Questions tagged [combinatorial-optimization]
Combinatorial optimization typically deals with optimizing over a finite set of objects that have some combinatorial structure (e.g. trees, matchings, matroids). Approximation algorithms, polyhedral methods, and integer programming are all on topic.
437
questions
1
vote
1
answer
78
views
Do we really need degree constraints for ILP formulations of TSP problems
The Dantzig-Fulkerson ILP-formulation of the symmetric TSP is
$$\min\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^n c_{ij}x_{\lbrace i,j\rbrace}\quad\text{s.t.}\\ \sum\limits_{j\ne i,\,j=1}^n x_{\lbrace ...
- 12.7k
1
vote
0
answers
41
views
Vertex cover via maximally unbalanced spanning trees
The vertex cover problem asks for a smallest subset $U\subseteq V$ that is adjacent to all edges of a symmetric graph $G(V,E)$.
Inspired by the observation that led to this question Perfectly balanced ...
- 12.7k
5
votes
1
answer
170
views
Graph combinatorial optimization problem
Let $G$ be a simple graph with vertex set $V$, such that for any two vertices $u,v\in V$, we have at least $k$ edge-disjoint paths of length $2$ (i.e., formed by $2$ edges) connecting $u$ with $v$. ...
- 1,677
1
vote
1
answer
156
views
Combinatorial graph optimization problem on integer adjacency matrices
We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.
Let $z_{i,j}:=\frac{M_{i,j}}{M_{i,j}+\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and
$z:=\...
- 1,677
5
votes
1
answer
425
views
Menger's theorem with restrictions on where the paths can begin and end
Let $k\in\mathbb N$. Given a finite graph with two subsets of vertices $X$ and $Y$, Menger's Theorem gives a criterion for when there are $k$ pairwise disjoint paths starting in $X$ and ending in $Y$.
...
- 1,388
2
votes
1
answer
136
views
Generating short Hamilton cycles from complete graphs
Let $G(V,E)$ be a complete symmetric graph without self-loops or parallel edges; depending on the context the edges may however be interpreted as a pair of antiparallel arcs of equal weight.
A vertex ...
- 12.7k
1
vote
1
answer
379
views
Knapsack like problem with nonnegative weight constraint
I am dealing with a knapsack-like problem with one difference from the conventional problem: the “weights” can be positive or negative and the constraint is $\sum w_i x_i \ge 0$ instead of $\sum w_i ...
- 13
-1
votes
1
answer
93
views
Finding a $k$-subset which maximizes a matrix sum
Let $M\in \mathbb{R}^{N\times N}$ be a given matrix and $k\ge 2$ be a given integer. Then my question is the following optimization problem:
Is there a polynomial-time solution to the following ...
9
votes
1
answer
285
views
Partitioning a set of lattice points in the plane into rectangles
The "long comment" by Pietro Majer on Reference for puzzle on dividing piles and scoring products suggests the
following problem. Let $S$ be a finite subset of $\mathbb{Z}\times
\mathbb{Z}$. ...
- 49.5k
3
votes
1
answer
120
views
Upper bound for the crossed-terms of a sum of multinomial coefficients
I am trying to upper bound the variance of a centered tree and I would like to get an upper bound which would look like : $$\sum\limits_{\substack{ (l_1, ..., l_d) \neq (k_1, ..., k_d), \\ \sum_{j=1}^...
- 55
0
votes
0
answers
49
views
Does LKH perform best with $\mathrm{1\unicode{x2013}trees}$
The LKH heuristic essentially generates sequence connected graphs with $n$ edges by means calculating minimum-weight spanning trees of $n-1$ of the vertices and connects the unspanned vertex to the ...
- 12.7k
0
votes
1
answer
102
views
Mapping problem reminiscent of Mastermind
Given 2 finite sets $S$ and $M$, with $\operatorname{card}(S) \geq \operatorname{card}(M)$, and an item $z \not\in M$. There is an unknown function $f: S \to M \cup \{z\}$, which is known to be one-to-...
1
vote
0
answers
67
views
Facility location and traveling salesman
This question is based on Distributing points evenly on a sphere and Facility location on manifolds
The 'dispersal problem' (which can be mapped to packing disks in many cases) places $n$ points in a ...
- 5,493
0
votes
1
answer
117
views
Integer programming for bin covering problem
I encounter an integer programming problem like this:
Suppose a student needs to take exams in n courses {math, physics, literature, etc}. To pass the exam in course i, the student needs to spend an ...
0
votes
1
answer
407
views
Finding a subset of vectors whose sum is close to a given vector
Given a set of vectors $x_1,...,x_n\in\mathbb{R}^d$ and a vector $y$, find a subset $I\subset\{1,2,...,n\}$ such that $\|\sum_{i\in I} x_i-y\|$ is as small as possible. Here $\|.\|$ can be any norm, ...
- 31
3
votes
1
answer
518
views
Magic circle (instead of magic square)
Motivation. I stumbled over this riddle (unfortunately in German): the goal is to fill the numbers $1\ldots7$ (or, equivalently, $0\ldots6$) into the $7$ little circles so that the sums of all numbers ...
- 45.5k
7
votes
1
answer
169
views
Metric TSP with integer edge cost
Given a metric TSP with integer edge cost upper-bounded by a constant $C_{\max}$, can we find an poly-time algorithm solving this TSP instance?
- 436
0
votes
0
answers
52
views
What do optimal tours tell about finite point sets?
Let $T_{\mathrm{MIN}}$ and $T_{\mathrm{MAX}}$ denote the shortest, resp. longest Hamilton cycle through a set of $n=2k+1$ points.
Let further $S_{\mathrm{MIN}}$ and $S_{\mathrm{MAX}}$ be the "...
- 12.7k
0
votes
0
answers
36
views
Approximabilty of submodular over modular maximization
Given a non-decreasing, normalized, submodular function $f : 2^{[n]}\mapsto \mathbb{R}_+$ and a modular non-decreasing function $g$, I am wondering what is the best approximation ratio I can hope for ...
- 171
3
votes
1
answer
286
views
Is anything written about winning the "Dollar Game" in the minimal number of moves?
I run some Master's projects on Chip-Firing games, using the Holly Krieger's Numberphile video on the topic as an initial motivation, and going on to prove the main theorem stated there (that you can ...
- 2,362
0
votes
1
answer
42
views
Maximize this score function on a directed tree
Let $\mathbb N_0^\ast$ be the set of all finite words/sequences over $\mathbb N_0$ and $\varepsilon$ the empty word. For a word $a=(a_1,\ldots,a_n)$ we define $\operatorname{len}a:=n$, $\Sigma a:=\...
- 237
1
vote
0
answers
74
views
Gadgets that reduce NP hard problems to the MAX-3-DCC Problem
A while ago I had asked for benchmark instance for the MAX-3-DCC Problem but did not receive any feedback.
This question is a follow up aimed at concrete examples of reductions of NP hard problems to ...
- 12.7k
0
votes
1
answer
191
views
How do you call a linear programming problem when the solution should be "constrained" to a norm?
(apologies for the n00b question)
Let's say we have a vector of length $n$, with to-be-determined values: $a_1, a_2, ...,a_n$.
And we have information that partial sums of these elements are equal to ...
- 143
1
vote
1
answer
280
views
Is this variant of knapsack problem strongly NP-hard?
Suppose we have a sequence of containers each of which contains multiple items. Each item $I_i$ is associated with an nonnegative weight $w_i$, a nonnegative value $v_i$, and $I_i(C)$ denotes the ID ...
2
votes
1
answer
234
views
Minimizing the degree of outgoing edges in a digraph, does this problem have a name?
I have a problem which can be rephrased in this way.
Suppose $G = (V,E)$ is a digraph (directed graph) and for each $v \in V$ we denote with $\delta^+(v)$ the number of outgoing edges of the vertex $v$...
- 103
0
votes
0
answers
210
views
k-secretary problem: not knowing the length of the queue
The secretary problem is a famous and old problem. You can find the basic definition of this problem here: https://en.wikipedia.org/wiki/Secretary_problem
Now I'm concerned with the k-secretary ...
- 109
4
votes
2
answers
296
views
High degree differences in bipartite graphs
Consider a finite, simple and undirected graph $G=(V,E)$ with $V=\{v_1,\dots, v_n\}$. Let us define the quantity:
$$\mathcal{I}_k(G) := \sum_{1\le i,j \le n} \mathbb{1}{\Big\{|\mathrm{deg}(v_i)-\...
user153000
1
vote
1
answer
230
views
Knapsack problem with capacity constraint
The traditional knapsack problem is that: given a sequence of $i$ items with positive weights $w_1,w_2,...,w_i$, positive values $v_1,v_2,...,v_i$, and a bag with capacity $B$, we want to insert items ...
15
votes
2
answers
894
views
n sets, each is large, the intersection of every three is small, what is the size of the union?
Let $A_1, A_2, \ldots, A_n$ be $n$ sets such that:
(1) for each $i\in [n]$, $\frac{n}{3}\leq |A_i|\leq n$;
(2) for any $1\leq i<j<k\leq n$, $|A_i\cap A_j\cap A_k|\leq a$, where $a$ is a constant ...
- 373
4
votes
0
answers
217
views
Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$
Crossposted at Theoretical Computer Science SE
A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$....
- 113
1
vote
1
answer
360
views
Knapsack problem with value range constraint
The traditional knapsack problem is that: given a sequence of $i$ items with positive weights $w_1,w_2,...,w_i$, positive values $v_1,v_2,...,v_i$, and a bag with capacity $B$, we want to insert items ...
1
vote
0
answers
154
views
Fastest algorithm to construct a proper edge $(\Delta(G)+1)$-coloring of a simple graph
A proper edge coloring is a coloring of the edges of a graph so that adjacent edges receive distinct colors. Vizing's theorem states that every simple graph $G$ has a proper edge coloring using at ...
- 1,130
2
votes
1
answer
67
views
What's the meaning of this inequality in the lot-sizing and scheduling problem
I learned about the MILP models proposed by Pochet and Wolsey. Here are the formulations of one of these models(MILP3).
So the decision variables and the primary formulation are as following:
Based ...
- 21
9
votes
3
answers
442
views
Pairs of vertices with high degree difference
Consider a finite, simple and undirected graph $G=(V,E)$ with $V=\{v_1,\dots, v_n\}$. Let us also fix an integer $k> n/2$. What are we able to say about the following quantity:
$$\mathcal{I}_k(G) :=...
user153000
1
vote
1
answer
103
views
$\mathrm{LP}$ formulation for $\mathrm{k}$-$\operatorname{opt}$ moves
$\mathrm{k}$-$\operatorname{opt}$ moves are an idea to improve non-optimal Hamilton cycles in weighted symmetric graphs by exchanging $\mathrm{k}$ tour-edges with $\mathrm{k}$ edges that do not belong ...
- 12.7k
2
votes
1
answer
75
views
Column subset selection with least angle optimization
Given a matrix $A$ I need to select a "representative" subset of its columns so that the each non-selected column is as close as possible to a selected one.
Formally:
Given $A \in \mathbb{R}...
- 6,882
0
votes
1
answer
124
views
What is the computational complexity of the calculation of $ \Psi(x) $?
What is the computational complexity of the calculation of $ \Psi(x) $ described below:
Let $\left\{ f_i : \{0,1,\dots,m\} \to \mathbb{R} \right\}_{i=1}^n$. For each $x \in \{0,1,\dots,m\}$ we ...
2
votes
0
answers
128
views
Maximum number of regions in a disk partitioned by pairs of parallel chords
We are given a disk $D$ in $\mathbb{R}^2$. Let $C$ be its boundary (i.e., the circle bounding $D$ on its plane). Let $P(n,d)$ be a set of $n$ pairs of chords of $C$ such that for each $\{c,c'\}\in P(n,...
- 1,677
0
votes
1
answer
52
views
Relation of 1-trees to optimal tours
Question:
given a complete symmetric graph $G(V,E)$ with $n$ vertices and edges $e_{ij}$ having weight $\omega_{ij}$, does there always exists a vector of vertex potentials $(\pi_1,\,\dots,\,\pi_n)$ ...
- 12.7k
2
votes
0
answers
69
views
Monotone rearrangements of function, constrained optimization in $\mathcal{L}^p$
By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy:
$$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)...
user153000
3
votes
0
answers
110
views
Matrix inequality $a X \succeq arcsin(X)$ for some $a > 0$
Let $X \in S^{n}_{+}$ be a positive semi-definite matrix with $X_{ii} = 1$ for all $i \leq n$ (thus $X$ is a correlation matrix).
Since $X$ is positive semi-definite, we have $|X_{ij}| \leq 1$ for any ...
- 205
1
vote
0
answers
35
views
Term or reference for a set of integer edge weights to guarantee distinct weighted degrees
I am looking for a term or reference describing sets $S$ of $\binom{n}{2}$ non-negative integers such that, for every bijection $w: E(K_n)\to S$ and every pair of distinct vertices $u$ and $v$ in $V(...
- 11
0
votes
1
answer
76
views
A question on graph partitioning
Given a connected un-directed simple graph $G=(V,E)$, is there a polynomial time algorithm to find the smallest subset $S$ of $V$ such that each node in $V \setminus S$ has at least 50% of its ...
- 1,196
0
votes
0
answers
23
views
Complexity of heaviest 2-optimal vertex-disjoint cycle covers
Calculating lightest vertex-disjoint cycle covers of finite complete symmetric graphs with weighted edges can be done efficiently and also renders the edge set of the calculated cycles free of pairs ...
- 12.7k
1
vote
0
answers
27
views
Optimality of trees generated via edge exchanges
let MST be the minimum spanning tree of a weighted finite graph; what can be said about the weight-optimality of the trees generated from the MST by sequentially exchanging a tree edge with a non-tree ...
- 12.7k
1
vote
0
answers
75
views
Generating triangulations with given topology
I am looking for information about the problem of identifying the heaviest minimal subset $F\subset E$ of the edgeset $E$ of a complete symmetric graph $G(V,E)$ with randomly weighted edges such that ...
- 12.7k
3
votes
0
answers
81
views
Additional symmetries of the Traveling Salesman Polytope
Given the complete graph $K_n=(V,E)$, the Traveling Salesman Polytope is a convex polytope in $\Bbb R^E$ obtained as the convex hull of the indicator vectors of (edge-sets of) Hamiltonian cycles in $...
- 12.6k
0
votes
1
answer
338
views
Maximize sum of edge weights on spanning tree
Problem: Given a complete graph with n vertices, the edge weight between vertex $i$ and vertex $j$ is $b[i]\times b[j]$.
Under the condition that the degree of point $i$ on spanning tree is DEG $[i]$, ...
- 11
1
vote
1
answer
198
views
Maximum number of colors for an optimal tiling which guarantees infinite paths
This question is a more applicable version of the question I've asked in mathexchange recently:
What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square
block ...
- 289
1
vote
1
answer
68
views
Complexity of calculating the optimal amalgamation of an optimal cycle-cover
Let $G(V,E)$ be a complete symmetric graph with positive edge weights and let further $\mathcal{C}=\lbrace C_1,\,\cdots\,C_k\rbrace$ be the minimum-weight vertex-disjoint cycle cover.
The set $E$ of ...
- 12.7k