All Questions
Tagged with co.combinatorics linear-programming
47 questions
3
votes
1
answer
154
views
How to maximize the variance of a subset of integers?
$\DeclareMathOperator{\Var}{Var}$Given the set of numbers $\Omega := \{1, \ldots, n\}, n \in \mathbb{Z}^+$, how can I choose a subset, $A$ of $\Omega$ , such that $\min(\Var(A), \Var(\Omega \setminus ...
- 1,432
1
vote
0
answers
29
views
Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
1
vote
1
answer
107
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 128k
1
vote
0
answers
99
views
Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
- 1,000
5
votes
1
answer
176
views
Efficient counting of integer solutions to linear system
In my research, I have a particular 18x18 matrix $\mathbf{A}$ which defines the linear system $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{-1}$ over the nonnegative integers. And I'm interested in ...
- 5,429
25
votes
3
answers
2k
views
Is the Ford-Fulkerson algorithm a tropical rational function?
The Ford-Fulkerson algorithm
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
- 33.8k
4
votes
0
answers
46
views
Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
0
votes
0
answers
145
views
Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
- 439
1
vote
0
answers
336
views
Closed-form solution of a particular linear program
(Note: I asked a similar question at math.stackexchange but the present one is more precise.)
I have a linear program of the form:
$$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$
$...
- 4,219
1
vote
1
answer
169
views
Best projection on non-convex discrete set with two constraints
I want to compute the projection of a vector $\left( x\right) _{1\leq
i,j\leq n}\in \lbrack 0,1]^{n\times n}$ on the following discrete set
$$
S=\left\{ x\in \{0,1\}^{n\times n}:x_{i,j}+x_{j,i}\leq 1;\...
- 128k
2
votes
1
answer
61
views
Counting the number of pair of d-uplets with upper bounded distance
Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_i = k$$ with $k$ ...
2
votes
2
answers
418
views
Lovasz theta function - uses
Lovasz theta function bounds the Shannon capacity of graphs. What are some other uses of the function - especially in asymptotic coding theory and optimization problems?
- 32.1k
16
votes
3
answers
1k
views
Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 32.1k
1
vote
0
answers
283
views
total unimodularity of a matrix
Let G be the node-arc incidence matrix of a given directed network (rows of $G$ correspond to nodes and its columns correspond to arcs). Let $B_1,\dots, B_K$ denote a partition of the nodes of the ...
- 393
5
votes
1
answer
424
views
What is the LP gap of vertex cover in planar graphs?
What is the LP gap of vertex cover in planar graphs?
The LP I refer to is min $\sum_{e \in E } c_e x_e \ \ $ subject to $ \ \ x_v + x_u \geq 1 \ \ \ \forall uv \in E $
$ c_e \geq 0 $ are ...
- 26
19
votes
4
answers
1k
views
Applications of linear programming duality in combinatorics
So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
- 761
3
votes
2
answers
437
views
convex polytope integer points
is there a simple proof for the following lemma:
An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.
- 1,991
5
votes
1
answer
394
views
Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$
The problem of giving an explicit formula for $A_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A_q(n,d)$ is given by the maximum number of codewords in a q-ary ...
- 297
9
votes
1
answer
295
views
Definition of packing property
Definition 1:
A clutter $C$ is said to have the packing property if $C$ and all of its minors satisfy the König property.
where,
vertex cover of $C$ is a set of vertices that have non-empty ...
- 30.5k
10
votes
0
answers
722
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
1
vote
1
answer
73
views
minimize number of unique elements in a vector
I was wondering if there is a simple or known way to minimize the number of unique elements in a decision variable (vector). Note that I'm not asking for minimization of nonzero elements (rank ...
- 14k
27
votes
5
answers
2k
views
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 43.2k
1
vote
1
answer
206
views
Show $0-1$ Knapsack is polynomially reducible to this problem
I have already posted this question here but have not received an answer so I am cross-posting with hope to reach a larger amount of mathematicians:
Let $T=\{1,\cdots,n\}$ and consider the ...
CommunityBot
- 1
0
votes
1
answer
81
views
Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
- 13.4k
8
votes
2
answers
246
views
Are sums of 0-1 Pareto efficient vectors Pareto efficient?
Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:
The entries of $A$ are $\in \{0, 1\}$.
For all pairs of columns $u, v$ of $A$ the entries of $u - ...
- 11
1
vote
0
answers
1k
views
Number of different combinations in a 0-1 knapsack problem with integer weights [closed]
My question is actually very similar to this other one: Given a vector of positive integers, count the number of combinations which have a sum that produces a different value. But, since this previous ...
CommunityBot
- 1
1
vote
0
answers
55
views
Separation on discrete set
Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$.
Define linear functions $f(x)= a_1x_1+ \...
- 61
0
votes
1
answer
204
views
Is the linear production game a convex game?
In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem.
Does anyone know if the LPG is a convex ...
- 19.6k
2
votes
0
answers
177
views
Formulating shortest path as submodular minimization
I'm curious about the general question of whether any combinatorial optimization problem with polynomial time solution can necessarily be reformulated as minimizing a submodular function.
The answer ...
- 21
2
votes
0
answers
71
views
Select n vectors from k vectors (in 3D) such that each component of the resultant vector >= each component of a given vector M
this was left unanswered for 1 week on MStackExchange, so I thought MOverflow would be more appropriate. Thanks :)
Let $R = (R_x, R_y, R_z)$ be the resultant vector of the n vectors and $M = (M_x, ...
- 21
3
votes
2
answers
2k
views
ILP for minimum edge coloring problem
We know that for a graph $G=(V,E)$, minimum edge coloring is a coloring of
$E$, i.e., a partition of $E$ into disjoint sets $E_1, E_2, \dots, E_k$ such
that, for $1 \leq i \leq k$, no two edges in $...
- 2,339
1
vote
1
answer
219
views
approximate diameter of polytopes in high dimensions
I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 \...
- 151k
1
vote
0
answers
171
views
Finding all feasible solutions
Let $u$ be a $n_{max} \times m$ matrix. Let $z$ be a $n_{max} \times s_{max} \times n_{max}$ cube. Let $w$ be a $n_{max} \times 1$ vector. All the three matrices can have values from the set $\{ 0, 1\}...
- 175
3
votes
1
answer
340
views
Name search for special Linear Integer Program
I am looking for a name for the following question in literature!
(and if you know it, then it would be great)
I couldn't find it and due to wide audience here, presumably you know more. Thank you
$...
CommunityBot
- 1
2
votes
1
answer
426
views
Network flows with shared capacities
Suppose we have a flow network, with capacity constraints on weighted sums of arc flows, such as:
$$2 f(1, 2) + 3 f(4, 5) + f(3, 7) \leq 10,$$
where $f(1, 2)$ denotes the flow through arc $(1, 2)$....
CommunityBot
- 1
6
votes
3
answers
11k
views
Maximum flow with negative capacities?
I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient ...
- 11
7
votes
1
answer
224
views
A polytope with a bound on the sum of any $k$ variables
Let $2\le k\le n-1$ and define the polytope
$$P_k(n) = \lbrace (x_1,\ldots,x_n) \in \mathbb{R}^n :
-1\le x_{i_1}+\cdots +x_{i_k} \le 1 \text{ for all } 1\le i_1\lt\cdots\lt i_k\le n\rbrace.$$
There ...
- 13.6k
1
vote
0
answers
140
views
Reduce a Combinatorial problem
It is given n sets with k vectors. (k is element-wise positive or zero)
Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal.
What i also know but is ...
- 11
4
votes
2
answers
212
views
combinatorial and linear duality
Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual $\...
- 24.8k
5
votes
0
answers
194
views
A linear optimization problem on a graph
Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
- 2,288
7
votes
1
answer
360
views
Does the Hirsch conjecture hold for $n < 2d$?
The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$.
After being open for decades, Francisco Santos has ...
- 1,502
2
votes
1
answer
137
views
Design constraint systems over the reals
This question is inspired by the discussion at this problem.
Suppose I have a design consisting of a finite point set $U$ of size $|U|=m_{\emptyset}$ and a family of $n$ subsets (sometimes called ...
CommunityBot
- 1
2
votes
0
answers
917
views
Guessing game with guess cost
This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
- 4,952
2
votes
0
answers
215
views
Number of breakpoints in parametric maximum flow problems
The parametric maximum flow problem can be formulated as
$$f(\lambda) = \min_{x\in\{0,1\}^n} \left( \sum_{i}(a_i + b_i\lambda)x_i + \sum_{i,j}c_{ij}x_ix_j \right),
$$
where all $c_{ij}<0$ (so that ...
- 567
1
vote
1
answer
531
views
Split sum into equal terms
Given a sum of $l$ integers $r_1+...+r_k+...+r_l$ and an integer $t$.
Find indices
$1 < p_1 <...< p_h <...< p_{t-1} < l$
such that in sum
$(r_1+...+r_{p_1})+...+(r_{p_{h-1}+1}+......
- 18.6k
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
10
votes
1
answer
2k
views
Sum of difference moduli vs. sum of modulus differences
This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.
Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...
- 61.9k