388 research outputs found

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

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    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 OPT\text{OPT} be the optimum and NN be the size of the input, is there an algorithm that runs in t(OPT)poly(N)t(\text{OPT})\text{poly}(N) time and outputs a solution of size f(OPT)f(\text{OPT}), for any functions tt and ff that are independent of NN (for Clique, we want f(OPT)=ω(1)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(OPT)o(\text{OPT})-FPT-approximation algorithm for Clique and no f(OPT)f(\text{OPT})-FPT-approximation algorithm for DomSet, for any function ff (e.g., this holds even if ff 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 2o(n)2^{o(n)}-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1ϵ)(1 - \epsilon)-satisfiable for some constant ϵ>0\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 ko(1)k^{o(1)}-FPT-approximation algorithm for Densest kk-Subgraph although this ratio does not yet match the trivial O(k)O(k)-approximation algorithm.Comment: 43 pages. To appear in FOCS'1

    Towards Effective Exact Algorithms for the Maximum Balanced Biclique Problem

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    The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP. Firstly, we introduce an Upper Bound Propagation procedure to pre-compute an upper bound involving each vertex. Then we extend an existing branch-and-bound algorithm by integrating the pre-computed upper bounds. We also present a set of new valid inequalities induced from the upper bounds to tighten an existing mathematical formulation for MBBP. Lastly, we investigate another exact algorithm scheme which enumerates a subset of balanced bicliques based on our upper bounds. Experiments show that compared to existing approaches, the proposed algorithms and formulations are more efficient in solving a set of random graphs and large real-life instances
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