Questions tagged [approximation-hardness]
Hardness of approximation, aka inapproximability.
174
questions
3
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0answers
60 views
Improving the approximation in Stockmeyer's counting theorem
Given a $\#P$ function $f(x)$, we can use Stockmeyer's counting theorem to get an approximation $g(x)$ such that
\begin{equation}
\left(1 - \frac{1}{\text{poly}(n)} \right) f(x) \le g(x) \le \left(...
2
votes
0answers
70 views
Consequences of an $o(\log n)$-approximation algorithm for a $\mathsf{\log\text{-}APX}$ hard problem
In [1], Feige proves that if there is a polynomial algorithm with approximation ratio in $o(\log n)$ for any $\mathsf{log\textit{-}APX}$ hard problem (say Minimum Dominating Set), then $\mathsf{NP}\...
1
vote
0answers
63 views
Hardness of Approximation of Continuous Metric k-Median
First let me describe the metric $k$-median problem.
Definition (Metric $k$-Median): Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ ...
0
votes
0answers
46 views
Balanced and general $MAXkSAT$ known approximation results and bounds from $UGC$
$MAX2SAT$ has a $0.9401$ to $0.9402$ approximation algorithm which is conjectured to be optimal by $UGC$ while there is a balanced $MAX2SAT$ bound of $0.943$ approximation which is conjectured to be ...
6
votes
1answer
205 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 ...
1
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0answers
145 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&...
1
vote
1answer
40 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 ...
1
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0answers
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 ...
0
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0answers
75 views
Hardness of approximation by reduction from MAX-E3SAT
For the MAX-E3SAT define $k=\sum_i^n (g_i-1)$, where $n$ is the number of variables and $g_i=\text{max}(\{\text{occurrences of }x_i, \text{occurrences of }\neg x_i\}), \text{ for each }i=\{1,\ldots,n\}...
1
vote
2answers
105 views
Where to find info on (polytime) approximability of various discrete optimization problems?
Where to find info on (polytime) approximability of various discrete optimization problems?
Sorry if this is stupid,but is there a site or reference that keeps up to date info on approximability of ...
6
votes
1answer
166 views
Good Survey paper for k-means/k-median/k-center/facility-location
I have stated 4 problems in the Question title.
All these problems are closely related and are studied in various variations. For example:
Space: Euclidean/metric/discrete/continuous/non-metric/2-...
9
votes
1answer
394 views
Is the following graph optimization problem approximable within a constant factor?
Let $G=(V,E)$ be an undirected graph, and let $\pi$ be a permutation of the vertices in $V$.
For a node $v\in V$, we denote by $\text{pred}_{\pi}(v)$ (respectively $\text{succ}_{\pi}(v)$) the set of ...
0
votes
0answers
33 views
Different version of approximation complexity and algorithm for densest-k-subgraph problem
In the densest-k-subgraph problem, we are given a graph $G =(V,E)$ and $k$, and we are asked to find a set $S \in V$ of vertices to maximize the number of edges in the induced graph of $S$, i.e. $|Ind[...
-1
votes
1answer
61 views
Is it possible to approximate the solution of NP-Hard problems in polynomial time using linear programming? [closed]
Suppose we have a NP-Hard problem such as the k-col, which is meant to determine if a graph may be colored using at most ...
0
votes
1answer
84 views
Optimization problems with same optimal value, but different approximation behavior
Context: related to this answer.
I would like to see an example to emphasize that approximation behavior depends not only on the optimal value but also the set of solutions. This makes sense ...
2
votes
0answers
100 views
Complexity of a scheduling problem with a fixed left bound of jobs
Consider the following scheduling problem. We have a finite set of jobs $\mathcal{J}= \{j_1, j_2, ..., j_n\}$ to do. Every job $j_i$ has its own value $c_i$ (the amount of money we are paid for ...
1
vote
1answer
210 views
Hardness of approximating acyclic chromatic number
I have some doubts on the inapproximability result for acyclic coloring presented in the paper New acyclic and star coloring algorithms with application to computing Hessians. They claim that there ...
0
votes
1answer
140 views
Problems that are NP-hard to approximate even when the input graph is regular
Are there graph problems which are NP-hard to approximate even when the input graph is regular?
For instance, are there optimization problems that are NP-hard to approximate within $O(n^\epsilon)$ ...
-1
votes
1answer
72 views
the shorstest cycle containing two given points
I am given a edge-weighted (multi)graph $G$ and two of its vertices, $u, v\in V(G)$. I want to find two edge-disjoint paths that connects $u$ and $v$ while minimizing the sum of the lengths of the ...
2
votes
0answers
77 views
How hard is it to approximate distance of linear code
I'm trying to figure out what is the current knowledge about how hard it is, given a generating matrix of a linear code over a field $F_{q}$, approximate it's distance.
I of course found that ...
1
vote
0answers
52 views
Sherali-Adams lowerbound instance of Unique Games constructed via CLT
The question comes from the following paper I have been reading:
[1] Integrality Gaps for Sherali–Adams Relaxations. SODA'09. Moses Charikar, Konstantin Makarychev, Yury Makarychev.
Theorem 6.1 of [...
2
votes
0answers
92 views
How hard is it to determine ex(n,G)?
Define the extremal Turán function $ex(n,G)$ of a graph $G$ as the most edges a graph on $n$ vertices can have without having a subgraph that is isomorphic to $G$. This function is known ...
3
votes
0answers
76 views
Example of a hardness-of-approximation proof which improves the approximation factor?
Are there any examples of hardness-of-approximation reductions where we get a better hardness bound for the problem we've reduced to than the problem we've reduced from? In the examples I've seen so ...
5
votes
0answers
52 views
Hard family for degree-$D$ MAX-3LIN
Max 3LIN is the problem where one is given a linear system of equations $C$ over $\mathbb{F}_2$ with $3$ variables per equation, and needs to determine the maximum number of equations that can be ...
2
votes
1answer
180 views
Maximum Positive Negative Set Cover Problem
I am considering the following problem.
Input: Given two disjoint subsets $A$ and $B$ and a collection $C$ of $k$ sets $S_1,S_2,\ldots,S_k$ where $S_i \subseteq A \cup B$ for all $i=1\ldots k$.
...
3
votes
0answers
57 views
Hardness of Approximation of Set Cover with Growing Size Bound
I'm considering the minimum set cover problem with the constraint that each set contains at most $k$ elements.
Here, $k$ depends on the size of the universe.
For example, $k$ may equal $\log n,\sqrt ...
6
votes
1answer
73 views
NP-intermediate approximation regimes for natural problems within the MAX-k-CSP family
I would like to know whether there are any examples of natural problems within the MAX-$k$-CSP family for which (under standard/reasonable conjectures) we believe the following:
There is a value $\...
5
votes
0answers
116 views
Is monotone 1-in-3 MAXSAT known to be APX hard?
Monotone 1-in-3 SAT is the problem where each clause of the SAT problem contains exactly 3 positive variables. The goal is to find an assignment such that exactly one variable is true in each clause ...
3
votes
0answers
78 views
Is #PP2DNF hard to approximate?
The problem #PP2DNF asks to count the number of satisfying assignments of a positive partitioned 2-DNF Boolean formula, i.e., a formula $\phi$ on variables $X_1, \ldots, X_n, Y_1, \ldots, Y_m$ of the ...
3
votes
2answers
143 views
Minimum relevant variables in linear system - additive approximation
In the problem Minimum Relevant Variables in Linear System (Min-RVLS), the input is a linear system, e.g.:
$$ A x = b $$
and the goal is to find a solution $x$ with as few nonzero variables as ...
2
votes
0answers
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 $n_0$, ...
3
votes
1answer
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 ...
7
votes
1answer
229 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 ...
1
vote
0answers
60 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 ...
5
votes
0answers
54 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 ...
9
votes
0answers
194 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 ...
0
votes
1answer
172 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 ...
3
votes
0answers
318 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 ...
3
votes
0answers
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 ...
1
vote
1answer
597 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 ...
1
vote
0answers
103 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 ...
3
votes
0answers
128 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", ...
6
votes
0answers
157 views
An optimal subspace projection problem
Suppose we have a $k$-dimensional subspace $V$ in $\mathbb{R}^n$ given by a basis $\{v_1,\cdots,v_k\in \mathbb{R}^n\}$, find an index set $I\subset [n]$ with $|I|=m$ where $k\le m\le n$, such that
$$\...
2
votes
1answer
390 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 (...
1
vote
2answers
305 views
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 ...
2
votes
0answers
65 views
On the impossibility of representing/approximating subadditive function using additive functions
I am trying to understand whether a monotone and subadditive function $f(S), S \subseteq 2^{[n]}$ and $ f : 2^{[n]} \rightarrow R_{\geq0}$ can be represented using an additive function $\hat{f}(S) = \...
1
vote
1answer
115 views
hardness of constant approximation of largest matching set
We say that $H$ is a matching graph if it contains $2n$ vertices and only $n$ vertex-disjoint edges, i.e. $H$ only contains those $n$ edges and no more.
Given a graph $G=(V,E)$ a subset $U\subseteq V$...
2
votes
0answers
177 views
Approximating the Radius of a (Dense) Graph
For a (dense) graph, computing its radius is as hard as computer "All Pairs Shortest Paths" (APSP) [1]. So we can focus on approximating the radius.
A $(1+\epsilon)$-approximating of APSP for a ...
2
votes
0answers
58 views
Studying the hardness of polynomial-time approximability using the concept of stability of approximation
In the Conclusion section, the author of this paper "Stability of Approximation Algorithms for Hard Optimization Problems" by Juraj Hromkovič, 1999 claims that
Using the notion of stability one can ...
3
votes
1answer
249 views
Looking for approximation class between NPO and Exp-APX
I'm trying to identify the approximation hardness of some maximization problem A. In problem A, finding a solution whose quality is 0 (i.e. such that the value returned by the objetive function is 0) ...
