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Questions tagged [co.combinatorics]

Questions related to combinatorics and discrete mathematical structures

13
votes
4answers
3k views

LP relaxation of independent set

I've tried the following LP relaxation of maximum independent set $$\max \sum_i x_i$$ $$\text{s.t.}\ x_i+x_j\le 1\ \forall (i,j)\in E$$ $$x_i\ge 0$$ I get $1/2$ for every variable for every cubic ...
0
votes
0answers
103 views

Star seperators to explain computational complexity of algorithms on a class of graphs?

A lot of NP-hard optimization problems on graphs which are perfect become solvable in polynomial time. Unfortunately, the class of graphs that arise in my problem are not perfect. The graphs can be ...
57
votes
5answers
4k views

The origin of the notion of treewidth

My question today is (as usual) a bit silly; but I would request you to kindly consider it. I wanted to know about the genesis and/or motivation behind the treewidth concept. I sure understand that ...
0
votes
0answers
110 views

Decomposition for a certain class of graphs

Suppose a graph, $G = (V,E)$ is characterized as a lattice/network of cliques as in the picture below. Does there exist some decomposition principle (i.e. on the right) for $G$, that yields some ...
0
votes
0answers
84 views

Maximum Weight Independent Set on a Changing Graph?

Suppose I have an optimal solution to the maximum weight independent/stable set problem on an arbitrary graph. If I were to induce a clique among a subset of its vertices (and perhaps add in some ...
0
votes
0answers
52 views

Internal information is at most expected number of bits

I am freely using known terms from communication complexity. Let $u$ be a distribution on $X\times Y$, and $P$ a randomized (private randomness) communication protocol. We define $IC(u,P) = I(P;X|Y)...
4
votes
1answer
78 views

Stable order on binary strings

I need some order on binary strings such that if I have a small (but superlinear in their length) number of sufficiently different strings, the order will stay the same if I change a few bits in the ...
0
votes
1answer
120 views

Subset Sum Problem and hard looking instances that are not really hard

I have been working in a subset sum solver (some new approach) and while working on the time complexity analysis I found what I describe below. Maybe this could explain why some "hard looking" ...
4
votes
1answer
107 views

Applications of Christol theorem

I'm looking forward to know about applications of Christol theorem mentioned in Jefrrey Shallit's Number theory and formal languages. One of them is purely algebraic: if $f, g \in \mathbb{F}_q[[z]]$ ...
1
vote
1answer
74 views

Pulling a graph across a partition

I am looking for the name for a particular graph property, if it has been studied, and efficient algorithms for computing it, if they exist. I realise that this may be a well known property that I am ...
9
votes
0answers
114 views

Random unbalanced bipartite graphs are good small set expanders

My question is about small set expansion properties of random unbalanced bipartite graphs. Fix a positive $\delta<1/2$, and a positive integers $n,m,d$. Let us call a bipartite graph $\mathcal{G}$...
2
votes
0answers
130 views

Smallest disjoint union chain containing a sequence of sets

Let $\mathcal{A}=\{A_1,\ldots,A_n\}$ be a family of sets, we have the property that $A_1=\emptyset$, and one can obtain $A_i$ from $A_{i-1}$ by adding or deleting a single element. A family $\...
3
votes
0answers
67 views

Linear optimization over intersection of totally unimodular matrices

I am currently dealing with a problem of the following form \begin{alignat}{2} &\underset{x, y \in \mathbb{R}^n}{{\text{min}}} && e^T x \nonumber\\ &\text{sub to} \hspace{0.05in}&&...
11
votes
1answer
467 views

Number of equivalence classes in regular languages as a function of DFA size

This question is related to a recent question by Janoma. Background In constraint programming, a regular global constraint $c$ over a domain $D$ is a pair $(s, M)$ with $s$ a tuple of variables (...
19
votes
2answers
770 views

Axioms for Shortest Paths

Suppose we have an undirected weighted graph $G = (V, E, w)$ (with non-negative weights). Let us assume that all shortest paths in $G$ are unique. Suppose we have these $\binom{n}{2}$ paths (sequences ...
8
votes
2answers
1k views

Faster pseudo-polynomial time algorithms for PARTITION

I want to partition N given numbers (may or may not be equal) into 2 subsets such that the 2 subsets have sum as close as possible and also the cardinality of the sets are equal (if n is even) or ...
5
votes
1answer
120 views

If $u\cdot v \in (z \cdot z')^*$ and $v \cdot u \in (z' \cdot z)^*$ then $u \in (z \cdot z')^* \cdot z$

I'm searching for a reference for the following property: Fact. Let $u, v \in A^*$. Write $w$ for the primitive root of $u\cdot v$, i.e., $u\cdot v = w^c$. There are unique words $z, z'$ such ...
1
vote
0answers
95 views

Growth of random square lattice trees

Consider the problem of growing a random tree on a $L\times L$ square lattice of initially disconnected vertices, starting from an isolated vertex on one of the corners of the lattice and proceeding ...
11
votes
1answer
838 views

A simple(?) funny combinatorial problem!

Let we fix $0<E<1$ and an integer $t>0$. for any $n$ and for any vector $\bar{c} \in [0,1]^n$ such that $\sum_{i\in [n]} c_i \geq E \times n$ $A_{\bar{c}} :=|\{ S \subseteq [n] : \sum_{i \...
6
votes
1answer
115 views

Average-case analogue of Small-bias Spaces

Recall that an $\epsilon$-biased space is a set $S \subset \{0,1\}^n$ such that for every non-zero linear test $\alpha \in \{0,1\}^n \setminus \{0\}^n$, the expected bias $$| \mathbb{E}_{x \in S} [ (-...
21
votes
2answers
677 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 ${\...
3
votes
0answers
69 views

Mutually exclusive replacement paths, an existing problem?

A replacement path is a simple path allocated to an edge $e \in G$ that connects the endpoints of $e$ in $G \setminus \{e\}$. The problem An undirected graph $G$ is given and the task is to allocate ...
4
votes
1answer
97 views

Probability of a $k$-path in a random graph

Assume that $G\in G(n,p)$; if $p=\frac{\ln n +\ln \ln n +c(n)}{n}$, the following fact is well known: \begin{eqnarray} Pr [G\mbox{ has a Hamiltonian cycle}]= \begin{cases} 1 & (c(n)\...
26
votes
1answer
580 views

Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?

We define a regular tree language as in the book TATA: It is the set of trees accepted by a non-deterministic finite tree automaton (Chapter 1) or, equivalently, the set of trees generated by a ...
11
votes
4answers
12k 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 ...
8
votes
3answers
525 views

Constant in Komlos conjecture

Given $n$ vectors $v_1,\dots,v_n\in\Bbb R^N$ with $\|v_i\|_2^2\leq1$ at every $i\in\{1,\dots,n\}$, Komlos conjecture states that, there is a $c\in\Bbb R$ (independent of $n,N$) such that at some $\...
1
vote
1answer
105 views

Total flow using minimum number of edges on a bipartite network

If I have a set of sources $S$ with total capacity $C$ and a set of sinks $T$ with the same capacity $C$, but not necessarily the same cardinality, is there an efficient way to find the minimum number ...
23
votes
3answers
1k views

Representing OR with polynomials

I know that trivially the OR function on $n$ variables $x_1,\ldots, x_n$ can be represented exactly by the polynomial $p(x_1,\ldots,x_n)$ as such: $p(x_1,\ldots,x_n) = 1-\prod_{i = 1}^n\left(1-x_i\...
1
vote
0answers
39 views

Counting vertex covers on a chain of k nodes that do not contain a sub-chain of length >=3

By a "chain of k nodes", I mean k nodes lined up like a linked-list: o-o-o.....-o . By "do not contain a sub-chain of length >=3", I mean that no cover should contain two edges that shares a node. ...
7
votes
1answer
254 views

Distinguishability a set of permutations

Given integers $d<n$, find the largest $k$ such that there exists a set of $k$ permutations $\sigma_1,\cdots,\sigma_k$ on $[n]$, such that any size-$d$ subset $T\subseteq [n]$ is ``distinguishable''...
1
vote
2answers
179 views

Is perfect matching for bipartite graph with no cycles unique?

Given a balanced bipartite graph that satisfies Hall's theorem (is non singular) then it shown that it has at least one perfect matching. My question is if the balanced bipartite graph is also ...
1
vote
0answers
28 views

Inductive definition of language operators like the set of all permutations of a word came from the shuffle operator

Let $X$ be a finite alphabet. Given two words $u, v \in X^{\ast}$ the shuffle operator is defined to be $$ u || v := \{ u_1 v_1 u_2 v_2 \ldots u_n v_n : 1 \le i \le n, u_i, v_i \in X^{\ast}, u = u_1 \...
-1
votes
1answer
145 views

Application of the inequality with expectations

Let $\Vert\cdot\Vert$ is a norm in $R^n$. Let $x_1,\dots,x_N$ non-independent Rademacher random variables random variables (variables which are uniform on $\{-1, 1\}$). . By $E$ we denote an ...
1
vote
0answers
113 views
22
votes
3answers
1k views

Number of distinct nodes in a random walk

Commute time in a connected graph $G=(V,E)$ is defined as the expected number of steps in a random walk starting at $i$, before node $j$ is visited and then node $i$ is reached again. It is basically ...
19
votes
0answers
617 views

Weighted Hamming distance

Basically my question is, what kind of geometry do we get if we use a "weighted" Hamming distance. This is not necessarily Theoretical Computer Science but I think similar things come up sometimes, ...
9
votes
2answers
308 views

Are there any 'graphical' algebras that can describe the 'shape' of graphs?

One of the main problems in graph enumeration is determining the 'shape' of a graph, e.g. the isomorphism class of any particular graph. I am fully aware that every graph can be represented as a ...
0
votes
0answers
48 views

Given $f \in \mathbb{Z}_2[x_1, …, x_n]$, are there properties of $f$ which can be used to bound $\mathbb{E}_{i \in [n]}|gap(f) - gap(f + x_i)$|?

Let $f \in \mathbb{Z}_2[x_1, ..., x_n]$ be arbitrary. We define $gap(f) = |f^{-1}(0)| - |f^{-1}(1)|$. I want to understand, as a function of any properties of $f$ you find relevant (number of terms, ...
6
votes
2answers
170 views

Difference Sets

Suppose we have a set $$P=\{p_1,p_2,...,p_K\}$$ where $$1\leq p_k\leq N , k=1,...,K \qquad \& \quad p_k \in \mathbb{N} $$ and $p_k$'s are distinct. We calculate the differences as: $$d=p_i-p_j\mod ...
1
vote
0answers
73 views

Nim-game variant - weird ending condition

the game is structured like this: two players the players alternate moves 4 heaps $h_1,h_2,h_3,h_4$ with sizes $n_1,n_2,n_3,n_4$ at each move, the player can either remove one or two elements from ...
21
votes
5answers
5k views

What is the maximum number of stable marriages for an instance of the Stable Marriage Problem?

Stable Marriage Problem: http://en.wikipedia.org/wiki/Stable_marriage_problem I am aware that for an instance of a SMP, many other stable marriages are possible apart from the one returned by the ...
6
votes
2answers
272 views

On the notion of positive rank

The positive rank of a square matrix is defined in Theorem $3$ of "Expressing Combinatorial Optimization Problems by Linear Programs" by Mihalis Yannakakis as follows: given a $n\times n$ matrix $A$, ...
3
votes
2answers
278 views

NP-hardness of finding 0-1 vector to maximize rows of {-1, +1} matrix

Consider the following discrete optimization problem: given a collection of $m$-dimensional vectors $\{ v_1, \dots, v_n \}$ with entries in $\{-1, +1\}$, find an $m$-dimensional vector $x$ with ...
-3
votes
1answer
114 views

Continous work distribution algorithm with failover

Imagine there's a system where there's N workers and M units of work, for example, N ≤ 64, M = 256. Is there an algorithm that ...
3
votes
2answers
2k views

How to solve the Shortest Hamiltonian Path problem on Sparse Graphs?

Problem: Given a positive-weighted undirected graph, find the shortest path (in terms of total sum of edges) that visits each node exactly once. For a subset $S$ of nodes and a node $i\in S$, let $D[...
-3
votes
1answer
95 views
6
votes
3answers
771 views

Is there any good and free Introduction to topological graph theory

My knowledge in topological graph theory is in low, I need some good reference which has two simple thing, Definition of new concepts (like genus,graph embedding in surface, ...) also contains related ...
3
votes
0answers
72 views

Finding the longest sub-permutation with bounded inversion number

Given a permutation $\sigma\in S_n$ and a positive integer $k$, find the largest integer $0\le m\le n$ such that there exist a subset $I\subseteq [n], |I|=m$, satisfying that the restriction of $\...
0
votes
0answers
46 views

Wavelet based Non linear optimization technique

I am outlining a method for solving Non Linear optimization problems. Consider the system of equations:--------------------------------- 1 f1(a0, a1, a2, a3 ......... an) = 0 f2(a0, a1, a2, a3 ........
5
votes
1answer
1k views

Is Hartmanis-Stearns conjecture settled by this article?

The paper "On the computational complexity of algebraic numbers: the Hartmanis--Stearns problem revisited" by Boris Adamczewski, Julien Cassaigne, Marion Le Gonidec https://arxiv.org/abs/1601....