Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs

We consider the question of approximating Max 2-CSP where each variable appears in at most $d$ constraints (but with possibly arbitrarily large alphabet). There is a simple $(\frac{d+1}{2})$-approximation algorithm for the problem. We prove the following results for any sufficiently large $d$: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{d}{2} - o(d)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{d}{3} - o(d)\right)$. Thanks to a known connection [Dvorak et al., Algorithmica 2023], we establish the following hardness results for approximating Maximum Independent Set on $k$-claw-free graphs: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{k}{4} - o(k)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{k}{3 + 2\sqrt{2}} - o(k)\right) \geq \left(\frac{k}{5.829} - o(k)\right)$. In comparison, known approximation algorithms achieve $\left(\frac{k}{2} - o(k)\right)$-approximation in polynomial time [Neuwohner, STACS 2021; Thiery and Ward, SODA 2023] and $(\frac{k}{3} + o(k))$-approximation in quasi-polynomial time [Cygan et al., SODA 2013]

Upper tails and independence polynomials in random graphs

The upper tail problem in the Erd\H{o}s--R\'enyi random graph $G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+\delta$. Chatterjee and Dembo showed that in the sparse regime of $p\to 0$ as $n\to\infty$ with $p \geq n^{-\alpha}$ for an explicit $\alpha=\alpha_H>0$, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where $H$ is a clique. Here we extend the latter work to any fixed graph $H$ and determine a function $c_H(\delta)$ such that, for $p$ as above and any fixed $\delta>0$, the upper tail probability is $\exp[-(c_H(\delta)+o(1))n^2 p^\Delta \log(1/p)]$, where $\Delta$ is the maximum degree of $H$. As it turns out, the leading order constant in the large deviation rate function, $c_H(\delta)$, is governed by the independence polynomial of $H$, defined as $P_H(x)=\sum i_H(k) x^k$ where $i_H(k)$ is the number of independent sets of size $k$ in $H$. For instance, if $H$ is a regular graph on $m$ vertices, then $c_H(\delta)$ is the minimum between $\frac12 \delta^{2/m}$ and the unique positive solution of $P_H(x) = 1+\delta$

Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs

In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. First, for any ``symmetric'' predicate $P:{0,1}^{k} \to {0,1}$ except \equ where $k\geq 3$, we show that every (randomized) algorithm that distinguishes satisfiable instances of CSP(P) from instances $(|P^{-1}(0)|/2^k-\epsilon)$-far from satisfiability requires $\Omega(n^{1/2+\delta})$ queries where $n$ is the number of variables and $\delta>0$ is a constant that depends on $P$ and $\epsilon$. This breaks a natural lower bound $\Omega(n^{1/2})$, which is obtained by the birthday paradox. We also show that every one-sided error tester requires $\Omega(n)$ queries for such $P$. These results are hereditary in the sense that the same results hold for any predicate $Q$ such that $P^{-1}(1) \subseteq Q^{-1}(1)$. For EQU, we give a one-sided error tester whose query complexity is $\tilde{O}(n^{1/2})$. Also, for 2-XOR (or, equivalently E2LIN2), we show an $\Omega(n^{1/2+\delta})$ lower bound for distinguishing instances between $\epsilon$-close to and $(1/2-\epsilon)$-far from satisfiability. Next, for the general k-CSP over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances $(1-2k/2^k-\epsilon)$-far from satisfiability requires $\Omega(n)$ queries. The matching NP-hardness is not known, even assuming the Unique Games Conjecture or the $d$-to-$1$ Conjecture. As a corollary, for Maximum Independent Set on graphs with $n$ vertices and a degree bound $d$, we show that every approximation algorithm within a factor d/\poly\log d and an additive error of $\epsilon n$ requires $\Omega(n)$ queries. Previously, only super-constant lower bounds were known

UG-Hardness to NP-Hardness by Losing Half

The 2-to-2 Games Theorem of [Subhash Khot et al., 2017; Dinur et al., 2018; Dinur et al., 2018; Dinur et al., 2018] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least (1/2-epsilon) fraction of the constraints vs. no assignment satisfying more than epsilon fraction of the constraints, for every constant epsilon>0. We show that the reduction can be transformed in a non-trivial way to give a stronger guarantee in the completeness case: For at least (1/2-epsilon) fraction of the vertices on one side, all the constraints associated with them in the Unique Games instance can be satisfied. We use this guarantee to convert the known UG-hardness results to NP-hardness. We show: 1) Tight inapproximability of approximating independent sets in degree d graphs within a factor of Omega(d/(log^2 d)), where d is a constant. 2) NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of 2/3+epsilon, improving the previous ratio of 14/15+epsilon by Austrin et al. [Austrin et al., 2015]. 3) For any predicate P^{-1}(1) subseteq [q]^k supporting a balanced pairwise independent distribution, given a P-CSP instance with value at least 1/2-epsilon, it is NP-hard to satisfy more than (|P^{-1}(1)|/(q^k))+epsilon fraction of constraints

Lower bounds for local approximation

In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. By prior work, there is a deterministic local algorithm that achieves the approximation factor of 4 − 1/⌊Δ/2⌋ in graphs of maximum degree Δ. This approximation ratio is known to be optimal in the port-numbering model—our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier. Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α < 1 and any r, k, and g.Peer reviewe

Packing 3-vertex paths in claw-free graphs and related topics

An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the domination number of G. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. Then (a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2) if v(G) = 1 mod 3, then G - x has an L-factor, (a3) if v(G) = 2 mod 3, then G - {x,y} has an L-factor, (a4) if v(G) = 0 mod 3 and G is either cubic or 4-connected, then G - P has an L-factor, (a5) if G is cubic with v(G) > 5 and E is a set of three edges in G, then G - E has an L-factor if and only if the subgraph induced by E in G is not a claw and not a triangle, (a6) if v(G) = 1 mod 3, then G - {v,e} has an L-factor for every vertex v and every edge e in G, (a7) if v(G) = 1 mod 3, then there exist a 4-vertex path N and a claw Y in G such that G - N and G - Y have L-factors, and (a8) d(G) < v(G)/3 +1 and if in addition G is not a cycle and v(G) = 1 mod 3, then d(G) < v(G)/3. We explore the relations between packing problems of a graph and its line graph to obtain some results on different types of packings. We also discuss relations between L-packing and domination problems as well as between induced L-packings and the Hadwiger conjecture. Keywords: claw-free graph, cubic graph, vertex disjoint packing, edge disjoint packing, 3-vertex factor, 3-vertex packing, path-factor, induced packing, graph domination, graph minor, the Hadwiger conjecture.Comment: 29 page

Local algorithms in (weakly) coloured graphs

A local algorithm is a distributed algorithm that completes after a constant number of synchronous communication rounds. We present local approximation algorithms for the minimum dominating set problem and the maximum matching problem in 2-coloured and weakly 2-coloured graphs. In a weakly 2-coloured graph, both problems admit a local algorithm with the approximation factor $(\Delta+1)/2$, where $\Delta$ is the maximum degree of the graph. We also give a matching lower bound proving that there is no local algorithm with a better approximation factor for either of these problems. Furthermore, we show that the stronger assumption of a 2-colouring does not help in the case of the dominating set problem, but there is a local approximation scheme for the maximum matching problem in 2-coloured graphs.Comment: 14 pages, 3 figure