Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

Inapproximability of Maximum Biclique Problems, Minimum kk-Cut and Densest At-Least-kk-Subgraph from the Small Set Expansion Hypothesis

The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose edge expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph $G$, find a complete bipartite subgraph of $G$ with maximum number of edges. - Maximum Balanced Biclique (MBB): given a bipartite graph $G$, find a balanced complete bipartite subgraph of $G$ with maximum number of vertices. - Minimum $k$-Cut: given a weighted graph $G$, find a set of edges with minimum total weight whose removal partitions $G$ into $k$ connected components. - Densest At-Least-$k$-Subgraph (DAL$k$S): given a weighted graph $G$, find a set $S$ of at least $k$ vertices such that the induced subgraph on $S$ has maximum density (the ratio between the total weight of edges and the number of vertices). We show that, assuming SSEH and NP $\nsubseteq$ BPP, no polynomial time algorithm gives $n^{1 - \varepsilon}$-approximation for MEB or MBB for every constant $\varepsilon > 0$. Moreover, assuming SSEH, we show that it is NP-hard to approximate Minimum $k$-Cut and DAL$k$S to within $(2 - \varepsilon)$ factor of the optimum for every constant $\varepsilon > 0$. The ratios in our results are essentially tight since trivial algorithms give $n$-approximation to both MEB and MBB and efficient $2$-approximation algorithms are known for Minimum $k$-Cut [SV95] and DAL$k$S [And07, KS09]. Our first result is proved by combining a technique developed by Raghavendra et al. [RST12] to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test [BK09] whereas our second result is shown via elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

Given a $k$-uniform hyper-graph, the E$k$-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that E$k$-Vertex-Cover is NP-hard to approximate within factor $(k-1-\epsilon)$ for any $k \geq 3$ and any $\epsilon>0$. The result is essentially tight as this problem can be easily approximated within factor $k$. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of $s$-wise $t$-intersecting families of subsets

From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting $\text{OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t(\text{OPT})\text{poly}(N)$ time and outputs a solution of size $f(\text{OPT})$, for any functions $t$ and $f$ that are independent of $N$ (for Clique, we want $f(\text{OPT})=\omega(1)$)? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no $o(\text{OPT})$-FPT-approximation algorithm for Clique and no $f(\text{OPT})$-FPT-approximation algorithm for DomSet, for any function $f$ (e.g., this holds even if $f$ is the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even $(1 - \epsilon)$-satisfiable for some constant $\epsilon > 0$. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm.Comment: 43 pages. To appear in FOCS'1

Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem.) Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main ingredient is a quantitative improvement of Razborov's rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure

Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs

This work revisits the PCP Verifiers used in the works of Hastad [Has01], Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable 4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to prove the following improved hardness results: Assuming that NP\not\subseteq DTIME(N^{O(loglog N)}), There is no polynomial time algorithm that, given an n-vertex 2-colorable 4-uniform hypergraph, finds an independent set of n/(log n)^c vertices, for some constant c > 0. There is no polynomial time algorithm that satisfies 7/8 + 1/(log n)^c fraction of the clauses of a satisfiable Max-E3-SAT instance of size n, for some constant c > 0. For any fixed k >= 4, there is no polynomial time algorithm that finds a partition splitting (1 - 2^{-k+1}) + 1/(log n)^c fraction of the k-sets of a satisfiable Max-Ek-Set-Splitting instance of size n, for some constant c > 0. Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs is an exponential improvement over the previous results of Guruswami et al.[GHS02] and Holmerin[Hol02]. Similarly, our inapproximability of (log n)^{-c} beyond the random assignment threshold for Max-E3-SAT and Max-Ek-Set-Splitting is an exponential improvement over the previous bounds proved in [Has01], [Hol02] and [Gur00]. The PCP Verifiers used in our results avoid the use of a variable bias parameter used in previous works, which leads to the improved hardness thresholds in addition to simplifying the analysis substantially. Apart from standard techniques from Fourier Analysis, for the first mentioned result we use a mixing estimate of Markov Chains based on uniform reverse hypercontractivity over general product spaces from the work of Mossel et al.[MOS13].Comment: 23 Page

Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat). The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem

On the NP-Hardness of Approximating Ordering Constraint Satisfaction Problems

We show improved NP-hardness of approximating Ordering Constraint Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, Maximum Acyclic Subgraph and Maximum Betweenness, we prove inapproximability of $14/15+\epsilon$ and $1/2+\epsilon$. An OCSP is said to be approximation resistant if it is hard to approximate better than taking a uniformly random ordering. We prove that the Maximum Non-Betweenness Problem is approximation resistant and that there are width-$m$ approximation-resistant OCSPs accepting only a fraction $1 / (m/2)!$ of assignments. These results provide the first examples of approximation-resistant OCSPs subject only to P $\neq$ \NP