Questions tagged [ds.algorithms]

Questions regarding well-defined instructions for completing a task, and relevant analysis in terms of time/memory/etc.

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3
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0answers
44 views

Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?

Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ is irreducible what is the best technique to factor such ...
8
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0answers
243 views

Algorithms to generate consecutive primes

The prime number theorem, states that the "average length" of the gap between a prime $p$ and the next prime is ln(p). I am looking for (preferably deterministic efficient) an algorithm that generates ...
2
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1answer
161 views

Is there a fast algorithm to quickly evaluate $a^{b^c}$ mod $n$?

I need to quickly evaluate $a^{b^c} \mod n$ where $c$ is pretty big. Using the usual repeated squaring trick, this can be performed in $O(\log(b^c)) = O(c)$ time. In my problem, $c$ is huge, (say, $&...
9
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1answer
302 views

Understanding performance of QFBV SMT solvers

SMT solvers such as Z3 or Boolector use a complex set of heuristics to solve problems. However, this also makes predicting the performance of such a solver for a given problem very hard. My question ...
-3
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1answer
118 views

What is wrong with this procedure to convert quadratic programming to convex quadratic programming?

Consider the feasibility quadratic program with constraint $$\sum_{i=1}^nc_{i1}x_{i}\leq \ell_1$$ $$\vdots$$ $$\sum_{i=1}^nc_{it}x_{i}\leq \ell_t$$ $$\sum_{i,j=1}^na_{ij}x_{i}x_{j}+\sum_{i=1}^nb_{i}x_{...
4
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0answers
918 views

Optimizing Maximum Weighted Matching (Edmonds Blossom)

Background: I've ported Edmonds Blossom Algorithm with Maximum Weighted Matching to Java: https://github.com/simlu/EdmondsBlossom/blob/master/src/Blossom.java The original Python implementation ...
1
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1answer
110 views

Practical/heuristic algorithm for multi set-cover

Consider a universe $N$ containing $n$ elements, and a collection of sets $\mathcal{C}$, over $N$. The $k$-multiset multicover (MSMC) problem is to cover all elements of the universe $N$ at least $k$ ...
1
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2answers
177 views

Check the match of the maximum of each subset

Given a number of vectors with $n$ elements, i.e., $S=(a_1, \cdots, a_n)$, $T_j=(b_1^j, \cdots, b_n^j)$ for $j=1,\cdots, m$ where each $a_i$ or $b^i_j$ is a natural number. Question: determine ...
4
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1answer
192 views

How to find largest supergroup in polynomial time?

Let $G \le S_n$, and G acts on set $[n]$ via a map $\pi$: $$\pi : G \times [n]\mapsto [n] $$ In Input generating set of $G$ is given. Question : I need to find the largest supergroup $G^{'}$ (...
0
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1answer
131 views

Different algorithms for longest increasing subsequence

The longest increasing subsequence problem has a simple and elegant $O(n \log n)$ time solution via patience sorting. Such a basic and well-studied problem, however, should have a number of different ...
4
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0answers
194 views

What exactly did Lenstra prove on mixed integer linear program?

I studied Lenstra's paper https://www.jstor.org/stable/3689168. I have no clue what complexity he provides on Mixed Integer Programming (it is too terse and it is not a stand alone paper as he assumes ...
6
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1answer
194 views

Cases of Linear programming known to be in $NC$?

Linear programming is $P$-complete. However are there special situations where we know an $NC$ algorithm?
3
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1answer
64 views

Complexity of Finding Optimal Synergistic Set Packings

Motivation: While developing tools for fast execution of machine learning workflows, we realized that many workflows require intermediate results -- sometimes we should cache these results, and ...
4
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1answer
1k views

Minimal Cost of Eulerian Path

Problem: Given a planar (undirected and mostly sparse) graph with an Eulerian Path, we introduce a cost function f: (v, e1, e2) for all two edges e1 and e2 that share a vertex v. The function also ...
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0answers
83 views

On permanent mod $3^t$ computation

By $'$ I mean transpose. I gather the info here from rjlipton.wordpress.com/2014/12/21/modulating-the-permanent. We know that if $U\in\Bbb F_{3^t}^{n\times n}$ satisfies $UU'=I_n$ in $\Bbb F_{3^t}$ ...
4
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1answer
378 views

Modifying Edmonds' Blossom Algorithm

Given a connected road network on an Island without one-way streets, where should I para-shoot in and what route should I take to deliver mail to all houses on the island (being picked up again by ...
0
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1answer
128 views

What is known about counting bipartite perfect matching with average degree in $[2,3]$ and max degree $3$?

We know counting perfect matching for bipartite graphs with vertex degree $2$ is in $P$ while counting perfect matching for graphs with vertex degree $3$ is in $\#P$. We also know there are degree $3$...
1
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1answer
207 views

Max-weight connected & co-connected subgraph problem

The max-weight connected subgraph problem (MWCS) may be described as follows: given a simple graph $G=(V,E)$ and a weight function $w:V\to\mathbb{R}$, one seeks for a subset $L\subseteq V$ for which ...
0
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1answer
504 views

Data Structure to calculate which interval a point lies in? [closed]

I have a list of $n$ non-overlapping intervals, namely $[a_1,a_2],[a_2,a_3],...,[a_n,n_{n+1}], a_i \in \mathbb{N}$. Each of these intervals has a corresponding value $v_i$ corresponding to it. Now ...
5
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1answer
92 views

What is known about learning a maximal independent set in a (very) sparse graph?

Maximal independent set is known to be hard in many meanings (hard to approximate, $W[1]$-hard, etc.). But if the number of edges is very small, then the problem becomes simpler. Here, I'm interested ...
4
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0answers
96 views

How to construct an $(n,m,k)$ ``separating set''?

This problem is probably known under some other name, if anyone has seen it before, a reference will be great. Given $n,m,k$ (for $m,k\ll n$), a $(n,m,k)$ separating set is a set of $n$-sized binary ...
3
votes
1answer
137 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, $k$...
2
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1answer
164 views

Complexity of $\{0,\pm1\}$ determinant in sparse cases?

If $M\in\{-1,0,+1\}^{n\times n}$ be a matrix with only $O(n)$ non-zero entries and hadamard product $M\odot M$ being symmetric can we compute $Det(M)$ in $O(n)$ bit complexity? Assume that the matrix ...
10
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2answers
295 views

Complexity of Homogenizing a String

Motivation: While developing tools for data versioning, we ended up looking into algorithms for "diff"ing two sets of integers, by coming up with a sequence of transformations that take one set of ...
1
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1answer
71 views

Finding an answer key given tests and scores

Consider multiple students taking a multiple-choice tests. The test has $q$ questions, and each question has $m$ choices. ($q$ can be large, but $m$ is rather small.) You are given $n$ tests, complete ...
3
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0answers
139 views

Efficiently computing the union of all minimal unsatisfiable constraint sets in a first-order unification problem

Suppose we are given a standard first-order unification problem, represented as a set $D$ of term equality constraints, such that the system $D$ as a whole is unsatisfiable. Consider the minimal ...
2
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0answers
85 views

Is berklemap-massey works on extention field

According to the articles that i read (also wikipedia [1]), they refer that the above algorithm works on finite field $GF(q)$. But I cannot find an evidence that it still works on extension field, i.e:...
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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 ...
4
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1answer
177 views

What is known about computing distinct count range queries?

Let $U=\{1,\ldots, u\}$ be a universe of elements for some $u\in \mathbb N$. Given some $n\in\mathbb N$, we are interested in computing some function $f:U^{\le n}\to\mathbb R$ over range queries. In ...
1
vote
1answer
138 views

Differences with Penalty

Given a sequence $a_1,\ldots,a_n$, find a $k$ and numbers $1\leq i_1< \dots< i_{2k}\leq n$ that maximizes $(\sum_{j=1}^{k}(a_{i_{2j}}-a_{i_{2j-1}}))-2k$. I can get an $n^3$ algorithm using ...
0
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0answers
62 views

Iterative algorithms and Lyapunov functions

Consider an iterative algorithm of the form $x^{t+1} = x^t - \eta g(x^t)$. (..if necessary feel free to assume that a function $L$ is explicitly known such that $g = \frac{\partial L}{\partial x}$..). ...
0
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1answer
84 views

Complexity status of restricted k-clique [closed]

Restricted $k$-clique: Input: $(G,v,k)$ where $v$ is vertex in $V$ Output: k-clique containing vertex $v$. What is the space and time complexity status of this Restricted $k$-clique problem? Is ...
3
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0answers
102 views

Restricted k-set cover is in NL or L

Restricted $k$-set cover: Input: $(U,S_1,S_2,\cdots, S_n, k)$, $U=[n]$ and $S_i\subseteq U$ for all $1\leq i \leq n$. Output: $\bigcap_{i\in I}S_i$ where $I=\{1,i_1,i_2\cdots,i_k\}, i_1=min(S_1),i_2=...
1
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0answers
113 views

Off-policy Monte Carlo Control

The off-policy Monte Carlo control algorithm to learn the optimal state-value function $V^*$ is given as follows, which is obtained from Sutton's book. I have three questions concerning this ...
6
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1answer
140 views

Merging Graph Edges to form a 2 color-able Graph (with weight constraints)?

Given an undirected graph $G$. Each vertex has a weight 1. We define shrinking an edge as merging and replacing 2 adjacent vertices $(A, B)$ with a new vertex $C$ such that all the vertices that were ...
3
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0answers
99 views

Notion of “total work” of a problem?

I apologize in advance if this question is outside the scope of this Exchange community; if so, perhaps someone can point me in the right direction. I am curious if there is a theoretical notion of "...
5
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1answer
127 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 ...
3
votes
1answer
119 views

Necessary and sufficient number of comparisons by every element to fully sort a set of n elements? [duplicate]

Given $n$ distinct elements. Is there a sorting algorithm which ensures that every element is compared atmost $\lg n$ time? Or is there a higher lower bound?
-3
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1answer
180 views

Is finding a solution harder than verifying a solution? [closed]

Is there any known problem in Comp science where determinisitically finding a "non-trivial" solution to that problem is asymptotically easier than verifying a solution?
7
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1answer
173 views

Rank-robustness of the parallel complexity of linear algebra problems

It is known that most computational problems related to linear algebra can be computed in $NC^2$ - i.e. for an $n\times n$ matrix $A$, over the reals or a finite field, we can compute the rank of $A$, ...
0
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1answer
99 views

Reachability in Dynamic Line Graph

Given a directed line graph $G = v_1 \rightarrow v_2 \rightarrow \cdots \rightarrow v_n$, there are two operators, namely $\mathsf{move}(v_i, v_j)$: this operator moves $v_i$ to the position ...
3
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0answers
254 views

A sequence wherein the Kolmogorov complexity of the terms does not increase

I am looking for an algorithm $A$ - which for any non-null input string $s_1$ produces a sequence $s_1, s_2...$ such that : It can be proved in some axiomtic system $S$ that: $\...
24
votes
2answers
714 views

If machine learning techniques keep improving, what's the role of algorithmics in the future?

Let's look at the future some 30 years from now. Let's be optimistic and assume that areas related to machine learning keep developing as quickly as what we have seen in the past 10 years. That would ...
1
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0answers
105 views

Can we make a matrix stable by changing its upper-left submatrix?

A matrix $A$ is called strictly stable if its eigenvalues have negative real parts. Given a matrix $A \in \mathbb{R}^{n \times n}$, suppose we can change its upper-left $k \times k$ submatrix at will (...
3
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0answers
113 views

Factorizing semiprime $n=pq$ with $p \mid q-1$

Could we find a fast integer factorization algorithm for any large semiprime $n=pq$, if we know that $p \mid q-1$?
1
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0answers
91 views

multi-agent pickup and delivery algorithm and conflict resolution

I am looking for a pathfinding algorithm handling the following issues: multiple agents the computed paths for agents may not lead to collisions or deadlocks in space-time a stream of activities ...
1
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0answers
32 views

Sparse coding and matching pursuit algorithms

Is it true that all known sparse coding algorithms which work efficiently in practice don't have convergence proofs and always use an intermediate step of a matching/subspace pursuit algorithm on the ...
5
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2answers
240 views

Log space algorithms for modular decomposition tree

Can we have log space algorithms for modular decomposition tree (see definition) for any graph? If not, can we have log space algorithms for modular decomposition tree for any particular graph class? ...
1
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0answers
61 views

What is known about data structures for encoding a set while considering approximate Rank queries?

Consider a universe $\mathcal U\triangleq \{1,2,\ldots n\}$, and assume that we are given a set $S\subseteq \mathcal U$. There are many data structures that allow storing $S$ while answering Rank ...
7
votes
1answer
243 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 ...

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