Questions tagged [exp-time-algorithms]

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46
votes
4answers
13k views

Approximation algorithms for Metric TSP

It is known that metric TSP can be approximated within $1.5$ and cannot be approximated better than $123\over 122$ in polynomial time. Is anything known about finding approximation solutions in ...
39
votes
2answers
953 views

How many distinct colors are needed to lower-bound the choosability of a graph?

A graph is $k$-choosable (also known as $k$-list-colorable) if, for every function $f$ that maps vertices to sets of $k$ colors, there is a color assignment $c$ such that, for all vertices $v$, $c(v)\...
33
votes
2answers
4k views

Do any quantum algorithms improve on classical SAT?

Classical algorithms can solve 3-SAT in $1.3071^n$ time (randomized) or $1.3303^n$ time (deterministic). (Reference: Best Upper Bounds on SAT ) For comparison, using Grover's algorithm on a quantum ...
32
votes
12answers
5k views

Problems that started out with hopelessly intractable algorithms that have since been made extremely efficient

This is somewhat of a meta-cstheory question, and is more historical in nature. What are some good examples of problems for which the literature followed the develpment below: The original algorithms,...
20
votes
0answers
981 views

Exact algorithm for NAE-3SAT

The NAE-3SAT problem is to determine whether a given 3CNF formula has a satisfying assignment that gives each clause at least one false (and at least one true) literal. The problem is NP-complete. One ...
18
votes
3answers
1k views

Solving Superstring Exactly

What is known about exact complexity of the shortest superstring problem? Can it be solved faster than $O^*(2^n)$? Are there known algorithms that solve shortest superstring without reducing to TSP? ...
17
votes
0answers
324 views

Intermediate problems between PSPACE and EXPTIME

Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves. Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, ...
16
votes
0answers
436 views

What is the evidence for average case separation between EXP and NEXP?

There is significant evidence from cryptography that there exist NP-complete problems that are hard in the average case (meaning that e.g. $AvgP \nsupseteq DistNP$). Namely, we have candidate one-way ...
15
votes
2answers
2k views

NP-hard problems with very fast exponential-time algorithms

NP-hard problems with very fast exact exponential-time algorithms, say with $O(1.01^n)$ time, are very rare. Is any fact like "For any constant $\epsilon>0$ there is an NP-hard 'natural' ...
14
votes
6answers
455 views

Problems with Unknown Single Exponential Time Agorithms

I'm looking for examples of problems for which it is easy to get algorithms running in time $2^{O(n\log n)}$, or $2^{O(n^c)}$ for some $c>1$ but for which it is not known whether there is an ...
14
votes
0answers
296 views

Exponential-time factorization of polynomials

Let an explicit field be a field for which equality is decidable (in some standard model of computation). I am interested in the factorization of univariate polynomials over an explicit field. It is ...
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. ...
13
votes
1answer
872 views

Exact algorithms for non-convex quadratic programming

This question is about quadratic programming problems with box constraints (box-QP), i.e., optimisation problems of the form minimise $f(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} + \mathbf{c}^T \mathbf{...
13
votes
1answer
413 views

The problem of deciding whether a monotone CNF implies a monotone DNF

Consider the following decision problem Input: A monotone CNF $\Phi$ and a monotone DNF $\Psi$. Question: Is $\Phi \to \Psi$ a tautology? Definitely you can solve this problem in $O(2^n \cdot \...
13
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0answers
198 views

Parameterized Algorithm to Speed up Exact Exponential-time Algorithm

The connection between $c^kn^{O(1)}$ for $c<4$ and exact exponential-time algorithms beating brute-force $O(2^n)$ algorithms has been known for a long time. However, when $c\geq 4,$ there are not ...
12
votes
2answers
384 views

$\overline{SAT} \in NTIME(subexp)$?

Is it possible that $\overline{SAT} \in NTIME(\exp(n^{0.9}))$ ? Are there interesting consequences of such containment? Would it contradict the Exponential Time Hypothesis?
12
votes
1answer
519 views

Computational Model in SETH

Impagliazzo, Paturi and Calabro, Impagliazzo, Paturi introduced Exponential-Time Hypothesis (ETH) and Strongly Exponential-Time Hypothesis (SETH). Roughly, SETH says that there is no algorithm which ...
10
votes
2answers
476 views

Hardness of a subcase of Set Cover

How hard is the Set Cover problem if the number of elements is bounded by some function (e.g, $\log n$) where $n$ is the size of the problem instance. Formally, Let $\mathcal{U}=\{e_1, \cdots, e_m\}$...
10
votes
2answers
600 views

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 one ...
10
votes
1answer
289 views

EXP-Complete Problems vs Subexponential Algorithms

Does the fact that a problem $A$ is EXP-time complete implies that $A$ is not in $DTIME(2^{o(n)})$? I'm aware that by the time hierarchy theorem, $EXP=DTIME(2^{n^{O(1)}})$ is not included in $E=...
10
votes
2answers
373 views

Subset Numbering

Fix $k\ge5$. For any big enough $n$, we would like to label all subsets of $\{1..n\}$ of size exactly $n/k$ by positive integers from $\{1...T\}$. We would like this labelling to satisfy the following ...
9
votes
2answers
394 views

Exact exponential-time algorithms for 0-1 programs with nonnegative data

Are there known algorithms for the following problem that beat the naive algorithm? Input: matrix $A$ and vectors $b,c$, where all entries of $A,b,c$ are nonnegative integers. Output: an optimal ...
9
votes
2answers
6k views

Time complexity of Held-Karp algorithm for TSP

When I looked through "A Dynamic Programming Approach to Sequencing Problems" by Michael Held and Richard M. Karp, I came up with the following question: why the complexity of their algorithm for TSP ...
8
votes
1answer
289 views

Two DFA intersection emptiness connections to SETH & L vs P

(re "fine grained complexity") Wehar has proved that Two DFA intersection emptiness in $O(n^{2-\epsilon})$ time → SETH false. does anyone see any particular key proof difficulty, challenge, ...
8
votes
1answer
315 views

On $\text{ETH}$ with $m$ as parameter: consequences of algorithm running in time $2^{\delta m}$ where $\delta \to 0$ as $k \to \infty$

It has been shown in [1] that $k\text{-SAT}$ has a $2^{o(n)}$ algorithm if and only if it has a $2^{o(m)}$ algorithm, $n$ being the number of variables and $m$ being the number of clauses. Being $s_k=\...
8
votes
1answer
433 views

Complexity of validity problem for Monadic First Order Logic?

Monadic First Order Logic is FOL with no function symbols, and predicate symbols restricted to arity 1. For this question, let's say that the = symbol is also forbidden. I want to know the complexity ...
8
votes
0answers
107 views

Reference for a circuit lower bound for slightly superexponential time

It is known that $EXP$ doesn't have circuits of size $n^k$. On the other hand proving $10 n$ lower bound on circuit size for $E$, $NE$ or even $E^{NP}$ is a known open problem. My question is ...
7
votes
2answers
368 views

Most general setting for fine-grained exponential-time complexity classes?

Consider the class of functions computable in time $(b+o(1))^n = 2^{\log_2{(b)} \times n + o(n)}$ on a $2$-tape Turing machine. By the Hennie-Stearns theorem, the same functions are computable in ...
6
votes
1answer
242 views

What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?

It is known that computing $(\sqrt 2 -\epsilon)$-approximation for VC is NP-hard and that UGC implies that even a $(2 -\epsilon)$-approximation is hard. There is also a parameterized algorithm for ...
5
votes
2answers
331 views

Questions regarding SETH

I read about the strong exponential time hypothesis, which states (as far as I understand) that SAT problem cannot be solved in running time $O(2^{\epsilon n})$ for any $\epsilon < 1$, where $n$ is ...
5
votes
2answers
220 views

Oracle comparing $EXP$ with $UP$

Heller (Theorem 6) gave an oracle relative to which $NP=EXP$, and Homer & Selman gave an oracle relative to which $P=UP$ and $\Sigma_2^P=EXP$. Beigel, Buhrman, Fortnow (freely available author's ...
5
votes
1answer
130 views

Reference request: complexity of $k$-partite $k$-SAT

Let's consider following variation of $k$-SAT that I will call $k$-partite $k$-SAT: given $n$ variables that are divided into $k$ groups and a $k$-SAT formula $\phi$ such that each clause has literal ...
5
votes
1answer
2k views

What is the worst-case runtime complexity to transform a NFA to DFA via Rabin-Scott's power set construction?

What is the worst-case runtime complexity to transform a NFA to DFA via Rabin-Scott's power set construction? Why? Details: http://en.wikipedia.org/wiki/Powerset_construction states that the worst-...
5
votes
1answer
200 views

Correctness of AKS algorithm for shortest vector problem

Short question In the end of section 1 of Regev's notes about the AKS algorithm for SVP, why is the following true? for each such $i$,$y_i− x_i$ remains $w$ with probability $1/2$ or otherwise ...
5
votes
0answers
243 views

Is MAX-SAT SETH (like) hard?

If we make a random assignment to the variables in $k$-sat ($m$ clauses), we are going to satisfy $(1-2^{-k})m$ clauses in expectation. In general satisfying fewer clauses is considered easy. There ...
4
votes
1answer
890 views

Example of context-free grammar that triggers exponential behaviour without memoization in RD parsers

It is often said that memoization brings the complexity of recursive-descent parsers from exponential to polynomial. However, I had a hard time finding an example grammar that triggers the exponential ...
4
votes
1answer
210 views

Is there a W[1]-hard problem that can be solved in $2^{o(n)}$ time?

This question is about subset problems (the solution is a subset of the instance, so trivially enumerable in $2^n \cdot n^c$ time), and the parameter is the solution size, so-called the standard ...
4
votes
1answer
90 views

Approximate matching in table of integer vectors

Disclaimer: This is my first question on cstheory.stackexchange.com so please be forgiving. I have a list of M (M is big, more than 1 million elements) vectors of integers. Each vector can contain 0-...
3
votes
4answers
3k views

Finding cliques in a big graph

I would like to find (all) cliques in a given graph with 8,568 vertices and 12,726,708 edges. The vertex with the lowes degree has 2000, the vertext with the highest degree has 4007. The cliques ...
3
votes
3answers
2k views

Pseudo-polynomial time algorithms

Consider the following algorithm: Given a natural number as input, say $N$, the algorithm runs a loop (in which the algorithm does $O(1)$ time operations) $N$ times. Now, by definition of time ...
3
votes
1answer
133 views

Consequences of faster parameterized integer programming

Integer programming in $k$ variables can be done in $k^{O(k)}$ time and $O(k^c)$ space. Is there any consequence if it can be done in $k^{O(k^\alpha)}$ time and $O(k^c)$ space for some $\alpha\in(0,1)...
3
votes
1answer
141 views

Is it possible to approximate Maximum Independent Set in $O(2^k\text{poly}(n))$ time?

We know that MIS is hard to approximate within a $n^{1-\epsilon}$ factor in polynomial time and that it is $W[1]$-hard and thus unlikely to admit a $f(k)\text{poly}(n)$ time exact algorithm. (here, $k$...
3
votes
0answers
132 views

If $\sf{E} = \sf{NE}$. Then $\sf{NP}-{P}$ contains no sparse sets [closed]

I am reading "The Complexity Companion" by Hemaspaandra & Ogihara, I have a question about lemma 1.21. In its proof, they suppose $L$ is some sparse language in $\sf{NP}$ ($||L^{=n}||&...
3
votes
0answers
909 views

What's the hardest problem with a non-trivial exact algorithm?

Exact algorithms for NP-complete problems are sometimes feasible, if the input is small enough. I’ve also came across some algorithms which are not practical even for very small inputs, and their ...
3
votes
0answers
247 views

Slightly Faster Exponential Algorithm for Integer Programming with Multi-linear Variables

Integer programing is one of the most narutal optimization tools. As an analogy of DNF or CNF in the Boolean function theory, we can consider the following equation. $x_{1}x_{2}x_{3}+$ $x_{3}x_{4}x_{5}...
2
votes
1answer
269 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
votes
0answers
157 views

Time complexity of Succinct-CVP

I want to know what is the best known lower time complexity of Succinct-CVP? The succinct version of many P-complete problems are EXP-complete and Succinct-CVP is EXP-complete too (It is because of ...
2
votes
0answers
91 views

A question about UE

Much has been written about the class UP see related (even more in literature) example question here. Much is understood about the class UP, and its place in collapsing the PH too. UP has a played ...
1
vote
1answer
688 views

Learning about EXPTIME and EXPSPACE

I'd like to know some good starting points (such as books, papers, lecture notes, etc.) on EXPTIME and EXPSPACE. I'd like to learn more about these two topics, but I'm not sure what the best approach ...
1
vote
0answers
201 views

anything hinting that EXPTIME $\subseteqq$ PSPACE?

Anything or evidence hinting that $$EXPTIME \subseteqq PSPACE$$?