Questions tagged [co.combinatorics]

Questions related to combinatorics and discrete mathematical structures

Filter by
Sorted by
Tagged with
8
votes
1answer
195 views

Does Horn SAT (Horn formula in CNF) have an integral polytope?

In some ways, my question is related to this: Is the matching polytope integral? Matching and Horn-SAT are both polynomial time solvable.. So I wonder if there is a Horn polytope, similar to the ...
0
votes
0answers
21 views

Going from one base packing to another using basis exchanges

Suppose I have a matroid $M = (E, \mathcal{I})$. It is a known fact that given any two bases $X_0$ and $X_n$, we can transform $X_0$ into $X_n$ by repeatedly applying the basis exchange axiom. So ...
5
votes
1answer
96 views

Short supersequence for many permutations

A shortest supersequence $c_n$ of all permutations on $[n]$ has length $\Theta(n^2)$: see this question on Mathoverflow. What if we force $c_n$ to be short? How many permutations can it cover? Let's ...
4
votes
2answers
219 views

Complexity of "can we get a cycle by stacking directed bipartite graphs?"

Preliminaries We consider directed bipartite graphs of the form $G = (V,V',E)$, in which the nodes are partitioned into $V = \{1,\ldots,n\}$ and $V'=\{1',\ldots,n'\}$, with $|V|=|V'|=n$, and $E\...
1
vote
1answer
91 views

Partition the edges of a bipartite graph into perfect $b$-matchings

Any $r$-regular bipartite graph can be partitioned into $r$ disjoint perfect matchings. I want to know whether a version of this extends to perfect $b$-matchings. Suppose we have a bipartite graph $G =...
10
votes
2answers
301 views

Finding subgraphs with high treewidth and constant degree

I am given a graph $G$ with treewidth $k$ and arbitrary degree, and I would like to find a subgraph $H$ of $G$ (not necessarily an induced subgraph) such that $H$ has constant degree and its treewidth ...
0
votes
1answer
109 views

Does an upper bound on the integrality gap imply an approximation algorithm with the same ratio?

Often, we can model combinatorial optimization problems with an Integer Program. Then there is an associated Linear Relaxation which drops the integrality constraints on the variables. Let's say we ...
6
votes
1answer
277 views

Hardness of maximizing $x^TAy$ with $\{-1,1\}$ entries

My question concerns the NP-hardness of the following discrete optimization problem: Given a matrix $A \in \{ \pm 1 \}^{m\times n}$, $$\begin{array}{ll} \underset{x \in \{ \pm 1 \}^m ,\, y \in \{ \pm ...
1
vote
1answer
80 views

Is there a regular bipartite graph where the minimum cuts are trivial?

My question is: Given integers $r$ and $k$, is there an $r$-regular bipartite graph $G = L \cup R$ with $|L| = |R| = k$, which is $r$-edge connected, and such that every minimum cut is trivial? We can ...
26
votes
3answers
797 views

The complexity of determining if a fixed graph is a minor of another

The result by Robertson and Seymour demonstrates an $O(n^3)$ algorithm for testing whether a fixed graph $G$ is a minor of $H$. I have two and a half questions on this topic: 1) It appears that there ...
2
votes
1answer
120 views

Do there exist two equivalent objective functions one of which can be approximated but another one cannot?

I have two equivalent problems A and B, meaning that the optimal solution of one must be the optimal solution of another one. However, it seems that problem A can be approximated but B cannot. Below ...
3
votes
1answer
338 views

How hard is this combinatorial optimisation problem?

Suppose we have multiple intervals $R_1,R_2,...,R_i$ of non-negative integers. These intervals may overlap and we use $R_h(\mathrm{median})$ to denote the median integer in the $h$-th interval $R_h$, ...
3
votes
1answer
149 views

TSP with "enemy" nodes

I am curious if the following variation of the traveling salesman problem (TSP) (or a vehicle routing problem (VRP) version) occurs in the literature and has a name I could search for. The story/idea ...
5
votes
0answers
147 views

Minimum spanning tree, but with an unusual objective function

This is a problem that came up in my study of rumour networks. I was wondering if anyone had thoughts or references on this problem. If we have a rooted tree $T = (V,E)$ with root $r$, I first label ...
2
votes
1answer
70 views

Covering a binary relation as a union of rectangles

Given finite sets $X$ and $Y$ and a subset $R\subset X\times Y$, I want to express $R$ as a union $R=\bigcup_{i=1}^n X_i\times Y_i$ with $n$ as small as possible. Here, each $X_i\subset X$ and $Y_i\...
4
votes
1answer
174 views

Does such a bipartite graph exist?

In the course of my studies on graphs I sometimes use gadgets. I recently came upon a need for a certain bipartite graph with the following properties, and I am wondering if anyone knows if such a ...
0
votes
0answers
79 views

Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$

Crossposted at MathOverflow 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 $a_{ij}\in\{0,1\}$. $\mathbf{1}$ is ...
3
votes
0answers
82 views

Boltzmann sampling for containers/dependent polynomials?

I’d like to randomly sample from dependently-typed data structures. Has anyone looked at extending Boltzmann sampling to containers or dependent polynomials?
0
votes
0answers
54 views

Prove that this linear relaxation has half-integral extreme points

Given a graph $G=(V,E)$, here is a Linear Relaxation of the edge cover polytope: (1) For each $v \in V, \sum_{e \in \delta(v)} x_e \geq 1.$ (2) For each $e \in E$, $0 \leq x_e \leq 1.$ Here $\delta(S)$...
3
votes
2answers
168 views

XOR Resilient Encoding

I am looking for an efficient encoding algorithm $E:\{0,1\}^n\to\{0,1\}^m$ such that given $E(x)\oplus E(y)$ for $x\neq y$, we can reconstruct $x$ and $y$. One example of such an algorithm would be ...
2
votes
1answer
34 views

Maximum weight matching with classes of edges in a multi-edge bipartite graph

Posted a similar question in mathoverflow, have tried to reduce this to Ford Fulkerson, but been stuck. Thought I'd ask TCS community to see if there are any ideas from individuals, here. Consider a ...
-2
votes
1answer
108 views

Subset sum for lists [closed]

Given a target list a = [2,4,1,4]` and a list of lists ...
0
votes
0answers
27 views

Unique naming/labeling of $40$-node strongly regular graphs

Brendan McKay's webpage lists all possible $40$-node strongly regular graphs. Is there a standard way to name them uniquely?
0
votes
0answers
40 views

are there approximation algorithms that use primal-dual with LP values and/or rounding?

Are there approximation algorithms that use primal-dual with LP values and/or rounding? e.g. An algorithm that during any iteration first tries to see an extreme point to the LP has any value above a ...
7
votes
1answer
157 views

Inferring the Kolmogorov complexity of a string from its substrings' complexity

I know that the Kolmogorov complexity of a substring $v$ of an incompressible string $x$ has $C(v)\geq |v|-O(\log{|x|})$ , but I'm wondering if it is also possible to infer the complexity of a string ...
2
votes
1answer
87 views

"Parity testing set" for disjoint pairs of sets

I'd like a construction of the following description. Let $V$ be a set of $n$ elements. I'd like a collection $X$ of subsets of $V$ such that for any pair $(P,Q)$ of disjoint subsets of $V$, there ...
0
votes
0answers
90 views

How many maximal planar graphs are there?

We denote by a triangulation a (simple) maximal planar graph. How many triangulations on $n$ vertices are there? How many triangulations are there if we cannot distinguish the vertices, i.e. ...
0
votes
1answer
134 views

Upper bound on Independence Number of Random Regular Graph with degree $\Theta(\sqrt{|V|} \log^2 |V|)$

Let $G=(V,E)$ be a random $\Delta$-regular graph with $\Delta \in \Theta(\sqrt{|V|} \log^2 |V|)$. I'm analysing an algorithm having asymptotic running time crucially depending on the Independence ...
2
votes
0answers
98 views

$k$-XOR collision free families

Given parameters $n,k\in \mathbb N^+$, I'm interested in finding a set of binary vectors $V_{n,k}=\{v_1,\ldots,v_n\}$ of length that satisfies: $\forall i: v_i\in\{0,1\}^{z_{n,k}}$. The bitwise xor ...
2
votes
0answers
56 views

Damerau–Levenshtein distance with transposition of non-adjacent characters?

Wondering if it's possible to calculate Damerau–Levenshtein distance with transposition of non-adjacent characters (DL distance allows transposition of immediately adjacent characters only). I want ...
0
votes
0answers
34 views

Combinatorial object for a rational generating function

The is a well-established connection between generating functions and the specification of combinatorial objects via symbolic and/or analytic combinatorics. A number of packages (e.g. SAGE) allow the ...
2
votes
1answer
134 views

Is this homework problem on T-joins wrong? [closed]

In Question 9.3a, it states that if $T=V$, then the minimum cost perfect matching is the minimum cost T-join. Is this actually true? I think I have a counterexample which I have drawn below.
0
votes
0answers
70 views

Graph-based backjumping vs Conflict-based backjumping comparison in CSP

Graph-based backjumping vs Conflict-based backjumping in CSP I have been studying techniques to find solutions in backtracking to model and solve a problem in c++ with this methods and I have the ...
28
votes
5answers
4k views

Counting words accepted by a regular grammar

Given a regular language (NFA, DFA, grammar, or regex), how can the number of accepting words in a given language be counted? Both "with exactly n letters" and "with at most n letters" are of ...
1
vote
0answers
114 views

Quantum error correction and graph codes

I was reading combinatorial approach towards quantum correction. A lot of work in this is on finding diagonal distance of a graph. Let me add definition of diagonal distance so that this remains self-...
1
vote
2answers
107 views

Where to find info on (polytime) approximability of various discrete optimization problems?

Where to find info on (polytime) approximability of various discrete optimization problems? Sorry if this is stupid,but is there a site or reference that keeps up to date info on approximability of ...
1
vote
1answer
86 views

Name of (and solution to) this generalization of linear assignment

I would like to know if the following problem is known and has any efficient solution. Given an $n\times n$ score matrix $S$. Find the best $a$ elements, in terms of their sum of scores, such that no ...
-4
votes
1answer
46 views

does within the "range a and b" include a and b?

I have not found the answer to this doubt of mine elsewhere, hence posting it here. It may be a silly question but I just want to be sure :P would be great if someone could help me out with this ...
2
votes
1answer
96 views

NP hard proving: separate graph into a set of the same size disjoint parts by maximizing the shared neighbours of each part

Given a graph $G=\{V,E\}$ where $V$ denotes the nodes and $E$ denotes edges. The size of the node $|V| = nk$. The target is to separte the graph into $n$ disjoint parts $P=\{V_i\}_{i=1}^n$ and the ...
11
votes
3answers
547 views

Applications of sunflower lemma in theoretical computer science

In one lecture by Kewen Wu who is one of the authors of paper Improved bounds for the sunflower lemma, it is said that the sunflower lemma can be applied to many fields like circuit lower bounds ...
2
votes
2answers
121 views

On the coloring number of small graphs with small cliques

Given a parameter $k$, and a graph $G$ with $O(k^2)$ vertices that has a maximum clique with $\le k$ vertices, I want to investigate the maximum number of colors $C(k)$ needed to properly color $G$, i....
0
votes
1answer
53 views

weights in low density codes

Generally, low density parity codes are decoded using sum product decoder (also known as decoding under belief propagation). Such codes are usually decoded nicely if there are no short length cycles ...
14
votes
5answers
14k views

Why can machine learning not recognize prime numbers?

Say we have a vector representation of any integer of magnitude n, V_n This vector is the input to a machine learning algorithm. First question : For what type of representations is it possible to ...
1
vote
1answer
78 views

Existence of graphs of every order related to Barnette’s conjecture

Consider the class C3CBP of $3$-connected cubic bipartite planar graphs. They form the class on which the (in)famous Barnette’s conjecture is based. My interest in C3CBP graphs is somewhat orthogonal ...
23
votes
2answers
831 views

Are shift-chains two-colorable?

For $A\subset [n]$ denote by $a_i$ the $i^{th}$ smallest element of $A$. For two $k$-element sets, $A,B\subset [n]$, we say that $A\le B$ if $a_i\le b_i$ for every $i$. A $k$-uniform hypergraph ${\...
6
votes
1answer
290 views

Re-packing of integers into fixed-size bins

We are given a constant $B \geq 3$, a set of positive integers $I$ s.t. $\forall i \in I, i \leq B$ and $\sum_{i \in I} = N$, and 2) a packing of these numbers into $N/B > 1$ bins of size $B$. An ...
4
votes
1answer
206 views

Level $k$ bounds in Analysis of Boolean functions

In Ryan O'Donnell's book Analysis of Boolean functions, following Corollary 9.25 the following appears: If $f\colon \{-1,1\}^n \to \{0,1\}$, and we have $\mathbb{E}[f] = \alpha$, then for any integer ...
4
votes
1answer
252 views

How tight is the XOR lemma?

The XOR lemma states that if you have a distribution $D$ on $\{0,1\}^n$, and all the Fourier coefficients of $2^n D$ are small, then it is close in $L_1$ to the uniform distribution. Specifically, ...
3
votes
1answer
190 views

Dual to hypercontractive inequality

Recall the hypercontractive inequality: Let $\rho = \sqrt{\frac{p-1}{q-1}}$, then $||T_\rho(f)||_q \leq ||f||_p$ In https://www.cs.cmu.edu/~odonnell/papers/analysis-survey.pdf it is stated that the ...
-1
votes
1answer
66 views

Chomsky-Schutzenberg Hierarchies explained for physicist (general) [closed]

I am classically trained in physics, however I have been interested in the use of information theory in studying some classical systems. As someone who is somewhat unfamiliar with the language of ...

1
2 3 4 5
13