Questions tagged [approximation-hardness]

Hardness of approximation, aka inapproximability.

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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 ...
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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 ...
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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 ...
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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 ...
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Algorithmic gap for greedy algorithm for (metric) uncapacitated facility location

In Jain et. al (2003), at the bottom of page 801, they construct an instance of (metric) uncapacitated facility location for which they claim the greedy (Hochbaum's) algorithm has gap $\Omega(\frac{\...
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Hard family for degree-$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 ...
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139 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 $i=1\ldots k$. ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 $n_0$, ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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568 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 ...
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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 ...
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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", ...
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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 $$\...
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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 (...
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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 ...
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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) = \...
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1answer
109 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 $U\subseteq V$...
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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 ...
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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 ...
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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) ...
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Is the current best approximation ratio for Vertex Cover problem also a lower bound?

In textbook "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein. in pp.1110-1111, they argue that the vertex-cover problem is a 2-approximation algorithm and it is lower bound so we ...
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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 ...
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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 ...
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1answer
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From Lasserre maps to pseudo-distributions

Let me define a ``Lasserre map of degree $d$" as a linear map $L : \mathbb{R}_n[x] \rightarrow \mathbb{R}$ i.e a real valued linear map on polynomials over $n$ variables with real coeffients. This is ...
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What are the hardness results known for CSP over $\mathbb{F}_q$?

I found two related papers, There is a UGC hardness result here, https://www.cs.cmu.edu/~venkatg/pubs/papers/qaryCSP.pdf A kind of a stronger result might be found in these two other papers, http://...
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When is the duality gap of semidefinite programming (SDP) zero?

I haven't been able to find in the literature a precise characterization of the vanishing of the SDP duality gap. Or, when does "strong duality" hold? For example, when one goes back and forth ...
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Given oracle for Max-3SAT compute clauses that cannot be satisfied

We know that Max-3SAT is NP-hard to compute exactly (and also hard to approximate to a particular constant multiplicative factor). However, suppose you are given an oracle for Max-3SAT that tells you ...
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Proving hardness of approximation with reduction in terms of 1/$\epsilon$

I have a reduction that proves that a problem is NP-hard to approximate to a factor $1 + \epsilon$ for any $0 < \epsilon < 1$. The reduction is polynomial in $n$ (the size of the instance of the ...
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About increasing the objective values of certificates for Max-Clique SDP

Say you write a round-k Lasserre (or any other hierarchy!) SDP relaxation of the Max-Clique problem. Lets say one now finds (or knows how to sample with high probability) a graph $G_1$ of size $n$ and ...
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Petrank's proof for the APX-hardness of MAX k-VERTEX COVER on subcubic graphs

My question is about the following maximization problem, which is the "fixed cardinality" version of MIN VERTEX COVER. I am interested in the restriction to subcubic graphs (i.e. of maximum degree 3). ...
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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 ...
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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 $n$...
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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 ...
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2answers
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Finding a minimal context free grammar that recognizes a finite set of strings of bounded length

Problem: Given a finite set of strings $\{x_1, x_2, ..., x_n\}$ of length $\ell$ or less from some finite alphabet $\Sigma=\{a_1, a_2, ..., a_k\}$, find the minimal context free grammar that ...
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Maximizing a monotone supermodular function s.t. cardinality

I've tried to comb the literature and seen a lot of references to results that almost but don't quite seem to address this. Question: Is it known to be true or is there a hardness result ...
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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 ...
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1answer
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Approximation algorithms for the maximum $2$-independence set problem

I am interested in approximating the maximum $2$-independent set problem in arbitrary graphs. In a graph $G$ a set $I$ of vertices is called $2$-independent if the distance between any two distinct ...
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On approx-preserving P- and A-reducibilities

Let $X$ and $Y$ be two NPO problems. Let $(f,g)$ be a reduction between $X$ and $Y$, in particular, assume that $(f,g)$ is both P-reduction and A-reduction, i.e., there exist two poly-time ...