Questions tagged [approximation-hardness]
Hardness of approximation, aka inapproximability.
65
questions with no upvoted or accepted answers
16
votes
0answers
434 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 ...
13
votes
0answers
496 views
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 ...
12
votes
0answers
239 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 ...
10
votes
0answers
192 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 ...
10
votes
0answers
348 views
Gap hardness of Multi-Dimensional Cover
Given a finite set $X$ and a collection $F$ of subsets of $X$, we define a cover of $X$ in $F$ as a subset of $F$ whose union is equal to $X$. A cover $C$ of $X$ in $F$ is said to be exact if the ...
8
votes
0answers
204 views
NP-hardness of approximation for unconstrained submodular maximization
The problem of unconstrained submodular maximization can be phrased as follows:
Given a non-negative submodular function $f$ on a domain $D$ find a set $S \subseteq D$ maximizing $f(S)$.
Here a ...
8
votes
0answers
166 views
Is the dominating set problem constant-factor-approximable in undirected path graphs?
I am interested in the complexity of the minimum dominating set problem (MDSP) in some specific graph class.
A graph is an undirected path graph if it is the vertex-intersection graph of a family of ...
8
votes
0answers
1k views
Is Exact Cover by Equally-Sized Sets reducible to Multi-Dimensional Matching in a certain nice way?
This question is motivated by my other question “Gap hardness of Multi-Dimensional Cover,” which is in turn motivated by the question “Set Cover for Permutation Matrices” by Brayden Ware.
Informal ...
7
votes
0answers
325 views
Does Dinur's proof of PCP Theorem imply a procedure for reconstructing a witness?
In Section 3.2 of On Syntactic versus Computational Views on Approximability by Khanna, et al., the authors state that an adaptation of the results from Proof Verification and Hardness of ...
7
votes
0answers
649 views
Hardness of Approximation results for Special Set Packing Problem Wanted
Is there any inapproximability result for the following NP-hard problem, which is a special case of the weighted Set Packing Problem?
The general Set Packing Problem would be:
Given A Collection of ...
7
votes
0answers
328 views
Approximating the diameter of a convex set defined by semidefinite constraints
A convex subset $C$ of $\mathbb{R}^{n^2}$ is given as the set of positive semidefinite $n\times n$ matrices whose coefficients fulfill some affine equations.
Now, if you want to minimize a linear ...
7
votes
0answers
475 views
Is there a constant approximation algorithm for longest path for 3-connected cubic planar graphs or maximal planar graph?
optimization problem
Input: a 3-connected cubic planar graph
feasible solution: A simple path
measure to optimize: length of the simple path
Is there a constant approximation algorithm for this ...
7
votes
0answers
261 views
Results regarding Bounded Diameter Minimum Spanning Tree
Given edge weighted undirected graph the problem asks to output a spanning tree $T$ of minimum weight such that the path between any two vertices in the tree $T$ is bounded by the input $k$. One of ...
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
$$\...
6
votes
0answers
700 views
Approximation results for 3-partition
The 3-partition as defined here is a strongly NP-complete decision problem. Consider one optimization problem of 3-partition where the $m$ subsets each have at most three elements and a sum of not ...
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 ...
5
votes
0answers
114 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 ...
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 ...
5
votes
0answers
243 views
Maximizing the number of selected edges with opposing requirements
Consider the following problem:
Input: a complete bipartite graph $G$ with its edges colored either white or black, a number $k$.
Output: a subset of vertices $W$ of size $k$ which maximizes the ...
5
votes
0answers
241 views
Is MAX-SAT SETH (like) hard?
If we make a random assignment to the variables in $k$-sat ($m$ clauses), we are going to satisfy $(1-2^{-k})m$ clauses in expectation. In general satisfying fewer clauses is considered easy.
There ...
5
votes
0answers
116 views
The distribution on the solution space induced by randomized rounding
Consider the Goemans-Williamson algorithm for the MAX-CUT problem.
It is known, that if $maxcut(G) \geq 1-\epsilon$, then
the algorithm returns a cut $S$ of fractional size at least $1-\sqrt{\epsilon}$...
5
votes
0answers
211 views
Approximate c-chromatic number, each color class is P4-free (cograph)
The classic chromatic number of graph, $\chi(G)$, describes the minimum number of colors needed so that each color class is an independent set. There are many other graph coloring definitions. One of ...
5
votes
0answers
140 views
Approximation for accumulative set cover
Let $S_1,\ldots,S_m\subseteq U$ be subsets of a set $U$ of size $\lvert U\rvert=n$. Over all permutations $\pi$ of the set $\\{1,\ldots,m\\}$, I want to maximize the quantity
\begin{equation}
\sum_{k=...
5
votes
1answer
214 views
Hardness of approximating chromatic number of triangle-free graphs
The chromatic number of graph, $\chi( G)$ is hard to approximate for general graphs.
Are there results of hardness of approximating $\chi(G)$ for triangle-free graphs?
4
votes
0answers
158 views
Hardness of approximating |V|+(size of vertex cover)
I know that UGC implies a hardness of 2 for vertex cover, but is there a way to have this hardness on instances where the size of the vertex cover is at least $(1-\epsilon)\frac{|V|}2$? More generally ...
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 ...
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 ...
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
0answers
317 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 ...
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", ...
3
votes
0answers
143 views
Maximize number of edges covered by an independent set of vertices
Smallest vertex cover which is also an independent set asks about finding an independent set that covers all edges. This problem is known as the independent vertex cover problem and is equivalent to ...
3
votes
0answers
121 views
Set cover approximation ratio as a function of m (number of sets)
Feige's well known result (and more recent results) show that set cover cannot be approximated within a factor of $(1 - o(1)) \ln n$, where $n$ is the number of variables. What if we want an ...
3
votes
0answers
104 views
$\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ problems
All the problems which are either $\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ cannot admit a $\mathsf{PTAS}$, unless $P=NP$. I would like to know whether $\mathsf{MaxSNP}$ ...
3
votes
0answers
59 views
APx hardness of Multiterminal Cut Problem
In a Multiterminal Cut problem input is a graph G=(V,E) and a set of k terminals T which is a subset of vertex set V. There is a weight w(e) associated with each edge in the graph. The question is to ...
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 ...
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 ...
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 ...
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 $n_0$, ...
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) = \...
2
votes
0answers
175 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 ...
2
votes
0answers
59 views
General hardness results for convex problems
What are the hardness results that we know of for generally convex problems? In particular, I know of the result that every convex problem is in $P$ when we have an oracle for generating a separating ...
2
votes
0answers
93 views
About representability of optimization versions of NP-complete questions as polynomial optimization over the hypercube
Naively, in my very limited awareness, it feels that the Max-CUT is a very "special" NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a $n$...
2
votes
0answers
64 views
About complexity of recovering or learning Bayesian networks
Are there complexity theoretic results about recoverability or learnability of the marginals (of the source vertices) and the conditionals (along each of the edges) of a Bayesian network from having ...
2
votes
0answers
124 views
Hardness of approximately counting independent sets with a PRAS, rather than FPRAS
It is known that approximately counting the independent sets of a graph is hard, even if randomness can be used, and even if we restrict ourselves to bounded degree graphs with degree bound at least 6....
2
votes
0answers
84 views
Approximating BLEDP on restricted graph classes
In the edge-disjoint paths (EDP) problem, we are given an undirected graph $G$, and a set $\{ (s_i,t_i) \mid 1 \leq i \leq k \}$ of $k$ source-sink pairs. The objective is to maximize the number of ...
1
vote
0answers
52 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
59 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 $F$ ...
1
vote
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&...
