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Questions tagged [optimization]

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

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1 vote
2 answers
337 views

What is the fastest static comparison sort? What is the proper term for "static"?

In a standard comparison sort, you perform a comparison and your next action is based off of the result of that comparison. What if this was not allowed, and you had to request all the results at the ...
1 vote
0 answers
86 views

Non-convex optimization with correlated minima

I am thinking of non-convex optimization problems where the minima are somehow correlated. Maybe there are symmetry relationships among minima or maybe there is regularity in spacing among minima in ...
1 vote
1 answer
112 views

How to calculate processor throughput boundary in CSAPP?

In Chapter 5.7 of CSAPP, author list out the throughput boundary of Intel Core i7 Haswell while executing the operation Integer addition and multiplication, floating points addition and multiplication....
0 votes
0 answers
69 views

What work on min max connectivity problems has there been?

For instance has min max spanning/steiner/prize-collecting tree been studied. i.e. each edge $e$ has costs $c_{v,e}$ of each resource $i$. And we wish to find a spanning tree minimizing the maximum ...
0 votes
0 answers
64 views

Has multiobjective prize collecting steiner tree or TSP been studied?

Suppose we have a graph $G$ a root $r$ and each node $v$ has some amount of $c_{v,i}$ of each resource $i$. I connect a set of nodes to the root that maximizes the minimum amount of any resource using ...
4 votes
1 answer
1k views

Knapsack with dependent profits (pairs of items)

I'm working on a problem which MAY be reduced to the following version of Knapsack: Suppose two items $e_i$ and $e_j$ have profit $p_i$ and $p_j$ respectively. However, if both items are present in ...
0 votes
0 answers
47 views

cutting plane method for convex optimization

The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector, we can cut the feasible set into ...
-1 votes
1 answer
52 views

Computability/Complexity of optimization problems in general

Dear StackExchange community, I have a question, or better phrased I am confused and would like to be enlightened by you! So assume we have a (optimization) problem like that: Instance: Let $f:\...
1 vote
0 answers
39 views

Convergence rates for the iterates of SGD on Lipschitz convex functions

Let $f:X \rightarrow \mathbb{R}$ be a convex and $L$-Lipschitz continuous function. Suppose $f^* = \min_{x \in X} f(x) \in \mathbb{R}$ and let $X^* = \{x \in X : f(x) = f^*\}$. For a non-negative ...
1 vote
0 answers
37 views

Placing a circle in a point cloud

I need to place a circle with fixed radius in a cloud of points. The circle also must lay in a polygon (the points are also in that polygon) This circle has to contain as many points as possible. Are ...
3 votes
0 answers
72 views

Hardness of Approximation for Three Matroid Intersection

I am searching for the best known hardness of approximation bound for three matroid intersection. The input is three matroids on the same ground set which are accessible using three different ...
4 votes
1 answer
213 views

Is Optimal Swap Sorting NP-Hard?

Given an array of integers with duplicates, find the minimum number of swaps to sort the array. According to this question, the problem is NP-Complete but the reference given proves NP-Completeness ...
3 votes
2 answers
166 views

Given a weighted graph with $pk$ nodes find a min weight forest with $p$ components each containing exactly $k$ nodes

Given a weighted graph with $pk$ nodes find a min weight forest with $p$ components each containing exactly $k$ nodes. Does this have a constant approximation? ($p,k$ and the graph are all part of the ...
2 votes
1 answer
101 views

Approximating the utilitarian welfare minus a constant

Assume we have $n$ agents and $m$ indivisible goods that need to be allocated among the agents such that their sum of utilities is maximized. Denote the set of allocations by $\mathcal{A}$ and the ...
0 votes
1 answer
292 views

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

Crossposted at Mathematics SE and MathOverflow 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) $\...
2 votes
1 answer
117 views

Is beta normalization used for program optimization?

Beta normalization reduces a lambda term to its beta normal form, if it exists. The beta normal form is a computationally equivalent term with no "redundant" computation, in a sense; for ...
2 votes
1 answer
96 views

Maximum cardinality disjoint cycle cover in undirected graphs

I call a maximum cardinality disjoint cycle cover of a graph a vertex-disjoint cycle cover containing the maximum possible number of cycles in the graph. What is known about the complexity of this ...
0 votes
0 answers
60 views

Is there an algorithm that finds a minimum vertex cover with an approximation factor of 3/2 for a planar graph?

Is there an algorithm that finds a minimum vertex cover with an approximation factor of 3/2 for a planar graph?
22 votes
1 answer
3k views

Solving semidefinite programs in polynomial time

We know that linear programs (LP) can be solved exactly in polynomial time using the ellipsoid method or an interior point method like Karmarkar's algorithm. Some LPs with super-polynomial (...
23 votes
3 answers
1k 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 ...
0 votes
0 answers
32 views

Nonlinear GAP similar to Min-GAP but with minimum quantities and without capacity

I have $m$ items and $n$ bins where each item $i$ and bin $j$ has a value $v_{i,j}$. Each bin $j$ has a value $V_j$. I want to pack the items into the bins such that (1) I minimize the ratio of the ...
2 votes
2 answers
133 views

Linear Programming Sensitivity to Matrix

Consider a linear program in the following standard form: \begin{align*} &\max c^T x &\\ &\mbox{subject to:}\\ &A x \preceq b\\ &x \succeq 0 \end{align*} Its dual is \begin{align*}...
4 votes
0 answers
134 views

Which variant of the ellipsoid method was used for the Santa Claus problem?

As one of the steps in the article The Santa Claus problem (Bansal and Sviridenko, 2006) the following linear problem was considered (at the end of the second page, as the dual): \begin{align*} &\...
0 votes
1 answer
185 views

Solving Grouped Weighted Job Scheduling with Release Times and Deadlines on a Single Machine with Multiple Availability Intervals

I'm working on a scheduling problem where I need to schedule a set of n weighted jobs that are partitioned into m groups, where ...
2 votes
0 answers
43 views

Submodulare welfare maximization: is an additive approximation algorithm known?

Sudmodular welfare maximization is the problem of allocating items among agents with different valuations, represented by submodular set functions, such that the sum of agents' values is as large as ...
-1 votes
1 answer
74 views

Find Combinations of fibonacci values to approximate a target value given $F(A,B,C,D) = (A + B + C) / D$

I am able to solve this using brute force but curious if there is a better approach. Given the function $F(A,B,C,D) = (A + B + C) / D$ where each variable is in the first 7 distinct values of the ...
1 vote
0 answers
25 views

Optimization: Turning a sparse graph of probabilities into the maximum likelihood DAG

I have a sparse matrix of probabilities that I want to turn into a DAG. If x[m,n] = pr it means that m is a descendent (direct or transitively) of n with probability pr. I want to construct a DAG over ...
1 vote
0 answers
32 views

Justification of smoothness of component functions

In stochastic optimization, the common objective is to minimize $f(x) = \frac 1N \sum_{i=1}^N f_i(x)$ (there is a different formulation, but it doesn't matter for this question). For nonconvex ...
2 votes
1 answer
135 views

Optimization problems where the solver can choose which variables are continuous

A typical optimization problem looks like the following, where $f$ represents the objective and $g$ the constraints: $$ \text{maximize}~~~f(x1,\ldots,x_n)~~~\text{subject to}: \\ g(x_1,\ldots,x_n)=0, \...
-2 votes
1 answer
256 views

What is the reason to believe that quantum heuristic algorithms can solve NP-Complete problems?

There is an ever going trend to believe that a large number of NP-Complete or NP-Hard problems can be solved using quantum heuristics. I have observed, a common trend, to take any sort of ...
26 votes
3 answers
1k views

Optimization problems with minimax characterization, but no polynomial-time algorithm

Consider optimization problems of the following form. Let $f(x)$ be a polynomial-time computable function that maps a string $x$ into a rational number. The optimization problem is this: what is the ...
3 votes
1 answer
229 views

Complexity class of optimization problems whose fractional relaxation is polynomial-time solvable

It is known that the problem of integer linear programming is NP-hard, but its fractional relaxation can be solved in polynomial time. The concept of fractional relaxation can be applied to any ...
0 votes
1 answer
76 views

An inequality about median of points in higher dimensions

Let $S$ be a set of points in $\mathbf{R^d}$ and let $m$ be the median of this set of points, i.e. $\sum_{x \in S} || x - y||$ is minimized when we have $y=m$. Now let $z$ be an arbitrary point in $\...
6 votes
2 answers
923 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
1 answer
225 views

Can the ellipsoid method be used with a randomized separation oracle?

Suppose we are trying to solve the following optimization problem: $$ \text{maximize } ~~ c\cdot y \\ \text{subject to } ~~ y\in S $$ where the region $S$ is described by an exponential number of ...
4 votes
0 answers
81 views

Submodular welfare maximization: what is the best known approximation ratio of a deterministic algorithm?

In the submodular welfare maximization problem, there is a set $M$ of items that should be partitioned among $n$ agents. Each agent $i$ has a value function $v_i: 2^M\to \mathbb{R}_+$. All value ...
0 votes
0 answers
95 views

How to prove that a given class of convex programs cannot be solved by linear programming?

Given the following program, where $f, g$ are convex functions: $$ \text{minimize}~~ f(x) \\ \text{subject to}~~ g(x)\leq 0 $$ the problem can be solved by convex programming algorithms, but it would ...
8 votes
3 answers
780 views

Is that edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{u\in V} ~\left( d_{...
3 votes
1 answer
339 views

Is this node permutation optimization NP-Hard?

Let $G=(V,E)$ be an undirected graph and let $\pi$ be a permutation of the vertices in $V$. For a node $v\in V$, we denote by $\text{succ}_{\pi}(v)$ the set of neighbors of $v$ that occur after $v$ in ...
0 votes
0 answers
40 views

Computational complexity of CVaR calculation

I am currently looking for literature discussing the computational complexity of CVaR calculation. At this point the only work I have found is the following. Mavronicolas, Marios, and Burkhard Monien. ...
3 votes
1 answer
195 views

Solving linear programs with special structure

We have an application and at some point we need to solve a linear programming problem that looks like this: $$ \min\ w_{1,2} + w_{3,4} + w_{5,6}\\ x_i - x_j \leq c_{ij},\ \forall\ (i,j) \in C\\ x_1 - ...
3 votes
2 answers
223 views

What is this graph problem, and how hard is it?

My problem is quite simple to state, so it surely must have a name: Given a graph $G=(V,E)$ with edge weights $w(e) \in \mathbb{Z}$, find a $V' \subseteq V$ that maximizes $\sum_{e \in E' } w(e)$, ...
1 vote
1 answer
82 views

A bound that follows from submodularity

I am studying Lemma 1 of this paper: The Adaptive Complexity of Maximizing a Submodular Function. The proof appears on page 11. I got stuck on this inequality: where $f$ is a monotone submodular set ...
1 vote
0 answers
93 views

How to solve the following continuous optimization problem?

Consider a function $f: X\times Y\times N$, where $X, Y \subseteq \mathbb{R}^m$ are convex sets, and $N = \{1,2,\dots,n\}$. We additionally know that $f(\cdot,y,S)$ is convex for fixed $y,S$ $f(x,\...
1 vote
0 answers
66 views

Unbounded Knapsack Instance with a Single Optimum that takes each Item Once?

Consider the Unbounded Knapsack Problem (UKP): We are given a set of $n$ items $I = \{1,\ldots,n\}$ of integral weights $w_1, \ldots, w_n \in \mathbb{N}$, integral profits $p_1, \ldots, p_n \in \...
0 votes
0 answers
59 views

Uniformly redistributing items across bins. What problem is this?

I'm trying to find reading material on a particular problem I'm interested in, but I don't know the terms to search. Problem assumptions/definitions: We have finite number of items I with weights [0, ...
1 vote
0 answers
46 views

Will the maximum entropy joint distribution given a known set of marginal distributions have the maximum plausible support?

Define $[n] = \{1, 2, ..., n\}$. Given a distribution $P : \{0, 1\}^{[n]} \rightarrow [0, 1]$ and a subset $S \subseteq [n]$, we can define the $S$-marginal of $P$, $P_S : \{0, 1\}^S \rightarrow [0, 1]...
23 votes
2 answers
4k 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
0 answers
83 views

Packing k vertex trees

Consider a graph $G=(V,E)$ with $n$ vertices. What do we know about packing of $k$ vertex trees, Both integral and fractional packing are interesting. $k=2$, it is just the number of edges, hence ...
0 votes
0 answers
26 views

Quantifying the cost of procedures

Is there any research on quantifying the cost of a procedure, with regard to compiler optimization? I.e. assigning some kind of cost in terms of CPU time or memory to a procedure, either so the ...

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