Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut

In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the "hard" instances of the Arora-Rao-Vazirani lemma [Sanjeev Arora et al., 2009; James R. Lee, 2005], we show that the Sum-of-Squares hierarchy can be adapted to provide "fast", but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here n denote the number of vertices of the graph and r can be any positive real number greater than 1 (possibly depending on n). - A (2 - 1/(O(r)))-approximation algorithm for Vertex Cover that runs in exp (n/(2^{r^2)})n^{O(1)} time. - An O(r)-approximation algorithms for Uniform Sparsest Cut and Balanced Separator that runs in exp (n/(2^{r^2)})n^{O(1)} time. Our algorithm for Vertex Cover improves upon Bansal et al.\u27s algorithm [Nikhil Bansal et al., 2017] which achieves (2 - 1/(O(r)))-approximation in time exp (n/(r^r))n^{O(1)}. For Uniform Sparsest Cut and Balanced Separator, our algorithms improve upon O(r)-approximation exp (n/(2^r))n^{O(1)}-time algorithms that follow from a work of Charikar et al. [Moses Charikar et al., 2010]

Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut 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 set of vertices whose expansion is almost zero and one in which all small sets of vertices have expansion almost one. In this work, we prove conditional 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. We show that, assuming SSEH and that NP != BPP, no polynomial time algorithm gives n^{1 - epsilon}-approximation for MEB for every constant epsilon > 0. - Maximum Balanced Biclique (MBB): given a bipartite graph G, find a balanced complete bipartite subgraph of G with maximum number of vertices. Similar to MEB, we prove n^{1 - epsilon} ratio inapproximability for MBB for every epsilon > 0, assuming SSEH and that NP != BPP. - Minimum k-Cut: given a weighted graph G, find a set of edges with minimum total weight whose removal splits the graph into k components. We prove that this problem is NP-hard to approximate to within (2 - epsilon) factor of the optimum for every epsilon > 0, assuming SSEH. The ratios in our results are essentially tight since trivial algorithms give n-approximation to both MEB and MBB and 2-approximation algorithms are known for Minimum k-Cut [Saran and Vazirani, SIAM J. Comput., 1995]. Our first two results are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani [Raghavendra et al., CCC, 2012] to avoid locality of gadget reductions with a generalization of Bansal and Khot\u27s long code test [Bansal and Khot, FOCS, 2009] whereas our last result is shown via an elementary reduction

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

New Tools and Connections for Exponential-Time Approximation

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of 1. r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r))) time, 2. r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r))) time, 3. (2−1/r) for minimum vertex cover in O∗(exp(n/rΩ(r))) time, and 4. (k−1/r) for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr))) time. (Throughout, O~ and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. in Discret Appl Math 159(17):1954–1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by exp(n1−o(1)/r1+o(1)) lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking O∗(2n/r) bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new randomized branching rule. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan’s PCP (Chan in J. ACM 63(3):27:1–27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016)

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search

We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size n which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized ?-approximation algorithm that runs in c^k?n^?(1) time, where k is the solution size, can be used to derive an ?-approximation randomized algorithm that runs in d??n^?(1) time, where d is the unique value in (1, 1+{c-1}/?) such that ?(1/??{d-1}/{c-1}) = {ln c}/? and ?(a?b) is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for ? = 1, and is strictly better when ? > 1, for any c > 1. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2??n^?(1) exhaustive search can be adapted to an ?-approximate exhaustive search that runs in time (1+exp(-???(1/(?))))??n^?(1), where ? is the entropy function. Furthermore, we provide a lower bound stating that the running time of this ?-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any ? ? 1, c > 1. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, 3-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a 1.1-approximation algorithm for Vertex Cover with running time 1.114??n^?(1), improving upon the previously best known 1.1-approximation running in time 1.127??n^?(1) by Bourgeois et al. [DAM 2011]

Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes

We study two optimization problems on simplicial complexes with homology over ??, the minimum bounded chain problem: given a d-dimensional complex ? embedded in ?^(d+1) and a null-homologous (d-1)-cycle C in ?, find the minimum d-chain with boundary C, and the minimum homologous chain problem: given a (d+1)-manifold ? and a d-chain D in ?, find the minimum d-chain homologous to D. We show strong hardness results for both problems even for small values of d; d = 2 for the former problem, and d=1 for the latter problem. We show that both problems are APX-hard, and hard to approximate within any constant factor assuming the unique games conjecture. On the positive side, we show that both problems are fixed-parameter tractable with respect to the size of the optimal solution. Moreover, we provide an O(?{log ?_d})-approximation algorithm for the minimum bounded chain problem where ?_d is the dth Betti number of ?. Finally, we provide an O(?{log n_{d+1}})-approximation algorithm for the minimum homologous chain problem where n_{d+1} is the number of (d+1)-simplices in ?

Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search

We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for {\em monotone subset minimization} problems. In a {\em monotone subset minimization} problem the input implicitly describes a non-empty set family over a universe of size $n$ which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use {\em approximate monotone local search} to show that a parameterized $\alpha$-approximation algorithm that runs in c^k \cdot n^{\OO(1)} time, where $k$ is the solution size, can be used to derive an $\alpha$-approximation randomized algorithm that runs in d^n \cdot n^{\OO(1)} time, where $d$ is the unique value in $d\in \left (1, 1+\frac{c-1}{\alpha} \right)$ such that \D{\frac{1}{\alpha}}{\frac{d-1}{c-1}} =\frac{\ln c }{\alpha} and \D{a}{b} is the Kullback-Leibler divergence. This running time matches that of Fomin et al.\ for $\alpha=1$, and is strictly better when $\alpha >1$, for any $c >1$. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2^n \cdot n^{\OO(1)} exhaustive search can be adapted to an $\alpha$-approximate exhaustive search that runs in time \left ( 1+ \exp\left (-\alpha \cdot \entropy\left (\frac{1}{\alpha}\right)\right)\right)^n \cdot n^{\OO(1)}, where \entropy is the entropy function. Furthermore, we provide a lower bound stating that the running time of this $\alpha$-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any $\alpha \geq 1,~c >1$. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for {\sc Vertex Cover}, {\sc $3$-Hitting Set}, {\sc Directed Feedback Vertex Set}, {\sc Directed Subset Feedback Vertex Set}, {\sc Directed Odd Cycle Transversal} and {\sc Undirected Multicut}. For instance, we get a $1.1$-approximation algorithm for {\sc Vertex Cover} with running time 1.114^n \cdot n^{\OO(1)}, improving upon the previously best known $1.1$-approximation running in time 1.127^n \cdot n^{\OO(1)} by Bourgeois et al.\ [DAM 2011]

Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search

We generalize the monotone local search approach of Fomin, Gaspers,Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection betweenparameterized approximation and exponential-time approximation algorithms formonotone subset minimization problems. In a monotone subset minimizationproblem the input implicitly describes a non-empty set family over a universeof size $n$ which is closed under taking supersets. The task is to find aminimum cardinality set in this family. Broadly speaking, we use approximatemonotone local search to show that a parameterized $\alpha$-approximationalgorithm that runs in $c^k \cdot n^{O(1)}$ time, where $k$ is the solutionsize, can be used to derive an $\alpha$-approximation randomized algorithm thatruns in $d^n \cdot n^{O(1)}$ time, where $d$ is the unique value in $d \in(1,1+\frac{c-1}{\alpha})$ such that$\mathcal{D}(\frac{1}{\alpha}\|\frac{d-1}{c-1})=\frac{\ln c}{\alpha}$ and$\mathcal{D}(a \|b)$ is the Kullback-Leibler divergence. This running timematches that of Fomin et al. for $\alpha=1$, and is strictly better when$\alpha >1$, for any $c > 1$. Furthermore, we also show that this result can bederandomized at the expense of a sub-exponential multiplicative factor in therunning time. We demonstrate the potential of approximate monotone local search by derivingnew and faster exponential approximation algorithms for Vertex Cover,$3$-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback VertexSet, Directed Odd Cycle Transversal and Undirected Multicut. For instance, weget a $1.1$-approximation algorithm for Vertex Cover with running time $1.114^n\cdot n^{O(1)}$, improving upon the previously best known $1.1$-approximationrunning in time $1.127^n \cdot n^{O(1)}$ by Bourgeois et al. [DAM 2011].<br