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Unanswered Questions

338 questions with no upvoted or accepted answers
47 votes
0 answers
1k views

Problem unsolvable in $2^{o(n)}$ on inputs with $n$ bits, assuming ETH?

If we assume the Exponential-Time Hypothesis, then there is no $2^{o(n)}$ algorithm for $n$-variable 3-SAT, and many other natural problems, such as 3-COLORING on graphs with $n$ vertices. Notice ...
19 votes
1 answer
2k views

Complexity of interval cover problem

Consider the following problem $Q$: We are given an integer $n$, and $k$ intervals $[l_i,r_i]$ with $1\leq l_i\leq r_i\leq 2n$. We are also given $2n$ integers $d_1,…,d_{2n}\geq 0$. The task is to ...
18 votes
0 answers
550 views

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 ...
17 votes
0 answers
984 views

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 ...
17 votes
0 answers
438 views

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 ...
16 votes
0 answers
281 views

Does small circuits for a NP-complete problem contradict ETH?

The remarks of the Theorem 4 in the paper "On the complexity of circuit satisfiability" claims that: if circuit satisfiability (CktSat) problem can be decided by deterministic circuits of $2^{o(n)}$ ...
16 votes
0 answers
506 views

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 ...
16 votes
0 answers
495 views

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 ...
16 votes
0 answers
1k views

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 ...
15 votes
0 answers
522 views

Williams' Method, Natural Proofs and Constructivity

I have some questions on the previous question which is written bellow. Natural Proof and Constructivity : The topic of the previous question Recently, Ryan Williams proved that Constructivity in ...
14 votes
0 answers
193 views

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 ...
14 votes
0 answers
699 views

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, ...
14 votes
0 answers
494 views

DPLL and Lovász Local Lemma

Let $\varphi$ be a CNF formula. Suppose that each of $\varphi$'s clauses consist of exactly $t$ literals (and, moreover, all literals within one particular clause correspond to different variables). ...
13 votes
0 answers
633 views

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 ...
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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