18

This is NP-hard. See: Joan Boyar, Philip Matthews, René Peralta. Logic Minimization Techniques with Applications to Cryptology. http://link.springer.com/article/10.1007/s00145-012-9124-7 The reduction is from Vertex Cover and is very nice. Given a graph $(\{1,\ldots,n\},E)$ with $m=|E|$, define an $m \times (n+1)$ matrix $A$ as: $A[i,j] = 1$ if $j < n+...


13

On the positive side, it is decidable whether a one-tape Turing machine runs in time $n \mapsto C \cdot n + D$ for given $C, D \in \mathbb{N}$, see: David Gajser: Verifying whether One-Tape Non-Deterministic Turing Machines Run in Time $Cn+D$, arXiv:1312.0496


12

We're interested in additive approximations to #3SAT. i.e. given a 3CNF $\phi$ on $n$ variables count the number of satisfying assignments (call this $a$) up to additive error $k$. Here are some basic results for this: Case 1: $k=2^{n-1}-\mathrm{poly}(n)$ Here there is a deterministic poly-time algorithm: Let $m=2^n-2k = \mathrm{poly}(n)$. Now evaluate $\...


12

The answer for (1) is "unlikely". It is simple to show (reduce from $Partition$) there exists no $\alpha$-approximation for Bin Packing, for any $\alpha<\frac{3}{2}$, unless $P=NP$. That said, Crescenzi et al. have shown that unless the polynomial hierarchy collapses, Bin Packing is not APX-Hard. As for (2), perhaps you could phrase it as "Does not ...


12

The problem is very similar to Min Uncut. In Min Uncut, given a graph $G = (V, E)$, we need to find a subset of edges $E'$ s.t. $G - E'$ is bipartite; the objective is to minimize the size of $|E'|$. For brevity, let me call you problem $\cal P$ and Min Uncut $\cal U$. Observation. An instance $G$ of $\cal P$ has a solution of cost 0 if and only if $G$ is ...


11

There are much stronger results for approximate graph coloring. S. Khot, Improved inapproximability results for maxclique, chromatic number and approximate graph coloring. show that for all sufficiently large constants $k$ it is NP-hard to color a $k$-colorable graph with $k^{\Omega(log k)}$ colors A very recent result of S. Huang Improved Hardness of ...


11

Here are the references: S. Khanna, N. Linial, and S. Safra, On the hardness of approximating the chromatic number, Combinatoria, 20 (2000), pp. 393–415. C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, J. ACM, 41 (1994), pp. 960–981.


11

Yes, a PTAS for Max-Planar-3-SAT can be constructed by using Brenda Baker's approach. This has been observed, for instance, in Theorem 17 in Pierluigi Crescenzi and LucaTrevisan: "Max NP-completeness made easy" Theoretical Computer Science 28, (1999), Pages 65-79


10

One paper that gives an answer to this question is Chalermsook, Laekhanukit, & Nanongkai (2013). There are also related works in the context of Fixed Parameter Tractability such as Hajiaghayi, Khandekar, & Kortsarz (2013) and Chitnis, Hajiaghayi, Kortsarz (2013). These hardness results are proven under various assumptions such as ETH or existence of ...


10

If I understood well the problem, perhaps this is an idea for a reduction from the Hamiltonian path problem: given $G$ with $|V| = n$, a source and target node $s, t \in V$; you can extend it adding a $(n-1) \times n$ "full" grid graph having the bottom-left node of the last row connected to $s$ and the bottom right node of the last row connected to $t$. ...


10

A particularly striking example of a phase transition is the maximum degree bound for Exactly-$k$-SAT (X$k$SAT), in which each clause contains exactly $k$ distinct literals. The problem flips from being trivially easy (always satisfiable) to being NP-complete by adding one to the associated parameter. Let $f(k)$ denote the largest number such that any X$k$...


10

As David pointed out, Khot's paper, "Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring", Theorem 1.6, says it is NP-hard to color $K$-colorable graph with $2^{\Omega((\log K)^2)}$ colors for graphs with degree at most $2^{2^{(\log K)^2}}$, for sufficiently large constant $K$. In other words, for graphs of ...


10

Let me see if I can clarify this, on a high level. Assume the UG instance is a bipartite graph $G = (V \cup W, E)$, bijections $\{\pi_e\}_{e \in E}$, where $\pi_e\colon \Sigma \to \Sigma$, and $|\Sigma| = m$. You want to construct a new graph $H$ so that if the UG instance is $1-\delta$ satisfiable, then $H$ has a large cut, and if the UG instance is not ...


10

There is a more complicated theory of duality for SDPs that is exact: there is no 'extra condition' like Slater's condition. This is due to Ramana. (For another take on this involving SOS, see [KS12].) To be honest, I've never tried to understand these papers and would be happy if someone dumbed them down for me. One notable consequence of this work is ...


10

Cormen, Leiserson, Rivest and Stein say that this algorithm that achieves ratio of 2 is tight. They did not exclude the possibility of another algorithm achieving better. Vertex Cover is NP-hard to approximate with a factor better than 1.36: http://annals.math.princeton.edu/wp-content/uploads/annals-v162-n1-p08.pdf Also check the following paper which give ...


9

There is an inapproximability result for coloring bounded degree graphs in Khot's FOCS'01 paper, "Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring" — it's probably weaker than you want, but at least it's in the right direction. He proves that, for a parameter $k$ (assumed to be constant), and for $k$-...


9

If $p$ is constant, then the size of the maximum clique in the $G(n,p)$ model is almost everywhere a constant multiple of $\log n$, with the constant proportional to $\log (1/p)$. (See Bollobás, p.283 and Corollary 11.2.) Changing $p$ should therefore not affect the hardness of planting a clique with $\omega(\log n)$ vertices as long as the clique is too ...


9

For the SDP in standard form $$ \min\{ \mathrm{tr}(C^T X): \mathrm{tr}(A_1^T X) = b_1, \ldots, \mathrm{tr}(A_m^T X) = b_m, X \succeq 0\}, $$ Slater's condition reduces to the existence of a positive definite $X\succ 0$ that satisfies the affine constraints $\mathrm{tr}(A_i^T X) = b_i$. I would guess this is satisfied for any SDP you can find in the ...


8

The best known hardness of approximating the chromatic number of $3$-colorable graphs with bounded maximum degree is due to Venkatesan Guruswami and Sanjeev Khanna, On the Hardness of 4-Coloring a 3-Colorable Graph: There is a constant $\Delta$ such that given a $3$-colorable graph with maximum degree at most $\Delta$, it is NP-hard to color it using ...


8

Here is a summary of what is known about approximability of $k$-CSP over a domain of size $q$: The best known approximation algorithms for the problem give an $\Omega(q \max(k, \log q)/q^k)$ approximation [MM14 and MNT16]. For $k = \Omega(q)$, there is a matching hardness of $O(kq/q^k)$ by Håstad (UGC-hardness) and Chan (NP-hardness) [Chan13]. For $k$ ...


8

The powering step fails. After the powering, each vertex is labeled with a neighborhood of the original graph. each edge checks that its endpoints agree on the intersection of their neighborhoods, and that this labeling satisfies the edges in this intersection. However, the edge cannot check anything about part of the labeling that lies outside the ...


7

A similar hardness result is known for the inapproximability of chromatic number [1]. Independent set is hard to approximate as well, since it is $\mathsf{MAX\text{-}CLIQUE}$ on the complement graph. Some problems that are easier to approximate include vertex cover, (connected) dominating set, feedback vertex set, TSP and Steiner tree. [1] Zuckerman, David. ...


7

Let me sketch the relation between the PCP theorem and 2-provers 1-round game. For concreteness let's consider the MAX-3-SAT problem. In this problem we are given a 3-CNF formula, and our goal is to find an assignment that maximizes the number of satisfied clauses. This problem is NP-hard, and (the CSP view of) the PCP theorem says that given a satisfiable ...


7

As suggested by Kaveh, the relevant chapters in Arora-Barak are a good starting point. Soon you'll want to sink your teeth into a significant paper. One difficulty you may encounter is that recent papers in this area build on two decades worth of intense research, which can be intimidating to a beginner. IMHO, the best balance between importance of results ...


7

I recommend Lectures on Proof Verification and Approximation Algorithms, E.W.Mayr and H.J.Prömel and A.Steger (Eds.), Lecture Notes in Computer Science, vol. 1367, Springer Verlag 1998. These lectures are very suitable for studying these topics from the basic.


7

Looking at the Goemans–Williamson algorithm in the SOS framework yields no technical advantages: it is exactly the same algorithm and the same ideas are used in the analysis. The only advantages in doing so are: Arguably the algorithm seems less "magical" in that viewpoint, though of course that's a matter of taste. It's a good basic case to get intuition ...


6

For a constant $d$ the $(k,d)$-hitting set problem is not harder than the original $d$-hitting set (i.e. $k=1$) in view of both approximation and parametrized complexity. There is a simple reduction from $kd$-HS to $d$-HS. For an instance $(U,\mathcal{F},d,k)$ of the first problem we get an instance of $(U',\mathcal{F'},d)$ of the second one in which every ...


6

Basically everything that is known about the Quantum PCP conjecture has been collected in this survey by Dorit Aharonov, Itai Arad, and Thomas Vidick: The Quantum PCP Conjecture See also Thomas' blog post on the topic.


6

See the paper "A parallel approximation algorithm for positive linear programming." by Luby and Nisan. (Some kinds of) linear programs can be approximated in log^(O(1)) n time.


6

This problem is way harder than set cover. Here is why... Intuitively, you can encode independent set as a problem of this type. Indeed, you are given an instance of independent set - a graph $G$ with $n$ vertices, and a number $k$, and the question is whether the graph $G$ has an independent set of size $k$. We assume that $k$ is large (say, polynomial in ...


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