Abstract The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only $$(1-\varepsilon )$$ ( 1 - ε ) -satisfiable (where the parameter is the number of variables) for some constant $$0<\varepsilon <1$$ 0 < ε < 1 . We consider a minimization version of CSPs (Min CSP), where one may assign r values to each variable, and the goal is to ensure that every constraint is satisfied by some choice among the $$r \times r$$ r × r pairs of values assigned to its variables (call such a CSP instance r list satisfiable). We prove the following strong parameterized inapproximability for Min CSP: For every $$r \ge 1$$ r ≥ 1 , it is $${\mathsf {W[1]}}$$ W [ 1 ] -hard to tell if a 2CSP instance is satisfiable or is not even r list satisfiable. We refer to this statement as “Baby PIH," following the recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the combinatorial arguments underlying the Baby PCP theorem, overcoming some basic obstacles that arise in the parameterized setting. Furthermore, our reduction runs in time polynomially bounded in both the number of variables and the alphabet size and thus implies the Baby PCP theorem as well.

Finding sparse vectors is a fundamental problem that arises in several contexts including codes, subspaces, and lattices. In this work, we prove strong inapproximability results for all these variants using a novel approach that even bypasses the PCP theorem. Our main result is that it is NP-hard (under randomized reductions) to approximate the sparsest vector in a real subspace within any constant factor; the gap can be further amplified using tensoring. Our reduction has the property that there is a Boolean solution in the completeness case. As a corollary, this immediately recovers the state-of-the-art inapproximability factors for the shortest vector problem (SVP) on lattices. Our proof extends the range of $\mathbf{l}_{\_} \mathbf{p}$ (quasi) norms for which hardness was previously known, from ‘p at least one’ to ‘p at least zero’, answering a question raised by (Khot, JACM 2005).Previous hardness results for SVP, and the related minimum distance problem (MDP) for error-correcting codes, all use lattice/coding gadgets that have an abundance of codewords in a ball of radius smaller than the minimum distance. In contrast, our reduction only needs many codewords in a ball of radius slightly larger than the minimum distance. This enables an easy derandomization of our reduction for finite fields, giving a new elementary proof of deterministic hardness for MDP. We believe this weaker density requirement might offer a promising approach to showing deterministic hardness of SVP, a long elusive goal. The key technical ingredient underlying our result for real subspaces is a proof that in the kernel of a random Rademacher matrix, the support of any two linearly independent vectors have very little overlap.A broader motivation behind this work is the development of inapproximability techniques for problems over the reals. Analytic variants of sparsest vector have connections to small set expansion, quantum separability and polynomial maximization over convex sets, all of which appear to be out of reach of current PCP techniques. We hope that the approach we develop could enable progress on some of these problems.

The Parameterized Inapproximability Hypothesis (PIH), which is an analog of the PCP theorem in parameterized complexity, asserts the following: there is a constant ϵ> 0 such that for any computable function f:λ.,•→λ.,•, no f(k)· nO(1)-time algorithm can, on input a k-variable CSP instance with domain size n, find an assignment satisfying 1-ϵ fraction of the constraints. A recent work by Guruswami, Lin, Ren, Sun, and Wu (STOC'24) established PIH under the Exponential Time Hypothesis (ETH). In this work, we improve the quantitative aspects of PIH and prove (under ETH) that approximating sparse parameterized CSPs within a constant factor requires nk1-o(1) time. This immediately implies, for example, that finding a (k/2)-clique in an n-vertex graph with a k-clique requires nk1-o(1) time (assuming ETH). We also prove almost optimal time lower bounds for approximating k-ExactCover and Max k-Coverage. Our proof follows the blueprint of the previous work to identify a "vector-structured"ETH-hard CSP whose satisfiability can be checked via an appropriate form of "parallel"PCP. Using further ideas in the reduction, we guarantee additional structures for constraints in the CSP. We then leverage this to design a parallel PCP of almost linear size based on Reed-Muller codes and derandomized low degree testing.

The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an ɛ fraction of constraints for some absolute constant ɛ > 0. PIH plays the role of the PCP theorem in parameterized complexity. However, PIH has only been established under the Gap Exponential Time Hypothesis (ETH), a very strong assumption with an inherent gap. In this work, we prove PIH under the ETH. This is the first proof of PIH from a gap-free assumption. Our proof is self-contained and elementary. We identify an ETH-hard CSP whose variables take vector values, and constraints are either linear or of a special parallel structure. Both kinds of constraints can be checked with constant soundness via a “parallel PCP of proximity” based on the Walsh-Hadamard code.

The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E) and a set {D}subseteq V x V of k demand pairs. The aim is to compute the cheapest network N subseteq G for which there is an s -> t path for each (s,t)in {D}. It is known that this problem is notoriously hard as there is no k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k. For the bi-DSN_Planar problem, the aim is to compute a planar optimum solution N subseteq G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN_Planar, unless FPT=W[1]. Additionally we study several generalizations of bi-DSN_Planar and obtain upper and lower bounds on obtainable runtimes parameterized by k. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N subseteq G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists when parameterized by k [Chitnis et al., IPEC 2013]. We show a tight inapproximability result: under Gap-ETH there is no (2-{epsilon})-approximation algorithm parameterized by k (for any epsilon>0). To the best of our knowledge, this is the first example of a W[1]-hard problem admitting a non-trivial parameterized approximation factor which is also known to be tight! Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k.

The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an ε fraction of constraints for some absolute constant ε > 0. PIH plays the role of the PCP theorem in parameterized complexity. However, PIH has only been established under Gap-ETH, a very strong assumption with an inherent gap. In this work, we prove PIH under the Exponential Time Hypothesis (ETH). This is the first proof of PIH from a gap-free assumption. Our proof is self-contained and elementary. We identify an ETH-hard CSP whose variables take vector values, and constraints are either linear or of a special parallel structure. Both kinds of constraints can be checked with constant soundness via a “parallel PCP of proximity” based on the Walsh-Hadamard code.

In this paper, we prove that assuming the exponential time hypothesis (ETH), there is no $f(k) \cdot n^{k^{o(1 / \log \log k)}}$-time algorithm that can decide whether an n-vertex graph contains a clique of size k or contains no clique of size $k / 2$, and no FPT algorithm can decide whether an input graph has a clique of size k or no clique of size $k / f(k)$, where $f(k)$ is some function in $k^{1-o(1)}$. Our results significantly improve the previous works [1], [2]. The crux of our proof is a framework to construct gap-producing reductions for the k-CLIQUE problem. More precisely, we show that given an error-correcting code $C: \Sigma_{1}^{k} \rightarrow \Sigma_{2}^{k^{\prime}}$ that is locally testable and smooth locally decodable in the parallel setting, one can construct a reduction which on input a graph G outputs a graph $G^{\prime}$ in $\left(k^{\prime}\right)^{O(1)} \cdot n^{O\left(\log \left|\Sigma_{2}\right| / \log \left|\Sigma_{1}\right|\right)}$ time such•if G has a clique of size k, then $G^{\prime}$ has a clique of size K, where $K=\left(k^{\prime}\right)^{O(1)}$.•if G has no clique of size k, then $G^{\prime}$ has no clique of size $(1-\varepsilon) \cdot K$ for some constant $\varepsilon \in(0,1)$.We then construct such a code with $k^{\prime}=k^{\Theta(\log \log k)}$ and $\left|\Sigma_{2}\right|=\left|\Sigma_{1}\right|^{k^{0.54}}$, establishing the hardness result above. Our code generalizes the derivative code [3] into the case with a super constant order of derivatives.

In this paper, we prove that it is W[2]-hard to approximate k-SETCOVER within any constant ratio. Our proof is built upon the recently developed threshold graph composition technique. We propose a strong notion of threshold graphs and use a new composition method to prove this result. Our technique could also be applied to rule out polynomial time ratio approximation algorithms for the non-parameterized k-SETCOVER problem with k as small as , assuming W[1] ≠ FPT. We highlight that our proof does not depend on the well-known PCP theorem, and only involves simple combinatorial objects.

Given a simple graph $G$ and an integer $k$, the goal of $k$-Clique problem is to decide if $G$ contains a complete subgraph of size $k$. We say an algorithm approximates $k$-Clique within a factor $g(k)$ if it can find a clique of size at least $k / g(k)$ when $G$ is guaranteed to have a $k$-clique. Recently, it was shown that approximating $k$-Clique within a constant factor is W[1]-hard [Lin21]. We study the approximation of $k$-Clique under the Exponential Time Hypothesis (ETH). The reduction of [Lin21] already implies an $n^{Ω(\sqrt[6]{\log k})}$-time lower bound under ETH. We improve this lower bound to $n^{Ω(\log k)}$. Using the gap-amplification technique by expander graphs, we also prove that there is no $k^{o(1)}$ factor FPT-approximation algorithm for $k$-Clique under ETH. We also suggest a new way to prove the Parameterized Inapproximability Hypothesis (PIH) under ETH. We show that if there is no $n^{O(\frac{k}{\log k})}$ algorithm to approximate $k$-Clique within a constant factor, then PIH is true.