Simple Combinatorial Construction of the $k^{o(1)}$-Lower Bound for Approximating the Parameterized $k$-Clique

We follow the definitions and notations from [FG06]. In the context of computational complexity, a decision problem is defined as a set of strings over a finite alphabet $\Sigma$, sometimes called a language, say $\Pi\subseteq\Sigma^{*}$. A parameterization of a problem is a polynomial-time computable function $\kappa:\Sigma^{*}\rightarrow\mathbb{N}$. A parameterized decision problem is then defined as a pair $(\Pi,\kappa)$, where $\Pi\subseteq\Sigma^{*}$ is an arbitrary decision problem, and $\kappa$ is its parameterization.

The parameterization $\kappa$ gives a characterization of an instance of the designated problem that depends on the instance’s property but not on the input size. That is, the function $\kappa$ maps an instance of a problem to a small positive integer, and we may think of a parameterized decision problem as a problem where each input instance $x\in\Sigma^{*}$ is associated with a number $\kappa(x)$ which is typically much smaller than the length $|x|$ of $x$.

We say that a parameterized problem $(\Pi,\kappa)$ is fixed-parameter tractable (FPT) if there is an algorithm that, given a string $x\in\Sigma^{*}$ decides whether $x\in\Pi$ in time $t(\kappa(x))\mathrm{poly}(|x|)$, where $\mathrm{poly}(n)=\bigcup_{c>0}n^{c}$ and $t(k)$ is a computable function that depends only on the parameter $k=\kappa(x)$. The running time of the form $t(\kappa(x))\mathrm{poly}(|x|)$ is called FPT-time, and the algorithm that decides a parameterized problem in FPT-time is called an FPT algorithm. The complexity class FPT is the class of all parameterized problems that admit FPT algorithms. Similar to the polynomial-hierarchy in the theory of NP-completeness, there exists the classes of W-hierarchy such that

The classes W[1] and W[2] contain many natural complete problems, in particular the $k$-Clique problem and the $k$-Dominating Set problem, respectively. Thus, $k$-Clique admits no FPT algorithm unless $\text{W[1]}=\text{FPT}$, and similarly, $k$-Dominating-Set admits no FPT algorithm unless $\text{W[2]}=\text{FPT}$ (which thus implies $\text{W[2]}=\text{W[1]}=\text{FPT}$).

FPT algorithms have been one of the tools for coping with NP-hard problems as when the parameter $\kappa(x)$ is much smaller than $|x|$, an FPT algorithm is simply a polynomial-time algorithm. However, as mentioned, assuming $\text{W[1]}\neq\text{FPT}$, many important optimization problems like the maximum clique problem and the minimum dominating set problem admit no FPT algorithm. This leads to the seek of an $f(k)$-FPT-approximation algorithm for those optimization problems, where $f$ is a computable function. That is, an FPT-time algorithm that, given an instance $x$ with the additional parameter $k$ such that cost of an optimal solution, denoted by $\mathrm{opt}(x)$, satisfies

produces a feasible solution $y$ with

The function $f$ is known as the approximation ratio of the algorithm. It should be clear that, for maximization problems, we are only interested in that ratio $f$ with unbounded and non-decreasing $k/f(k)$.

The development of FPT-approximation algorithms has been at a fast pace in both upper bound and lower bound (i.e., the FPT-inapproximability results). Please see, e.g., [FSLM20] for references therein. At present, two core problems in the area of parameterized complexity – the $k$-Clique problem and the $k$-Dominating Set problem – are known to be totally FPT-inapproximable, i.e., they admit no FPT-approximation algorithm parameterized by the size of optimal solution $k$ for any ratio, unless Gap-ETH is false [CCK⁺20], and the total FPT-inapproximability of the $k$-Dominating Set problem under $\text{W[1]}\neq\text{FPT}$ was later proved by Karthik C.S, Laekhanukit and Manurangsri in [SLM19]. There has been steady progress on the hardness of approximation for $k$-Clique and $k$-Dominating Set [LRSW23a]. Nevertheless, the question of whether $k$-Clique is totally FPT-inapproximable under $\text{W[1]}\neq\text{FPT}$ remains open.

To be formal, we say that a maximization (respectively, minimization) problem $\Pi$ parameterized by a standard parameter $k$ (e.g., the size of the optimal solution) is totally FPT-inapproximable if, for any computable non-decreasing functions $t(k)$ and $f(k)$, depending only on $k$, given an input of size $n$, there exists no algorithm running in time $t(k)\mathrm{poly}(n)$ that outputs a feasible solution with the cost at least $k/f(k)$ (respectively, at most $f(k)\cdot k$ for minimization problems).

A similar problem is also open for the $k$-Dominating Set problem.

Lastly, another conjecture has been established as a generic tool for deriving FPT-inapproximability result, namely the Parameterized Intractability Hypothesis (PIH). The conjecture involves asking whether a $2$-Constraint Satisfaction problem ($2$-CSP) on $k$ variables, where $k$ is a constant, while each variable, say $X_{i}$, takes a value from $[n]$ admits no constant FPT-approximation algorithms. More formally, $2$-CSP is a constraint satisfaction problem in which each constraint involves exactly two variables, and it is hypothesized that there is no constant-factor approximation for such a $2$-CSP parameterized by $k$ in FPT-time:

The result of Chalermsook et al.[CCK⁺20] implies that PIH holds under Gap-ETH, which has been improved to merely assuming ETH in a very recent breakthrough [GLR⁺24]. Nevertheless, it has been a major open problem whether PIH holds under the $\mathrm{W}[1]$-hardness:

Network coding is an information compression technique used in multi-input multicast networks. This approach was first introduced in [ACLY00], formalized in [HKM⁺03] for delay-free acyclic networks and then generalized to the general case in [HMS⁺03]. Please see [YLC06] for references therein.

In the model of random linear network coding, the data are transmitted from $k$ different source nodes to $k$ different destinations as $k$ vectors over the finite field $\mathbb{F}_{q}^{d}$, say $v_{1},\ldots,v_{k}\in{\mathbb{F}}^{d}_{q}$, where $q$ is a sufficiently large prime power. Whenever data packets meet at an intermediate node, the data are compressed and transmitted as a linear combination of the $k$ vectors with random coefficients, i.e.,

where $r_{i}\in\mathbb{F}_{q}$ for every $i\in[k]$

Given that a sink node receives enough packets, it can decode the information correctly by solving the linear system. More precisely, in the linear network coding with random coefficients, it was proved that $O(k)$ packets are enough to guarantee the existence of $k$ linearly independent vectors. Thus, there is a unique solution to the linear system, allowing the sink nodes to retrieve and correctly decode the information.

The advantage of network coding is in the efficiency of the throughput, which is close to the optimum, and the amount of memory required to store the packets.

The connection between the network coding and the hardness of approximating $k$-Clique is not known until the breakthrough result of Lin [Lin21], who implicitly applied the network coding approach to compress the information of $k$ vertices as one single vector. Once we have $k$ vectors with linear independent coefficients, then the information of the original $k$ vertices can be decoded uniquely, thus allowing one to check whether the encoded $k$ vertices form a $k$-clique or not. As every vector encodes information of $k$ vertices, every subset of $k$ linearly independent vectors gives a unique solution to the linear system, making the verifier reject a graph that has no $k$-clique by reading only a tiny portion of the vectors. This, consequently, gives an approximation hardness of the $k$-Clique problem through a very clever reduction from this specific probabilistic proof system to an instance of $k$-Clique.

Technically, the network coding is the same as Hadamard code, but it is more in line with our intuition of the self-reduction creating the gap for the $k$-Clique problem. See Section 3 for more detailed discussions.

A Sidon set is a subset $S$ of an abelian group such that the sum of any two distinct elements in $S$ are different, i.e., for any $x,y,x^{\prime},y^{\prime}\in S$ with $x\neq y$ and $x^{\prime}\neq y^{\prime}$, we have $x+y=x^{\prime}+y^{\prime}$ if and only if $\{x,y\}=\{x^{\prime},y^{\prime}\}$. Given a positive integer $n\geq 1$, a Sidon set $S\subseteq\mathbb{Z}/n\mathbb{Z}$ of size at least $n^{1/3}$ can be constructed in polynomial time using a greedy algorithm (the algorithm is attributed to Erdös): Start with a set $A=\emptyset$. Iteratively add to $A$ a number $x\in\{0,\ldots,n-1\}\setminus A$ such that $x$ cannot be written as $a+b-c$, for any $a,b,c\in A$ until no such number exists. Erdös and Turán [ET41] showed that a Sidon set of size $\sqrt{n}$ can also be constructed efficiently using quadratic residues. In particular, they showed that the set $S=\{2pk+(k^{2}):1\leq k\leq p\}$ is a Sidon set, where $p$ is a fixed prime, and all the operations are done under $\mathbb{F}_{p}$. Observe that if the vertices of a graph are from a Sidon set, then the endpoints of every edge sum to a unique element in the underlying abelian group. Thus, representing a graph vertices with a Sidon set makes it open to standard arithmetic operation, albeit under some finite fields.

There are a few generalizations of Sidon sets [O’B04]. For our purposes, we require the linear form of Sidon sets, which have been studied in the works of Ruzsa [Ruz93, Ruz95], which we call linear Sidon sets.

The above definition is a natural generalization of the sum-free set [CE16, Gre04] and the set excluding no three-terms arithmetic progression. Our linear Sidon set, on the other hands, is a special case of the $(M,b)$-free set [Gre05, Sha09] defined as a set $S$ in which there exists no non-trivial solution to a linear system $Mx=b$ takes value from $S^{\ell}$.

For the construction in Subsection 5.4, we need to consider linear combinations of more than two vectors in a given set $S$. This leads to the following definition.

We leave the details of the construction of $t$-term linearly independent sets to the reader, which is a straightforward generalization of Theorem 1.

Let $(G,k)$ be an instance of the multi-colored $k$-Clique problem. In particular,

and each $V_{i}$ is an independent set in $G$. We construct a graph $H$ as follows.

• •

  Let $\mathbb{F}_{q}$ be a finite field where $q$ is to be determined later. Moreover, let $n\coloneqq|V(G)|$ and $d\coloneqq\lceil\frac{3\log n}{\log q}+3\rceil$. So, by Theorem 1, we can assume without loss of generality that $V(G)\subseteq\mathbb{F}_{q}^{d}$ is a linear Sidon set of size $n$.

• •

  For every $r\in\mathbb{F}_{q}^{k}$ we define the column with index $r$ as

  $$C_{r}\coloneqq\Big{\{}\Pi_{r,\pi}\Bigm{|}\pi\in\mathbb{F}_{q}^{d}\Big{\}}.$$

  Here $\Pi_{r,\pi}$ is a unique vertex associated with $r$ and $\pi$. Then we set

  $$V(H)\coloneqq\bigcup_{r\in\mathbb{F}_{q}^{k}}C_{r}.$$

  (8)

• •

  We still need to define the edge set $E(H)$ for the graph $H$. Let $r,r^{\prime}\in\mathbb{F}_{q}^{k}$ and $\Pi_{r,\pi}\in C_{r}$, $\Pi_{r^{\prime},\pi^{\prime}}\in C_{r^{\prime}}$. We distinguish the following cases.

  1. (H1)

     If $r=r^{\prime}$, then there is no edge between $\Pi_{r,\pi}$ and $\Pi_{r^{\prime},\pi^{\prime}}$.

  2. (H2)

     If $\mathrm{Hamming}(r,r^{\prime})=1$, say $r^{\prime}=r+ae_{i}$²²2Recall $e_{i}$ is unit vector defined as (1). for some $a\in\mathbb{F}_{q}^{*}$ and $i\in[k]$. Then

     $\displaystyle\Pi_{r,\pi}\Pi_{r^{\prime},\pi^{\prime}}\in E(H)$

     $\displaystyle\iff\text{$\pi^{\prime}-\pi=av$ for some $v\in V_{i}\subseteq V(G)$}.$

     (9)

     That is, $\pi$ and $\pi^{\prime}$ pass vertex test with the vertex $v$ as described in Section 3.

  3. (H3)

     $a,b\in\mathbb{F}_{q}^{*}$ and distinct $i,j\in[k]$. Then

     	$\displaystyle\Pi_{r,\pi}\Pi_{r^{\prime},\pi^{\prime}}\in E(H)\iff$	$\pi^{\prime}-\pi=au+bv$
     		$\displaystyle\quad\text{for some $u\in V_{i}$ and $v\in V_{j}$ with $uv\in E(G)$}.$		(10)

     Hence, $\pi$ and $\pi^{\prime}$ pass the edge test with the edge $uv$. Note by Definition 1 the vertices $u$ and $v$, if exist, are unique.

  4. (H4)

     If $\mathrm{Hamming}(r,r^{\prime})\geq 3$, then $\Pi_{r,\pi}\Pi_{r^{\prime},\pi^{\prime}}\in E(H)$.