...
1
vote
0answers
99 views

What definition for $FPT$ algorithm for $KSUM$ gives $W[P]=FPT\implies KSUM$ is $FPT$?

In the definition on $KSUM$ problem we are given $n$ input integers and we have to decide if $K$ of them sum to $0$. $KSUM$ is $FPT$ if there is a $O(f(K)poly(n))$ algorithm for it. However Downey ...
1
vote
1answer
93 views

Can MONOTONE WSAT be in solved in polynomial time?

In the weighted monotone satisfiability problem (MONOTONE WSAT), the input is an n-variable MONOTONE CNF Boolean formula (when there is no a clause with a negated variable) and an integer k, and the ...
3
votes
1answer
121 views

On complexity of linear programming with quadratic equality/inequality constraints?

Feasibility test in Linear programming is in $P$ and in convex quadratic programming is in $P$. What is the maximum $k$ such that $n$-variable $m=poly(n)$ linear constraint feasibility test with $k$ ...
4
votes
0answers
112 views

Reduce $m$-clause 3SAT to PLANAR-3SAT in $O(m^{2-\varepsilon})$

The classic reduction from 3SAT to PLANAR-3SAT requires a removal of $O(m^2)$ crossings from a rectilinear representation of 3SAT with $m$ clauses. However, the crossing number inequality suggests ...
10
votes
1answer
131 views

On sparse complete sets and P vs L

Mahaney's Theorem tells us that if there is a sparse $NP$-complete set under polynomial-time many-one reductions, then $P = NP$. (See "Sparse complete sets for NP: Solution of a conjecture of Berman ...
14
votes
1answer
181 views

What are the consequences of $P \subseteq L/poly$?

A language is in $L/poly$ if there exists a logspace Turing machine that decides the language with polynomial amount of advice. See here for more info: https://en.wikipedia.org/wiki/L/poly ...
2
votes
0answers
83 views

On $\Sigma \Pi \Sigma \Pi(2,r)$-circuits

As I understand from the survey "Progress on Polynomial Identity Testing - II" a polynomial-time algorithm for solving PIT for $\Sigma \Pi \Sigma \Pi (2, r)$ is unknown. However, there exists paper ...
1
vote
0answers
67 views

Time complexity of polynomial regression with random coefficients

Suppose that I have $$\lim_{k,l,m \rightarrow \infty}(f(x,l) ~~=~~ \sum_{n=1}^{m} g(a_{n,k})h(x,n)$$ where the $a_{n,k}$ are pseudo-random real numbers with $k$ digits generated in an arbitrary way, ...
5
votes
0answers
340 views

Do we know some quasi-polynomial problem that is known to not be in NP?

The title pretty much says it all, but to explain how I got there: I think, that one of the reasons we are unable to prove or disprove , but mainly disprove $P=NP$ (and yes, I was provoked by the ...
6
votes
1answer
118 views

Bounded distance decoding beyond Babai

Consider a full-rank lattice in $\mathbb{R}^n$. Let $\lambda_1$ be the length of the shortest nonzero vector. Given a vector in $\mathbb{R}^n$ we wish to find the nearest lattice vector, as measured ...
13
votes
2answers
626 views

Counting the number of satisfying assignments in a POSITIVE CNF-SAT

We know the problem of counting the number of satisfying assignment in a given general boolean formula (CNF-SAT), a given DNF formula, or even a given 2SAT formula is a #P-complete problem. Now, ...
8
votes
1answer
238 views

Using an oracle to find a vector $b$ for which $Ax=b$ has a solution

There is an oracle built around a hidden $m\times n$ matrix $A$ all of whose entries are 0 or 1, where $m>n$. The oracle takes as input an integer vector $b$ with positive entries, and answers as ...
13
votes
1answer
514 views

Are there interesting graph classes where the treewidth is hard (easy) to compute?

Treewith is an important graph parameter that indicates how close a graph is from being a tree (although not in a strict topological sense). It is well known that computing the treewidth is NP-hard. ...
8
votes
0answers
240 views

Under what conditions does an Integer Programming problem run in polynomial time?

Given $AX\leq B$ where $A\in\Bbb Z^{m\times n}$,$B\in\Bbb Z^m$ finding $X\in\Bbb Z^n$ where $m\geq n$ is the integer programming problem. If $A$ is totally unimodular then the problem is solvable in ...
6
votes
2answers
316 views

On integer programming

Integer programming is NP-hard. What is the status of integer programming problem that decides between existence of $\leq1$ solution and $>1$ solutions (note $0$ solutions falls in $\leq1$ ...
10
votes
1answer
186 views

What is the conjectured relationship between P (PTime) and Type 1 (context-sensitive) languages?

It is unknown whether $P\subseteq CSL$ or $P\not\subseteq CSL$, where $P$ is the set of all languages decidable in polynomial time on a deterministic Turing machine, and $CSL$ is the class of context-...
-2
votes
1answer
196 views

questions on implications Babais quasi P time graph isomorphism result

Babai has reputedly repaired his proof of graph isomorphism in quasipolynomial time.[1] the proof hinges crucially on Johnson graphs. based on the proof, does this mean now that if Johnson ...
1
vote
1answer
629 views

Is the value of $\max_{f:V\rightarrow [\frac{-1}{2},\frac{1}{2}], \\ \sum_{v}{f(v)}=0} \frac{f^T L_G f}{n-f^Tf}$ polynomially computable?

For a graph $G$ on $n$ vertices, what is the value of following ratio: $$\max_{f:V\rightarrow [\frac{-1}{2},\frac{1}{2}], \\ \sum_{v}{f(v)}=0} \frac{f^T L_G f}{n-f^Tf} ,$$ where $L_G=D_G-A_G$ is ...
9
votes
0answers
167 views

Wide and shallow circuits for $\mathrm P$

The $\mathrm{NC} \stackrel?= \mathrm P$ question is not as famous as the $\mathrm P$ versus $\mathrm{NP}$ problem, but still a deep and interesting question. It is generally accepted that there are ...
5
votes
0answers
100 views

$\mathsf{P}$ is the closure of [finite set] under [operation between languages]

I am searching for statements of the above form, that is, asserting the existence of a finite set $F$ of languages and one or more operations $\otimes\colon \mathcal{P}(\Sigma^*) \times \mathcal{P}(\...
3
votes
1answer
301 views

Graph planarity testing via adjacency matrix

I have looked at several efficient graph planarity algorithms which rely on computing and traversing DFS trees (that add one vertex/edge/path at a time). I am looking for graph planarity algorithms ...
7
votes
1answer
200 views

Two extremely naive questions about the Kronecker problem from Geometric Complexity Theory

I was reading the GCT IV paper (http://arxiv.org/pdf/cs/0703110v4.pdf) and while the representation theory is clear enough (by which I do not mean to say 'easy'!) the relation to complexity theory as ...
0
votes
2answers
276 views

Packing $n$ objects into $m$ bins whose size is variable

Assume we have $n$ fixed size objects with sizes $O_1$ to $O_n$. Also, assume we have $m$ bins with size $a \times B_1$ to $a \times B_m$ in which $a$ is a real number and $a\ge1$. We want to put ...
4
votes
0answers
48 views

Ground Reachability in String and Term Rewriting Systems

I have two questions concerning ground reachability in string and term rewriting systems. String Rewriting Systems: Let $\Sigma$ be a finite alphabet. I have a set of rules $R$ of the form $a_ib_i =...
4
votes
0answers
169 views

Voronoi diagram in presence of polygonal obstacle

Suppose there is a set of convex polygons ($\mathbb{P}$) on the plane. For each convex polygon $P_i$ there is one "facility" $f_i$ placed on the boundary of $P_i$. The distance between a point $p \in ...
5
votes
1answer
425 views

Canonical way of coloring graphs (individualization) for isomorphism purpose

Stefan Kratsch and Pascal Schweitzer in their paper Graph Isomorphism for Graph Classes Characterized by two Forbidden Induced Subgraphs give a characterization of graphs on which Graph Isomorphism ...
1
vote
0answers
139 views

How does one sample uniformly at random from an uncountably infinite set?

I want to know if there are any examples of polynomial time algorithms which can sample uniformly at random from a given uncountably infinite set? (assuming it is possible) Does it help if the sample ...
10
votes
0answers
151 views

Hardness in P: methods to show optimality of $O(m^2n)$-like time?

In recent years, there has been exciting work in proving lower bounds for polynomial-time problems conditional on conjectures like SETH or "All-Pairs-Shortest-Paths (APSP) cannot be solved in $O(|V|^{...
8
votes
5answers
276 views

Problems in quasi-polynomial time that could conceivably be in P (without causing collapses or violating widely held beliefs)

What are some examples of problems with quasi-polynomial time ($QP$) algorithms that could conceivably be in $P$. In other words, they are in $QP$ for no obvious reason other than no one has found a ...
15
votes
4answers
1k views

Is there a natural problem in quasi-polynomial time, but not in polynomial time?

László Babai recently proved that the Graph Isomorphism problem is in quasipolynomial time. See also talk at University of Illinois, Chicago, note from the talks by Jeremy Kun GLL post 1, GLL post 2,...
8
votes
1answer
316 views

Are there more polynomial time problems with complexity lower bounds?

I'm looking for more problems in $P$ with classical time complexity lower bounds. Some people might wonder how you could prove such a lower bound. See below. Exponential Lower Bounds: Claim: If ...
1
vote
0answers
60 views

Multicuts (or multiway cuts) for 3 Terminals of Minimum Capacity

For each fixed $k \geq 3$, the following well-known problem is NP-complete. k-Multicut (aka "Multiway Cut", "k-Way Cut", "Multicut") Input: $(N,l)$ where $N=(V,E,T,c)$ is an undirected $k$-terminal ...
15
votes
2answers
761 views

Given a 4-cycle free graph $G$, can we determine if it has a 3-cycle in quadratic time?

The $k$-cycle problem is as follows: Instance: An undirected graph $G$ with $n$ vertices and up to $n \choose 2$ edges. Question: Does there exist a (proper) $k$-cycle in $G$? Background: For any ...
2
votes
1answer
180 views

Polynomial-time reductions between undecidable languages

The Turing degree $\mathbf{0}'$ is defined as all languages Turing-equivalent to the halting problem. In fact any recursively enumerable language is polynomial-time reducible to the halting problem. ...
21
votes
1answer
794 views

Time complexity with irrational exponent?

Is there any natural problem in P for which the best known running time bound is of the form $O(n^\alpha)$, where $\alpha$ is an irrational constant?
13
votes
2answers
368 views

What is an equivalent definition of mP/poly in terms of a Turing machine?

P/poly is the class of decision problems solvable by a family of polynomial-size Boolean circuits. It can alternatively be defined as a polynomial-time Turing machine that receives an advice string ...
11
votes
0answers
312 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 ...
0
votes
1answer
76 views

polytime transformation from a graph to a set of binary strings

$d_H$ denotes the Hamming distance between two binary strings of the same size. The problem is stated as follows. Given any undirected graph $(V, A)$, does there always exist a one-to-one ...
2
votes
0answers
113 views

What is the most “unbalanced” vector between two given vectors of numbers?

Let $\mathbb{R}_+$ be the set of non-negative real numbers. Let $m$ be a positive integer and $\leq_m$ the product ordering on $\mathbb{R}_+^m$. That is, $\leq_m$ is the partial ordering on $\mathbb{R}...
6
votes
0answers
90 views

Complexity of solving vs verifying in P

Thinking of (seemingly) very different complexity of finding a solution to a NP problem and verifying it as the basis of practical cryptography, I am wondering if such separation is possible among ...
6
votes
1answer
2k views

Magic constant to solve NP-complete problem in polynomial time

Let's suppose that $P\ne NP$. Is that possible to solve all the instances of size $n$ of an NP-complete problem in polynomial time using some "universal magic constant" $C_n$ that has a polynomial ...
13
votes
3answers
4k views

Examples of problems where exponential algorithms run faster than polynomial algorithms for practical sizes?

Do you know of any problems (preferably at least somewhat well known), where, for a practical problem size, an exponential algorithm runs much faster than a best-known polynomial time counterpart. ...
9
votes
3answers
287 views

Is there a generalization of information theory to polynomially knowable information?

I apologize, this is a bit of a "soft" question. Information theory has no concept of computational complexity. For example, an instance of SAT, or an instance of SAT plus a bit indicating ...
18
votes
5answers
570 views

What notable automaton models have polynomially-decidable containment?

I'm trying to solve a particular problem, and I thought I might be able to solve it using automata theory. I'm wondering, what models of automata have containment decidable in polynomial time? i.e. if ...