Capacitated min-k-cut problem
In the capacitated min-$k$-cut problem we are given a graph (hypergraph) with non-negative edge (hyperedge) weights. The task is then to find a partitioning of the graph’s vertices into $k$ sets of...
View ArticleProving NP-hardness of scheduling problem (total weighted completion time)
Consider the problem $P mid mid sum w_j C_j$. I want to prove that this problem is (strongly) NP-hard by reducing from $3$-Partition, but I am not really sure how to do this. Just to be precise, this...
View ArticleAbout representability of optimization versions of NP-complete questions as...
Naively, in my very limited awareness, it feels that the Max-CUT is a very “special” NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a $n$...
View ArticleDeeper look at Algorithmica?
Russell Impagliazzo published “A Personal View of Average-Case Complexity” (preprint) back in 1995. He presented five possible worlds we could be living in, depending on how P and NP were related. The...
View ArticleHuffman Tree Depth, Is there any theory?
I’d like to as a variation on this question regarding Huffman tree building. Is there any theory or rule of thumb to calculate the depth of a Huffman tree from the input (or frequency), without drawing...
View ArticleComplexity of digraph homomorphism to an oriented cycle
Given a fixed directed graph (digraph) $D$, the $D$-COLORING decision problem asks whether an input digraph $G$ has a homomorphism to $D$. (A homomorphism of $G$ to $D$ is a mapping $f$ of $V(G)$ to...
View ArticleRecognizing sequences with all permutations of ${1, ldots, n}$ as subsequences
For any $n > 0$, I say that a sequence $s$ of integers in ${1, ldots, n}$ is $n$-complete if, for every permutation $mathbf{p}$ of ${1, ldots, n}$, written as a sequence of pairwise distinct...
View ArticleWhat is the complexity of vertex cover on k-partite graphs?
Given a k-partite graph which is already partitioned into k parts, what is the complexity of finding a vertex cover of minimum size? I guess that it’s NP-hard, but couldn’t yet prove it or find...
View ArticleBalanced Max-2-SAT NP-Hardness
The Balanced Max-2-SAT is a special case of Max-2-SAT (each clause is a disjunction of exactly 2 literals) in which for every variable $x$, there is a $k$ such that $x$ appears positive exactly $k$...
View ArticleComplexity of minimizing monotone arithmetical formulas
Let’s say that I have a multi-variate arithmetical expression $A(x_1, ldots, x_n)$ that uses addition and multiplication operations and is also in a very simple form of sums of products, e.g.,...
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