general questions about selecting a best element from some set of available alternatives.

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5
votes
0answers
155 views

Maximizing difference of a submodular and a modular function

I'm considering a network planning problem which is stated as follows: From the given ground set $\mathcal{V}$, select $\mathcal{A} \subseteq \mathcal{V}$ such that \begin{equation} f(\mathcal{A}) - ...
3
votes
3answers
193 views

Facility location problem with a cost function

I'm struggling with a facility location problem. In its original form the problem is quite straightforward: Given a matrix of distances between cities, I have to pick a minimal number of centers from ...
3
votes
1answer
353 views

Maximizing a convex function with linear constraints

I have the following optimization problem: $$ \arg\max_{\{\phi_p\}_{p=1}^M}\sum_{i=1}^N \max_{p=1,\ldots,M}\{\phi_p a_{ip}\} \mbox{ such that }\sum_{p}\phi_p\leq 1\mbox{ and }0\leq \phi_p\leq ...
25
votes
3answers
784 views

Rounding to minimise the sum of errors in pairwise distances

What is known about the complexity of the following problem: Given: rational numbers $x_1 < x_2 < \dotso < x_n$. Output: integers $y_1 \le y_2 \le \dotso \le y_n$. Objective: minimise ...
-5
votes
1answer
234 views

Solving a system of linear inequations

Consider the following system of inequalities: $Ax=b$; $x\geq 0$; A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. a) How feasibility of this system can ...
0
votes
1answer
176 views

Boyd & Vandenberghe, question 2.31(d). Stuck on simple problem regarding interior of a dual cone

In Boyd & Vandenberghe's "Convex Optimization", question 2.31(d) asks to prove that the interior of the dual cone $K^*$ is equal to (1) $\text{int } K^* = \{ z \mid z^\top x > 0 $ for all $ x ...
0
votes
0answers
88 views

Control optimization for a black box model

Computer science is not my major, hence my question is two folded : how is defined the problem I have (if I can name it, it will be easier for me to look for references) ? what kind algorithm would ...
14
votes
1answer
405 views

Does zero integrality gap imply zero duality gap for certain problems?

We know that if the gap between the values of an integer program and its dual (the "duality gap") is zero, then the linear programming relaxations of the integer program and the dual of the ...
0
votes
1answer
169 views

putting objects in buckets — optimization problem

Let $\mathcal B$ and $\mathcal O$ denote finite sets ("buckets" and "objects", respectively) and let $$ E: \mathcal O \times \mathcal B \to \mathbb{R}_{>= 0} $$ be a function. Is there an ...
4
votes
1answer
194 views

Algorithms for Interval Coloring with Capacities and Demands

We are given a graph $G = (V,E)$, where each edge $e \in E$ has a capacity $c_e$. There are a set of $k$ requests $R_1,\ldots,R_k$. Request $R_i$ has a demand $d_i$, and has an interval $I_i = ...
8
votes
2answers
312 views

Hardest optimization problems in NC

When learning optimization problems, we usually consider linear programming (or more generally: convex optimization) as the simplest example. It is solvable in polynomial time, and has relatively easy ...
5
votes
1answer
233 views

Is it possible to include an optimization algorithm's own running time in the cost?

Fix $f(\_; \_)$, which represents a class of optimization problems, ie, for a specific $P$, $f(\_; P)$ is a function we'd like to optimize. Now, we have an algorithm $A$ that takes as input $P, a, ...
0
votes
1answer
119 views

Name of the problem to find the maximum number of characters covered by a set of strings

I am interested in the following problem. One has two collections $Q$ and $T$ of strings, and a set $A$ of alignments of strings in $Q$ to strings in $T$. I want to find a subset $A'$ of $A$ that (i) ...
10
votes
3answers
274 views

min hitting set of every base of a matroid

We are given a matroid. Our goal is to find a set of elements of minimum size that has non-empty intersection with every base of the matroid. Is the problem studied before? Is it in P? For example, ...
0
votes
1answer
180 views

PH and Optimization Problems

If we have a machine which can solve any problem in the second level of PH, can this machine solve optimization problems which is generalized version of NP-complete problems such as MAX-CLIQUE or ...
3
votes
0answers
290 views

Minimal context-free Grammar for a special one-letter Language

For natural numbers $n \geq 5$, $m \geq 2^{n-2} + 1$ the following context-free language is given: $$ L_{n,m} = \{ a^i | 2 \leq i \leq m \} \setminus \{a^{2^i}|2 \leq i \leq n-2\} $$ Find and ...
9
votes
2answers
308 views

A variation on discrepancy involving random graphs

Suppose we have a graph on $n$ nodes. We would like to assign to each node either a $+1$ or a $−1$. Call this a configuration $\sigma \in \{+1,−1\}^n$. The number of $+1$s that we have to assign is ...
14
votes
2answers
310 views

Complexity of optimization over unitary group

What is the computational complexity of optimizing various functions over the unitary group $\mathcal{U}(n)$? A typical task, arising often in quantum information theory, would be maximizing a ...
1
vote
1answer
427 views

Primal vs dual decomposition methods

I noticed that dual decomposition methods tend to be preferred over primal ones in the large scale optimization literature (here are some examples: (1), (2)). The reason seems to be, from what I ...
3
votes
1answer
215 views

Minimum length walk from s to t covering a subset of vertices

I want to find the current literature for the following problem (I have searched on google/asked friends/some Profs didn't get much useful results yet): Input: weighted undirected graph G = (V,E), ...
5
votes
1answer
871 views

Optimality of Greedy algorithm for minimization Knapsack Problem

Given items with weight $w_i$ and profits $p_i$, minimization Knapsack problem is to pick a subset of items $I$, s.t. $\sum_{i\in{I}}{w_i} \geq W$ and $\sum_{i\in{I}}{p_i}$ is minimized. The greedy ...
0
votes
1answer
520 views

Linear Programming with Modulo Linear Constraints

Given $G = (V,E)$ I can formulate a relaxation of graph $K$-coloring as: find feasible point s.t. $\min \sum_{ij}v_{ij}$ for all $(i,j)$ in $E$ $z_{ij} \le (c_i - c_j) \bmod k$ (i) $z_{ij} \le ...
4
votes
1answer
123 views

Minimum ($\ell_1$ or $\ell_2$) norm of sum of edge length in multigraph over linear ordering of vertices

Suppose we have a multigraph with vertex set $V$ where for each $v \in V$, $d_v > 0$ is the diameter of the vertex. We want to put a linear ordering on the set of vertices such it minimizes ($L_1$ ...
21
votes
3answers
755 views

Clique problem on fixed graphs

As we know, the $k$-clique function $CLIQUE(n,k)$ takes a (spanning) subgraph $G\subseteq K_n$ of a complete $n$-vertex graph $K_n$, and outputs $1$ iff $G$ contains a $k$-clique. Variables in this ...
8
votes
1answer
863 views

computing the minimal NFA for a DFA

Many years ago I heard that computing the minimal NFA (nondeterministic finite automaton) from a DFA (deterministic) was an open question, as opposed to the vice versa direction which has been known ...
1
vote
1answer
139 views

Distribution of variable sized images/boxes(only aspect ratio given) on a 2D area

I'm trying to find a solution for the following problem. You have a set of pictures or let us assume they are just boxes with a given aspect ratio. And you have a two-dimensional area with width and ...
5
votes
2answers
419 views

machine learning for code and compiler optimization?

I am looking into ML for generating more efficient code (i.e. compile time and run time heuristics). I have a phd (compilers, hpc), but very little ML experience. I would appreciate any references ...
2
votes
2answers
401 views

Is the following optimization problem NP-hard?

Set S, which is an non-empty finite subset of $\{ (i,j) : i, j \in N \land i \neq j \}$, is given. E.g. $S=\{(1,3), (2,3), (1,4), (2,4), (3,1), (3,4)\}$ . For each element $(i,j)$, we have weight ...
1
vote
0answers
177 views

Minimization on a binary matrix

Assume you are given a matrix $$ X= \begin{bmatrix} x_1^1 & x_1^2 & \dots & x_1^m \\ x_2^1 & x_2^2 & \dots & x_2^m \\ \vdots & \vdots & \ddots & \vdots \\ ...
2
votes
2answers
559 views

For efficient algorithm on “minimization” knapsack problem

Suppose there are two arrays of positive numbers, $a[\cdot]$ and $b[\cdot]$, and value $B>0$. How to pick a set of indexes $I$, so that $\sum_{i\in{I}}{b[i]}\geq{B}$ To minimize ...
18
votes
1answer
322 views

Finding good induced subgraph

You are given a graph $G = (V,E)$ with $n$ vertices. It might be bipartite if you want. There are $m$ sets of edges $E_1,\ldots, E_m \subseteq E$ (say disjoint). I am interested in the problem of ...
7
votes
1answer
207 views

Assignment problem with sum replaced by max

In the assignment problem, one tries to find $f$ such that the cost function $$ \sum_{a\in A} C(a,f(a)) $$ is minimized. Here $f$ is a bijection between sets $A$ and $B$ of equal finite ...
3
votes
1answer
111 views

What is the correct name for the space of genotypes and fitness?

I'm looking for a formal definition of the space that consists of the dimensions gene 1-N and the fitness. In literature there is often the search space mentioned, but it only contains all genes. If ...
2
votes
1answer
128 views

Parallel algorithms to find the optimum of polynomials

Are there any non-trivial parallel algorithms to find the optimum (local or global) of polynomials? By trivial, I mean something which is an obvious application of a serial algorithm. For example, one ...
6
votes
1answer
330 views

What kind of optimization problem is that?

I am new here as a writer but I read this group from time to time. I am thinking about the following problem. Problem description: Assume we have an area $A$ of size $1000 \times 1000$ cells. One ...
6
votes
1answer
269 views

Approximation schemes for P-complete problems?

What work has been done on approximation schemes for $\mathsf{P}$-complete optimization problems? Would the desired approximation algorithms here be "fully log-space approximation schemes" or "fully ...
4
votes
1answer
166 views

Minimal sum of matrix elements

Here's my attempt to explain the problem in mathematical language: $$ \text{Given square matrix A} $$ $$ \left( \begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,N} \\ a_{2,1} ...
4
votes
1answer
266 views

Stochastic Gradient Descent with integer arithmetics

Most implementations of stochastic gradient descent (SGD) rely on floating points. Is there implementations using infinite or finite precision integer arithmetics ?
5
votes
1answer
466 views

Tree width of a particular graph

What is the tree-width of the graph $G = (V_1 \cup V_2 \cup \dotsb V_n, E)$ where the connected components of an induced subgraph of any neighboring set of vertices (i.e. $G[V_i \cup V_j], i = j - ...
4
votes
1answer
457 views

Set optimization problem - is it np-complete?

Set $S=\{e_1,\cdots,e_n\}$ is given. For each element $e_i$, we have weight $w_i>0$ and cost $c_i>0$. The goal is findIng the subset $M$ of size $k$ that maximize the following objective ...
6
votes
0answers
231 views

View of Multiplicative Weights in contexts of combinatorial optimization, low-regret/online optimization, and entropy-regularized gradient descent?

Also called exponentiated gradient. I understand these are three places where multiplicative weights shows up (i.e. $w_{t+1} = w_{t}e^{- \text{loss}(w_{t})}$ or variations. And I understand a bit ...
15
votes
1answer
669 views

Solving a linear diophantine equation approximately

Consider the following problem: Input: a hyperplane $H = \{ \mathbf{y} \in \mathbb{R}^n: \mathbf{a}^T\mathbf{y} = {b}\}$, given by a vector $\mathbf{a} \in \mathbb{Z}^n$ and $b \in \mathbb{Z}$ in ...
0
votes
0answers
164 views

Find all the special graphs which can reduced to the shortest paths graph

I have a directed weighted graph $G = (V, E, W)$. $\forall i, j \in V, \exists e(i,j) \in E$ and $w(i,j)$ is an integer which could be $+\infty$, but never $-\infty$. There does not exist any ...
0
votes
0answers
303 views

Algorithm to maximize profit: ways to solve/approach? (Advanced NP-Complete)

This one's hard, so all help really appreciated! I know it is NP-Complete and thus cannot be solved in polynomial time, but looking for help in analysis, i.e. what type of NP-Complete problem it ...
7
votes
3answers
313 views

Hardness of MAX-CUT on sparse graphs

Let a weighted graph $G(V,E)$, where the weights are real (positve and negative). Assume that $G$ has $\mathcal{O}(n\log n)$ edges. How fast can we compute MAX-CUT on this graph? Can we compute ...
0
votes
0answers
116 views

covering an NxN grid using overlapping vs. non-overlapping windows residing k points in each

Let the problem, $P_{overlapping}$, be the following. We have an $N_1 \times N_2$ grid. Each cell of the grid can have the value either 0 or 1. Assume that we have $a \times b$ overlapping windows as ...
7
votes
2answers
469 views

Capacitated multiple vehicle routing problem with handovers

I'm looking for literature about a variant of the capacitated vehicle/fleet routing problem (a.k.a. VRP, CVRP, etc.) that takes into account the possibility of handovers between multiple vehicles, ...
7
votes
1answer
768 views

Sorting points such that the minimal Euclidean distance between consecutive points would be maximized

Given a set of points in a 3D Cartesian space, I am looking for an algorithm that will sort these points, such that the minimal Euclidean distance between two consecutive points would be maximized. ...
5
votes
0answers
136 views

On which classes of graphs is resource constrained shortest path (RCSP) NP-hard?

I'm looking to link a problem I'm working on to a known NP-hard problem. I think I can model my problem as a resource constrained shortest path problem. However, the structure of my graph is not ...
0
votes
1answer
379 views

Maximizing strictly increasing convex function

Let the objective be to maximize the sum of $f_i(x_i)$ where all $f_i$ are strictly increasing convex functions. Maximizing a convex function is hard as a local maximum might not be a global one. ...