Unanswered Questions

16
votes
0answers
368 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 ...
15
votes
0answers
279 views

Subset sum problem with many divisibility conditions

Let $S$ be a set of natural numbers. We consider $S$ under the divisibility partial order, i.e. $s_1 \leq s_2 \iff s_1 \mid s_2$. Let $\qquad \displaystyle \alpha(S) = \max \{|V| \mid V\subseteq S, ...
14
votes
0answers
649 views

Is it NP-hard to fill up bins with minimum moves?

There are $n$ bins and $m$ type of balls. The $i$th bin has labels $a_{i,j}$ for $1\leq j\leq m$, it is the expected number of balls of type $j$. You start with $b_j$ balls of type $j$. Each ball of ...
13
votes
0answers
276 views

Do functions with slower growth than inverse Ackermann appear in runtime bounds?

Some complicated algorithms (union-find) have the nearly-constant inverse Ackermann function that appears in the asymptotic time complexity, and are worst-case time optimal if the nearly constant ...
12
votes
0answers
181 views

Problems for which algorithms based on partition refinement run faster than in loglinear time

Partition refinement is a technique in which you start with a finite set of objects and progressively split the set. Some problems, like DFA minimization, can be solved using partition refinement ...
12
votes
0answers
237 views

Approximate minimum-weighted tree decomposition on complete graphs

Say I have a weighted undirected complete graph $G = (V, E)$. Each edge $e = (u, v, w)$ is assigned with a positive weight $w$. I want to calculate the minimum-weighted $(d, h)$-tree-decomposition. By ...
11
votes
0answers
167 views

Graph problem known to be $NP$-complete only under Cook reduction

The theory of NP-completeness was initially built on Cook (polynomial-time Turing) reductions. Later, Karp introduced polynomial-time many-to-one reductions. A Cook reduction is more powerful than a ...
11
votes
0answers
167 views

Solving divide & conquer reccurences if the split-ratio depends on $n$

Is there a general method to solve the recurrence of the form: $T(n) = T(n-n^c) + T(n^c) + f(n)$ for $c < 1$, or more generally $T(n) = T(n-g(n)) + T(r(n)) + f(n)$ where $g(n),r(n)$ are some ...
10
votes
1answer
122 views

Efficient algorithms for vertical visibility problem

During thinking on one problem, I realised that I need to create an efficient algorithm solving the following task: The problem: we are given a two-dimensional square box of side $n$ whose sides are ...
10
votes
0answers
90 views

Finding an st-path in a planar graph which is adjacent to the fewest number of faces

I am curious whether the following problems has been studied before, but wasn't able to find any papers about it: Given a planar graph G, and two vertices s and t, find an st-path $P$ which minimizes ...
10
votes
0answers
248 views

Complexity of deciding whether there is a winning strategy in the following game

The sum divider game for $n$ starts with the set $M_0 = \{1,\dots,n\}$. Player A chooses a number $m_1$ from $M_0 \setminus \{1\}$ and B has to choose a divider $m_2$ of $m_1$ from $M_1 = M_0 ...
9
votes
0answers
275 views

distributed alpha beta pruning

I am looking for an efficient algorithm that lets me process the minimax search tree for chess with alpha-beta pruning on a distributed architecture. The algorithms I have found (PVS, YBWC, DTS see ...
8
votes
0answers
80 views

Would $\sf RP = NP$ imply $\sf NP = coNP$?

If $\sf RP = NP$ then the hierarchy collapses to its second level (by the Karp-Lipton theorem). But what about $\sf NP$ and $\sf coNP$? I tried to prove that $\sf BPP$ is contained in $\sf NP$ (the ...
8
votes
0answers
115 views

“Essential” problem for MA

I am trying to understand different interactive proof systems. Is there a typical problem for the complexity class MA (like graph-nonisomorphism is for AM)? Is there a problem in MA but not known to ...
7
votes
0answers
97 views

Change in the distances in a graph after removal of a node

Given an undirected unweighted graph $G=(V,E)$ and a node $s \in V$, we are looking for a vector $\operatorname{diff}[]$, such that, $$\operatorname{diff}[v] = \sum_{u \in V \setminus \{v\}}{(d^{G ...

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