# Questions tagged [co.combinatorics]

Questions related to combinatorics and discrete mathematical structures

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11k views

### Counting the Number of Simple Paths in Undirected Graph

How can I go about determining the number of unique simple paths within an undirected graph? Either for a certain length, or a range of acceptable lengths. Recall that a simple path is a path with no ...
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### graphs from real-life problems

Where can I find graphs relevant to real-life problems? Two repositories I know of are: University of Florida's Sparse Matrix Collection Bodlaender's TreewidthLib
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### Grid $k$-coloring without monochromatic rectangles

Update: The obstruction set (i.e. the NxM "barrier" between colorable and uncolorable grid sizes) for all monochromatic-rectangle-free 4-colorings is now known. Anyone feel up to trying 5-colorings? ;...
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### Is optimally solving the n×n×n Rubik's Cube NP-hard?

Consider the obvious $n\times n\times n$ generalization of the Rubik's Cube. Is it NP-hard to compute the shortest sequence of moves that solves a given scrambled cube, or is there a polynomial-time ...
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### Maximal classes for which largest independent set can be found in polynomial time?

The ISGCI lists over 1100 classes of graphs. For many of these we know whether INDEPENDENT SET can be decided in polynomial time; these are sometimes called IS-easy classes. I would like to compile ...
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### 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 ...
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### 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 ...
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### Cover Time of Directed Graphs

Given a random walk on a graph the cover time is the first time (expected number of steps) that every vertex has been hit (covered) by the walk. For connected undirected graphs, the cover time is ...
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### Minimum number of transpositions to sort a list

In trying to devise my own sorting algorithm, I'm looking for the optimal benchmark to which I can compare it. For an unsorted ordering of elements A and a sorted ordering B, what is an efficient way ...
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### Can you identify the sum of two permutations in polynomial time?

There were two questions asked recently on cs.se which were either related to or had a special case equivalent to the following question: Suppose you have a sequence $a_1, a_2, \ldots a_n$ of $n$ ...
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### What kind of mathematical background is needed for graph theory?

It is going to be the first time for me to learn graph theory. What kind of mathematical background do I need to prepare master theses about this subject in following years? Which subjects should be ...
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### Cover a Concave Polygon with a minimum number of rectangles

I am trying to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle. I thought about ...
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### Maximal/maximum independent sets

Is there something known about the class of graphs with the property that all maximal independent sets have the same cardinality and are therefore maximum ISs? For example, take a set of points in ...
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### Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs

Crossposted from MO. Let $C$ be a graph class defined by a finite number of forbidden induced subgraphs, all of which are cyclic (contain at least one cycle). Are there NP-hard graph problems that ...
984 views

### Generating interesting combinatorial optimization problems

I'm teaching a course on meta-heuristics and need to generate interesting instances of classic combinatorial problems for the term project. Let's focus on TSP. We are tackling graphs of dimension $200$...
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### Application of Ramsey Numbers

The definition of Ramsey numbers is the following: Let $R(a,b)$ be a positive number such that every graph of order at least $R(a,b)$ contains either a clique on $a$ vertices or a stable set on $b$ ...
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### An interesting variant of maximum matching problem

Given a graph $G(V,E)$, the classic maximum matching problem is choosing the maximum subset of edges $M$ s.t., for each edge $(u,v) \in M$, $d(u)=d(v)=1$. Has anybody studied the following variant? ...
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### Is the the spectral norm of a Boolean function bounded by the degree of its Fourier expansion?

Let $f: \{-1,1\}^n \rightarrow \{-1,1\}$ be a Boolean function. The Fourier expansion of $f$ is $$f(T) = \sum_{S \subseteq [n]} \widehat{f}(S)\ \chi_S(T)$$ where $\widehat{f}(S)$ are real numbers ...
338 views

### Does faster exact algorithm for counting independent sets in comparability graphs than general graph exisits?

Sorry for not-precise question. :-( There are several papers concerning exact counting (maximum) independent sets in general graphs. Actually, they concerns counting of solutions of 2SAT. The best of ...
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### Information Theory used to prove neat combinatorial statements?

What's your favorite examples where information theory is used to prove a neat combinatorial statement in a simple way ? Some examples I can think of are related to lower bounds for locally decodable ...
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### 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 ...
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### Kolmogorov complexity applications in computational complexity

Informally speaking, Kolmogorov complexity of a string $x$ is a length of a shortest program that outputs $x$. We can define a notion of 'random string' using it ($x$ is random if $K(x) \geq 0.99 |x|$)...
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### Using error-correcting codes in theory

What are applications of error-correcting codes in theory besides error correction itself? I am aware of three applications: Goldreich-Levin theorem about hard core bit, Trevisan's construction of ...
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### Polynomial method for complexity results

Polynomial methods, say Combinatorial Nullstellensatz and Chevalley–Warning theorem are powerful tools in additive combinatorics. By representing a problem with proper polynomials, they can guarantee ...
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### Graph families which have polynomial time algorithms for computing the chromatic number

Post updated on 31st of August: I added a summary of the current answers below the original question. Thanks for all the interesting answers! Of course, everyone can continue posting any new findings. ...
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### A combinatorial version for the polynomial Hirsch conjecture

Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ . Suppose that (*) For every $i \lt j \lt k$ and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$, ...
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### Are there any applications of techniques in real analysis to theoretical computer science?

I have looked far and wide for such applications and have mostly turned up short. I can find plenty of applications of topology and similar structures on countable (or uncountable) sets, but rarely do ...
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### Social choice, arrow's theorem and open problems ?

Last few months I started to lecture myself on social choice, arrow's theorem and related results. After reading about the seminal results, I asked myself about what happens with partial order ...
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### Mathematical background needed for pursuing a PhD in TCS [duplicate]

Possible Duplicate: What kind of mathematical background is needed for complexity theory? I'm a final year undergrad and I'm looking to pursue a PhD in theoretical computer science in the long ...
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### What are the root difficulties in going from graphs to hypergraphs?

There are many examples in combinatorics and computer science where we can analyze a graph-theoretic problem but for the problem's hypergraph analog, our tools are lacking. Why do you think problems ...
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### Computational complexity of counting induced subgraphs which admit perfect matchings

Given an undirected and unweighted graph $G=(V,E)$ and an even integer $k$, what is the computational complexity of counting sets of vertices $S\subseteq V$ such that $|S|=k$ and the subgraph of $G$ ...
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### What are infinite graphs good for?

I have just read on the German Wikipedia that an infinite graph is a graph with an infinite number of nodes or an infinite number of edges. I only know applications and algorithms for finite graphs. ...
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### Partition a graph into node-disjoint cycles

Related problem: Veblen’s Theorem states that "A graph admits a cycle decomposition if and only if it is even". The cycles are edge disjoint, but not necessarily node disjoint. Put another way, "The ...
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### Decomposing graphs of genus one

Planar graphs are $K_{3,3}$-free. Such graphs can be decomposed into tri-connected components, which are known to be either planar or $K_5$ components. Is there such a "nice" decomposition of ...
Let $k\in\mathbb{N}$ and denote by $G_k$ the set of all graphs that can be embedded on a surface of genus $k$ such that all vertices are situated on the outer face. For instance, $G_0$ is the set of ...