Unanswered Questions
276 questions with no answers
18
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0
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550
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Complexity of the densest $k$-subgraph problem on planar graphs
In the densest $k$-subgraph problem, one is given an undirected graph $G$ and wants to find a set of vertices $N$ with $|N| = k$ such that the number of edges in the subgraph of $G$ induced by $N$ is ...
- 99
17
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0
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984
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Deeper 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 ...
CommunityBot
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17
votes
0
answers
438
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Is Node Multiway Cut NP-complete on planar graphs when all terminals lie on the outer face?
I am interested in the following problem.
Node Multiway Cut on Planar Graphs with terminals on the outer face
Instance: A plane graph G, and integer k, and a set $S \subseteq V(G)$ of terminals which ...
- 359
16
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0
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506
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a geometric variant of k-medians. NP-hard or in P?
The following problem is a special case of k-medians. Is it NP-hard? Is it in P?
Input: $n$ points $(x_1,y_1), (x_2,y_2), \ldots, (x_n, y_n)$ with each $y_i \ge 0$, and an integer $k$.
Output: a set ...
- 10.9k
16
votes
0
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495
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Is graph coloring complete for poly-APX?
Is the graph coloring problem complete for poly-APX under C-reductions
(alternatively, under AP-reductions)? For the graph coloring problem, speaking of a feasible solution means a proper coloring for ...
- 6,795
16
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0
answers
1k
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Phase Transitions in NP Hard Problems
SAT Problems have a phase transition that depends on the ratio $r$ of variables to clauses. Below $r$, SAT problems are solvable quickly; above, they become difficult. The same is true of NP ...
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14
votes
0
answers
193
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NP-Hardness of 4-cycle packing problem in complete bipartite digraph?
A directed complete bipartite graph is a bipartite graph where there is exactly one directed edge between any two vertices from its two different parts. In other words, it's an orientation of a ...
- 483
14
votes
0
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699
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Does solving matrix multiplication in quadratic time imply that SETH is false?
I have a little conjecture that if you could perform matrix multiplication (or solve 3-clique) in $O(n^2 \log(n))$ time, then you could solve CNF-SAT in $O(2^{(1-\epsilon)n})$ time.
In other words, ...
- 5,167
13
votes
0
answers
633
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Recent progress on the next-to-shortest-path problem for directed graphs?
In the paper "Computing strictly-second shortest paths" (1997), Lalgudi and Papaefthymiou consider the following problem:
Let $G$ be a directed graph with edge-weighting $w$. Let $u,v$ be ...
CommunityBot
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13
votes
0
answers
296
views
Which monotone DNFs are evasive?
A Boolean function $\phi$ on variables $X$ is evasive if every decision tree for $\phi$ has height $|X|$. In other words, for any strategy that picks variables of $X$ and asks for their value, an ...
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13
votes
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answers
530
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Approximating and bounding Ramsey numbers
Calculating the diagonal Ramsey numbers R(s,s) is hard. There is a famous quote from Joel Spencer:
Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and ...
CommunityBot
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12
votes
0
answers
357
views
NP complete problem help
I'm currently trying to find a reduction to this problem:
Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...
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11
votes
0
answers
191
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Complexity of $(\Delta-1)$-coloring graphs of maximum degree $\Delta$
Suppose we have a graph $G$ with maximum degree $\Delta$. Deciding if $G$ can be colored with $\Delta$ colors is easy: Brooks' theorem says that this is always possible, unless $G$ is a clique or an ...
- 3,722
11
votes
0
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347
views
Is unary $\Pi_2$-SUBSETSUM coNP-complete?
Consider the following problem:
for given integers $a_1, \ldots, a_{2n}$ and $A$ that are given in unary representation
define is it true that
for every $S \subseteq \{1, ..., 2n \}$ such that $|...
- 2,023
11
votes
0
answers
240
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Inapproximability of multiterminal cut
In the multiterminal cut the input is a graph $G$ and a subset $T$ of its vertices. The task is to remove the minimum number of edges from $G$ such that there is no path connecting any distinct ...
CommunityBot
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