# Questions tagged [approximation-hardness]

Hardness of approximation, aka inapproximability.

171 questions
Filter by
Sorted by
Tagged with
7 views

### Balanced and general $MAXkSAT$ known approximation results and bounds from $UGC$

$MAX2SAT$ has a $0.9401$ to $0.9402$ approximation algorithm (refer https://dl.acm.org/doi/10.1145/1250790.1250818) which is conjectured to be optimal by $UGC$ while there is a balanced $MAX2SAT$ ...
169 views

### Why does Dinur's proof of the PCP theorem fail to work for unique games?

What is the critical step where things go wrong if one attempts to use Dinur's proof the PCP theorem to prove the unique games conjecture by starting from a unique label cover instance and doing gap ...
132 views

### What does it mean by the statement: “a problem is hard to approximate ”?

In most of the research papers that I have read so far, I often come across the statement of the following form: "the problem is hard to approximate within any factor smaller than some constant&...
33 views

### Best approximations of Minimum Dominating Sets in chordal graphs

I am searching results and papers related to the (in)approximability of the Minimum Dominating Set problem in chordal graphs. In particular, what is the best approximation ratio achievable in polytime ...
54 views

### polytime approximability of directed multicut

Does anyone know how well directed multicut can be approximated in planar and minor free graphs? Also any survey of approximability of directed multicut and multicut in various graph classes would be ...
70 views

151 views

162 views

### Best polynomial-time approximation factor for NP-optimization problems

Let us say that a function $f(n)$ is the best approximation factor for an NP-optimization problem, if both of the following hold: There exist a polynomial-time algorithm $A,$ and an integer $n_0$, ...
64 views

### Distinguising between the cases of low or high cover number

Is there a known result saying that for some constants $0 < a < b < 1$, it is NP-hard to distinguish a graph having vertex cover number at most $a \cdot n$ from a graph having vertex cover ...
213 views

### Does Max Planar 3-SAT admit a PTAS?

Suppose we are given a formula $\phi$ of 3-SAT, with variables $x_1,\dots, x_n$ and clauses $C_1,\dots, C_m$. Consider the graph $G_\phi$ where there is one node for each clause $C_i$, for each ...
197 views

### Is there any research on approximation of reals with computable numbers

I was wondering if there is any research in the field of Diophantine Approximation using the computable numbers. It seems to be a good fit, a dense countable set with a variety of different potential ...
591 views

### Approximately counting paths and cycles in a graph

Counting cycles, and paths in graphs is a hard problem, see questions here, and related question for cycles for a given length $k$, here. The question is about the approximability of these graph ...
59 views

### Reasoning about NP hardness of optimization problems with closed form functions as input

(This may not be a research level question per se. I can delete this question if the community thinks this way too) I am trying to understand how to reason about hardness of optimization problems ...
238 views

### 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 ...
52 views

### Inapproximability Results for APX-hard Geometric Optimization Problems

A lot of Geometric Optimization problems are NP-hard and APX-hard to approximate (No PTAS unless P=NP). One example of the geometric set cover problem can be found here. However, it is not easy to ...
189 views

### Which version of KAKUTANI does lie in PPAD?

The seminal paper of Papadimitriou [1] claims that the computational search problem KAKUTANI is $\mathbf{PPAD}$-complete. Unfortunately, there are very few details. Many other papers and surveys cite ...
1k views

### Why the reduction from MINIMUM SET COVER to MINIMUM DOMINATING SET means $c \log n$-inapproximability for MINIMUM DOMINATING SET

There is a well-known reduction from MINIMUM SET COVER to MINIMUM DOMINATING SET provided at http://en.wikipedia.org/wiki/Dominating_set#L-reductions (attributed there to Kann 1992, but seen, for ...
152 views

### Is there any better than (2/k)-approximation algorithm for Independent Set in Coloring graph?

I would like to know any results in terms of approximation algorithm about Maximum weight Independent Set problem in coloring graph. Input: Given a graph G with non-negative vertex weights and valid ...
269 views

### Hardness of Approximation for minimum path cover in an undirected graph?

Given an undirected graph $G = (V,E)$, a path cover is a set of disjoint paths such that every vertex $v\in V$ belongs to exactly one path. The minimum path cover problem consists of finding a path ...
44 views

### Approximating a monotone submodular function using a concrete coverage function

Is it possible to approximate a monotone submodular function using a concrete coverage function, i.e. Given a ground set $U$ and a monotone submodular function, $f:2^U\to \mathbb{R}$, the goal is to ...
101 views

### Does the NP-hardness of finding any valid solution imply NPO-hardness?

Suppose we have an NP optimization problem $A$ such that the problem of finding any valid solution, regardless of whether it is good or bad, is NP-hard (as it happens in NPO-complete problems such as ...
365 views

### Is there any known Poly-APX-complete minimimization problem?

All Poly-APX-complete problems I know are maximization problems, e.g. Max Clique, Max Independent Set, Max One for some set of contraints, and even choosing the attributes of a product to maximize (...
125 views

### Approximation class of finding decision trees with minimal depth

We are given some sets $S_1, \cdots , S_n$ and two disjoint sets $A$ and $B$. A decision tree is a binary tree where each node asks "$x \in S_i?$" for some $i$, taking the left branch means "yes", ...
The question asked is whether the following question is decidable: Problem  Given an integer $k$ and Turing machine $M$ promised to be in P, is the runtime of $M$ ${O}(n^k)$ with respect ...
### Is Asymptotic PTAS $\subseteq$ APX?
The definition of asymptotic polynomial-time approximation scheme (Asymptotic PTAS) is defined as follows: A minimization problem $\Pi$ is Asymptotic PTAS if for all $\epsilon$ there exists an ...