30 research outputs found
Inapproximability of Maximum Biclique Problems, Minimum -Cut and Densest At-Least--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 , find a complete
bipartite subgraph of with maximum number of edges.
- Maximum Balanced Biclique (MBB): given a bipartite graph , find a
balanced complete bipartite subgraph of with maximum number of vertices.
- Minimum -Cut: given a weighted graph , find a set of edges with
minimum total weight whose removal partitions into connected
components.
- Densest At-Least--Subgraph (DALS): given a weighted graph , find a
set of at least vertices such that the induced subgraph on has
maximum density (the ratio between the total weight of edges and the number of
vertices).
We show that, assuming SSEH and NP BPP, no polynomial time
algorithm gives -approximation for MEB or MBB for every
constant . Moreover, assuming SSEH, we show that it is NP-hard
to approximate Minimum -Cut and DALS to within factor
of the optimum for every constant .
The ratios in our results are essentially tight since trivial algorithms give
-approximation to both MEB and MBB and efficient -approximation
algorithms are known for Minimum -Cut [SV95] and DALS [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
The Strongish Planted Clique Hypothesis and Its Consequences
We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time n^?(log n) (so that the state-of-the-art running time of n^O(log n) is optimal up to a constant in the exponent).
We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter k: Densest k-Subgraph, Smallest k-Edge Subgraph, Densest k-Subhypergraph, Steiner k-Forest, and Directed Steiner Network with k terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves o(k)-approximation for Densest k-Subgraph. This inapproximability ratio improves upon the previous best k^o(1) factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter k.
Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, improving the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis
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 be the optimum and be
the size of the input, is there an algorithm that runs in
time and outputs a solution of size
, for any functions and that are independent of (for
Clique, we want )?
In this paper, we show that both Clique and DomSet admit no non-trivial
FPT-approximation algorithm, i.e., there is no
-FPT-approximation algorithm for Clique and no
-FPT-approximation algorithm for DomSet, for any function
(e.g., this holds even if 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 -time algorithm can distinguish between a satisfiable
3SAT formula and one which is not even -satisfiable for some
constant .
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
-FPT-approximation algorithm for Densest -Subgraph although this
ratio does not yet match the trivial -approximation algorithm.Comment: 43 pages. To appear in FOCS'1
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
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
Parameterized Inapproximability for Steiner Orientation by Gap Amplification
In the k-Steiner Orientation problem, we are given a mixed graph, that is, with both directed and undirected edges, and a set of k terminal pairs. The goal is to find an orientation of the undirected edges that maximizes the number of terminal pairs for which there is a path from the source to the sink. The problem is known to be W[1]-hard when parameterized by k and hard to approximate up to some constant for FPT algorithms assuming Gap-ETH. On the other hand, no approximation factor better than ?(k) is known.
We show that k-Steiner Orientation is unlikely to admit an approximation algorithm with any constant factor, even within FPT running time. To obtain this result, we construct a self-reduction via a hashing-based gap amplification technique, which turns out useful even outside of the FPT paradigm. Precisely, we rule out any approximation factor of the form (log k)^o(1) for FPT algorithms (assuming FPT ? W[1]) and (log n)^o(1) for purely polynomial-time algorithms (assuming that the class W[1] does not admit randomized FPT algorithms). This constitutes a novel inapproximability result for polynomial-time algorithms obtained via tools from the FPT theory. Moreover, we prove k-Steiner Orientation to belong to W[1], which entails W[1]-completeness of (log k)^o(1)-approximation for k-Steiner Orientation. This provides an example of a natural approximation task that is complete in a parameterized complexity class.
Finally, we apply our technique to the maximization version of directed multicut - Max (k,p)-Directed Multicut - where we are given a directed graph, k terminals pairs, and a budget p. The goal is to maximize the number of separated terminal pairs by removing p edges. We present a simple proof that the problem admits no FPT approximation with factor ?(k^(1/2 - ?)) (assuming FPT ? W[1]) and no polynomial-time approximation with ratio ?(|E(G)|^(1/2 - ?)) (assuming NP ? co-RP)