Questions tagged [dynamic-programming]

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46 views

choosing the best subset where the metric is based on pairwise relationship

I want to model the competition between agents. Say there is a set of agents, and at each time step each agent $i \in \{0,1,\cdots, n-1\}$ can select a nonnegative number $x_i$ from a given range $[0,...
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0answers
201 views

Run Length eXtreme encoded length

In run length encoding (RLE) the code stream consists of pairs $(c_i,\ell_i)$, which is understood as writing the character $c_i$ repeatedly $\ell_i$ times. Consider the following "improvement" of ...
2
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1answer
83 views

Longest stack-sortable subsequence

Given an array of $n$ pairwise-different positive integers, the problem is to find the longest subsequence that is stack-sortable, i.e. avoiding the permutation pattern $231$. How fast can this ...
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1answer
111 views

A dominate vector subset sum problem

Let $k$ be some constants (e.g. one can take $k=2$ for simplexity), for any $u,v\in \mathbb{R}$, we say $u$ dominate $v$ if $\forall 1\le i\le k,~ u[i]\ge v[i]$, write it as $u\succ v$. Consider the ...
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1answer
84 views

Ordering of sub problems in dynamic programming

1) Can every dynamic programming question be solved using 3 different orderings or can there be more than 3 or less than 3 ( like unique ordering )? My understanding is that a) it might have a unique ...
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0answers
54 views

Probabilistic linebreaking algorithm

I'm currently trying to implement this paper: Bouckaert, Remco R., A probabilistic line breaking algorithm, Gedeon, Tamás D. (ed.) et al., AI 2003: Advances in Artificial Intelligence. 16th ...
5
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68 views

Series-parallel extension of a partial order respecting a given total order

Consider a partial order $P$, a series-parallel order $Q$ and a total order $R$, such that $P \subseteq Q \subseteq R$. Given $P$ and $R$, we are asked to find $Q$ of minimum length. An $O(n^3)$ ...
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0answers
103 views

Is scalable hardware support for LogCFL (= sAC^1) possible?

The (uniform) circuit classes $TC^0$, $NC^1$ and $sAC^1$ seem to lend themselves to efficient hardware implementation. But using an FPGA approach to create the circuits on the fly seems problematic, ...
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1answer
204 views

Max weight travel on a graph with deadline

Given a deadline $D>0$ and a complete graph $K_n$ (with loops) in which each edge $e_{ij}$ has a weight $w(e_{ij}) \ge 0$ and a travel time $l(e_{ij}) > 0$. Starting from one of the nodes, we ...
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0answers
49 views

Counting multiplicative closures

Given a set $S$, its multiplicative closure is the set $$ \mathcal{M}(S) = \{s_1s_2\cdots s_k: k\in\mathbb{N},s_i\in S\} $$ of products of zero or more elements of $S$. So the multiplicative closure ...
9
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472 views

How to prove “obvious” facts?

The title is somewhat "arrogant": say, most of us treat $P\neq NP$ as an "obvious" fact, albeit no proof is in sight. But my question is at a much, much lower level, is about a fact which "should be" ...
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134 views

Paper regarding the complexity of the longest path problem on weighted directed graphs of bounded treewidth

I would like to cite a paper/report/etc that solves the following problem polynomially in $n$: Given a weighted directed graph $G=(V,E)$, $|V|=n$, of bounded treewidth $k \in \mathbb{N}$ and a source-...
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3answers
446 views

Beating naive dynamic programming: examples similar to integer partitions?

Let $p(n)$ denote the number of partitions of $n\in\mathbb{N}$ (briefly, number of ways to split a pile of $n$ stones into $\geq1$ unordered nonempty parts). The classical dynamic programming ...
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0answers
267 views

Monotone circuit complexity of matroids?

Call a monotone boolean function $f$ a matroid function if its minterms are bases of some matroid. I am interested in monotone circuit complexity of such functions, even when we "tie hands" of these ...
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2answers
3k views

Dynamic Programming vs Greedy Algorithm

In (Sniedovich 2006) "Dijkstra's algorithm revisited: the dynamic programming connexion", Sniedovich provides us another interpretation of Dijkstra's algorithm as a dynamic programming implementation. ...
14
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1answer
763 views

Monotone arithmetic circuit complexity of elementary symmetric polynomials?

The $k$-th elementary symmetric polynomial $S_k^n(x_1,\ldots,x_n)$ is the sum of all $\binom{n}{k}$ products of $k$ distinct variables. I am interested in the monotone arithmetic $(+,\times)$ circuit ...
15
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1answer
597 views

Is Dynamic Programming never weaker than Greedy?

In the circuit complexity, we have separations between powers of various circuit models. In the proof complexity, we have separations between powers of various proof systems. But in the algorithmic,...
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131 views

Counting words of length $n$ in an inherently ambiguous CFG?

There is a polynomial-time algorithm for computing the number of words of length $n$ in an unambiguous CFG $G = (V, \Sigma, R, S)$ (via a dynamic programming approach). However, for ambiguous CFGs, ...
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1answer
359 views

Covering string by palindromes

Given a string $w=\sigma_1\sigma_2\ldots\sigma_n$, a palindrome cover is a sequence $p_1p_2\cdots p_m$ of words $p_i$ such that $p_1p_2\cdots p_m = w$ and such that each $p_i$ is a palindrome. How ...
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1answer
163 views

How can I rank paths through an HMM? [closed]

I have a profile hidden Markov model that I use to identify all instances of a user-defined pattern of symbols in a long sequence of symbols. I use the Viterbi algorithm to find the most probable path ...
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0answers
24 views

Which matrix of Q values is being used here?

This question refers to this paper: Using Free Energies to Represent Q-values in a Multiagent Reinforcement Learning Task In section 2.1, equations (5) and (6), I am wondering which Q values are ...
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146 views

What is the current “state-of-the-art” solver for quadratic knapsack problems?

New to this forum, so please let me know if my question format is incorrect. For linear KP with $n$ items and $c$ capacity, dynamic programming can find exact solutions in $\mathcal{O}(nc)$. I have ...
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0answers
942 views

How to solve such a graph optimization problem?

I have a graph optimization problem which is hard to describe in the title. There is a component based system which consists of components and data transmissions between components(components and ...
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0answers
320 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 ...
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444 views

Can short-distance connectivity be harder than connectivity?

Has anybody seen the following (or similar) question being considered: Can it be easier to determine the presence/absence of $s$-$t$ paths than to determine the presence/absence of short $s$-$t$ ...
3
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1answer
109 views

Locally sorted sequences

Let $S=s_1,\ldots,s_n$ be a sequence and $p$ be a permutation on the indices of $S$ such that $p$ sorts $S$. Define a sequence to be locally sorted with degree $k$ if $\forall s_i \in S |p(i) - i | \...
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0answers
216 views

Dynamic Programming with two optimization goals

I am working on the problem of distributed database query planning. Existing work [1] uses dynamic programming to search the potential query plan space and find the one with minimal cost. However, I ...
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1answer
819 views

Liner time complexity for wordwrap problem [closed]

Can some body explain me how to apply memoization technique to achieve linear time complexity for bellow. http://www.geeksforgeeks.org/dynamic-programming-set-18-word-wrap/
5
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0answers
145 views

Evidence of non P-hard problems that require polynomial space?

It is admitted that a $\mathsf{P}$-complete problem requires polynomial space and thus cannot be efficiently parallelized. One purpose of these problems is that they can be used to 'defeat' an (...
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0answers
191 views

Calculating exact/approximate solution to a formula

Suppose we have a set of variable $\mathbf{y} = \left(y_1, ..., y_n \right)$. Also consider the set of functions $g_i(y_i), 1 \leq i \leq n$. Note that $g_i()$ is dependent only on $y_i$. Consider ...
4
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1answer
79 views

Bellman principle and approximability

Does anybody know if a combinatorial optimzation problem that enjoys the Bellman's optimality principle can in automatic way be approximated?
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4answers
210 views

Efficiently generate list of lightest intervals of a vector

Suppose a vector of size $n$ is given. The goal is to compute, $\forall i \in [n]$ the lightest interval of size $i$ (i.e. the interval whose sum is minimal). For example, if we have the array: <...
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1answer
2k views

Dynamic programming and Divide and conquer approach [closed]

How does Dynamic Programming differ from Divide and conquer approach for solving problems? Can anyone explain the essential idea of Dynamic Programming. Thanks for any help.
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0answers
379 views

extension for Levenshtein distance

I am looking for an extension for Levenshtein distance (Edit distance) for multi dimensional strings (2D and 3D). I am not sure if there is a formal definition for multi dimensional or not, but here ...
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1k views

Euclidean TSP algorithms

Are there any known exact algorithms for Euclidean TSP that take advantage of the inherent structure of the problem? Do any of these algorithms have better asymptotics than $O(2^n n^2)$ of a DP ...
5
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1answer
418 views

Long Cycle in Bounded Tree-Width Graphs using DFS and Dynamic Programming

For fixed parameter $k$, I would like to find a long cycle of length $\geq k$ in an undirected graph $G(V,E)$. This can be done in $O(k!2^k|V|)$-time [2] using 1) depth-first search (DFS) and 2) ...
4
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1answer
353 views

Problem understanding “connectivity” characteristic for the $k$-connected subgraph problem

I am reading this article, and I am having trouble to understand the 11th definition (page 7) about the connectivity characteristic. I do understand the raw ...
2
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1answer
144 views

Computing unique subset intersections

Given a set S = {si : {zj : z ∈ N} }, what is a time-efficient algorithm for computing the unique sets of intersections of all of the subsets of S? As per @JeffE's comment below, there are edge ...
3
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2answers
282 views

Find two sequences of integers that have sum N but that don't have sub-sequences starting at the head of equal sum

This question arose from a discussion between a friend and I. $A$ is a sequence of length $T$ where for any $a_i$ in $A$, $a_i \in \left\{{1, 2, 3}\right\}$ $B$ is a sequence of length $U$ where ...
2
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0answers
396 views

How can I find all numbers for which the XOR-sum is 0?

Given a list of integers $[a_1, a_2, \dots a_n]$, I want to find the number of $n$-tuples $(x_1,\dots,x_n)$ of integers such that the following three conditions are satisfied: $x_1 \oplus x_2 \oplus \...
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2answers
848 views

Is there some mathematical closed form (or somewhat tight asymptotic one) for “Google Eggs Puzzle”?

The following brief description of the known "Google Eggs Puzzle" comes mainly from the web site Google Eggs: Google Eggs Puzzle: Given n floors and m eggs, what is the approach to find the highest ...
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0answers
4k views

Combinatorics of Bellman-Ford or how to make cyclic graphs acyclic?

Roughly speaking, my question is: How costly is to make a cyclic graph acyclic while preserving all simple $s$-$t$ paths? Let $K_n$ be a complete undirected graph on vertices $\{0,1,\ldots,n+1\}$. (...
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0answers
1k views

Monotone complexity of s-t connectivity

In the problem CONN, we obtain a directed $n$-vertex graph (encoded as a boolean string of $n^2$ bits, one for each potential edge), and want to decide whether there is a path between all $n^2$ pairs $...
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1answer
410 views

Does this bin packing problem have a name?

My problem is related to the standard bin packing problem (where you have bins of capacity $1$, items of capacity $(0,1]$, and want to pack the items into as few bins as possible), but there are a ...
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0answers
463 views

Converting a bounded knapsack problem to 0/1 knapsack problem

I originally posted this question at the programmers section of StackExchange (because that section is supposed to deal with data structures and algorithms), but they suggested posting in the math ...
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2answers
2k views

Dynamic programming and shortest path problem

Several months back, I asked in math.SE the following question I wonder if any dynamic programming problem can always be converted to a source-sink shortest path problem in a network with source ...
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0answers
397 views

Cannot understand the problem of Bitonic Euclidean Traveling-Salesman [closed]

I am referring to the problem in Introduction to Algorithms. I kind of fail to understand the problem. From what I see, I need to sort the x-coordinates of the given set of points and then form a ...
7
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0answers
373 views

Embedded dynamic programming (and planar subgraph isomorphism)

In Planar Subgraph Isomorphism Revisited, Frederic Dorn obtains an improved algorithm for Planar Subgraph Isomorphism, by using a technique he calls Embedded Dynamic Programming. This technique ...
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1answer
305 views

Duration Viterbi Algorithm

I am searching for some good resources to understand the Duration Viterbi algorithm. Does anyone knows a good resource to understand and learn how to model a Duration Viterbi Hidden Markov Chain ...
0
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1answer
283 views

Variable profit knapsack [closed]

sorry for bad formatting earlier We are given Cx,i, Cy,i, Cz,i ∈ ℕ and Px,i, Py,i, Pz,i > 0 for i=1,2,3 such that Px,1 < Px,2 < Px,3, Py,1 < Py,2 < Py,3, and Pz,1 < Pz,2 < Pz,3. We ...