# Questions tagged [approximation-hardness]

Hardness of approximation, aka inapproximability.

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422 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 ...
486 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 ...
235 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 ...
345 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 ...
184 views

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

The seminal paper of Papadimitriou  claims that the computational search problem KAKUTANI is $\mathbf{PPAD}$-complete. Unfortunately, there are very few details. Many other papers and surveys cite ...
191 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 ...
162 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 ...
986 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 ...
314 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 ...
620 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 ...
456 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 ...
259 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 ...
646 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 ...
323 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 ...
43 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 ...
104 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 ...
51 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 ...
110 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 \...
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 ...
235 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 ...
115 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 $1-\sqrt{\epsilon}$...
184 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 ...
137 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=...
211 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?
157 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 ...
69 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 ...
52 views

170 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) . So we can focus on approximating the radius. A $(1+\epsilon)$-approximating of APSP for a ...
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 ...
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 ...
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 $n$...
109 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....
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### 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 ...
337 views

### The non-metric k-median problem

It is well-known that the non-metric $k$-median problem cannot be approximated better than $O(\log(n))$ (by a gap preserving reduction from the set cover problem). Is there any logarithmic ...
46 views

### Sherali-Adams lowerbound instance of Unique Games constructed via CLT

The question comes from the following paper I have been reading:  Integrality Gaps for Sherali–Adams Relaxations. SODA'09. Moses Charikar, Konstantin Makarychev, Yury Makarychev. Theorem 6.1 ...
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 ...