Inapproximability of Additive Weak Contraction under SSEH and Strong UGC

11/30/2019 ∙ by Siddhartha Jain, et al. ∙ IIIT Delhi 0

Succinct representations of a graph have been objects of central study in computer science for decades. In this paper, we study the operation called Distance Preserving Graph Contractions, which was introduced by Bernstein et al. (ITCS, 2018). This operation gives a minor as a succinct representation of a graph that preserves all the distances of the original (up to some factor). The graph minor given from contractions can be seen as a dual of spanners as the distances can only shrink (while distances are stretched in the case of spanners). Bernstein et al. proved inapproximability results for the problems of finding maximum subset of edges that yields distance preserving graph contractions for almost major classes of graphs except for that of Additive Weak Contraction. The main result in this paper is filling the gap in the paper of Bernstein et al. We show that the Maximum Additive Weak Contraction problem on a graph with n vertices is inapproximable up to a factor of n^1-ϵ for every constant ϵ>0. Our hardness results follow from that of the Maximum Edge Biclique (MEB) problem whose inapproximability of n^1-ϵ has been recently shown by Manurangsi (ICALP, 2017) under the Small Set Expansion Hypothesis (SSEH) and by Bhangale et al. (APPROX, 2016) under the Strong Unique Games Conjecture (SUGC) (both results also assume NP⊈BPP).

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1 Introduction

Coping with very large networks has been a major issue in computer science in both theory and practice. Operations acting on extremely large networks are rather time-consuming and not quite satisfactory for most applications. The techniques of graph compression have been posited as a solution, as they reduce the size of the network and thus reduce the processing time. While compressing it is impossible to retain all the information about the original network. So we aim to retain specific properties of the input. This motivates the study of an object called Spanner that keeps only a sparse subset of the edges of a network, while preserving the distances between every pair of vertices up to some multiplicative (resp., additive) factor. To be formal, in graph theory terminology, a -spanner of a graph is a sparse subgraph of such that every pair of vertices in has distance at most a multiplicative (resp., additive) factor away from the original distance in .

Bernstein et al. [BDD18] recently proposed a compression operation called Distance Preserving Graph Contraction. Here we obtain a subgraph by contracting subsets of edges, while promising a lower bound on the distances, which results in a minor of the input graph. Observe that the contraction operation can only decrease the distance of each pair of vertices, whereas deleting edges as in the case of spanners can only increase the distances. The contracted graph can be thought of as a dual of spanners.

In [BDD18], the authors introduced the problem of finding the maximum number of edges whose contraction produces a minor that guarantees the distance between any two vertices to be shorter by a factor of at most , namely, the -Contraction problem. The authors also defined the relaxed variant, namely the Weak -Contraction problem. Note that, the Contraction problem is studied only in the case of additive contraction; this is because, in the multiplicative case, no edges can be contracted. In each variant, Bernstein et al. studied several important graph classes and either present polynomial-time algorithm or show NP-hardness results for the problems; see Table 1. We list their results with ours.

Inapproximability results
Tolerance Type Hardness
Additive () Contraction -inapx on bipartite graphs, [BDD18]
Additive () Contraction -inapx [BDD18]
Additive () Weak Contraction -inapx on bipartite graphs, [Thm 16]
Multiplicative () Weak Contraction -inapx [BDD18]
Table 1: Known Results on the (Weak) Contraction problems.

2 Preliminaries

2.1 Notations

We use or simply to refer to a simple, undirected graph with a non-negative distance function . Throughout, we assume that has veritces and edges. We assume further that the graph is connected since, otherwise, we can solve the subproblem on each connected component separately. Given a set of contracted edges (we will mostly use to denote contracted edges), the distance function induced by after the contraction is denoted by .

2.2 Distance-Preserving Contractions

Definition 1.

Given a graph and a distance function , an -contraction of is a set of edges such that for all vertices . We abbreviate -contraction by Cont.

The concept of Weak Contraction is defined to allow enough flexibility to make Multiplicative (Weak) Contraction non-trivial.

Definition 2.

Given a graph and a distance function , an -weak-contraction of is a set of edges such that the following two properties hold:

  • whenever for all

We abbreviate -weak-contraction by WeakCont.

We can now formulate the corresponding problems for these structures.

Problem 3.

Given a graph and a distance function , find an optimal Cont, of . is said to be optimal if there does not exist a Cont, of such that .

Problem 4.

Given a graph and a distance function , find an optimal WeakCont, of . is said to be optimal if there does not exist a WeakCont, of such that .

2.3 Complexity Assumptions

We now discuss the Complexity Theoretic assumptions that are required for the result by each of the previous works. For completeness, we reproduce the definitions here.

The Strong Unique Games Conjecture (SUGC).

This conjecture was first introduced by Bansal and Khot [BK09], but coined by Bhangale et al. [BGH16] with a slight modification.

Definition 5.

Given a bi-regular bipartite graph and a labelling , we say an edge is satisfied if . Moreover, let be the fraction of edges satisfied by the labelling .

Conjecture 6.

[BGH16] For all there exists a such that given , it is NP-Hard to distinguish between the following two cases:

  • There exists an such that

  • For all , . Moreover, for all such that , we have that where

The Small Set Expansion Hypothesis (SSEH).

This conjecture was introduced by Raghavendra and Steurer [RS10] to overcome the shortcomings of UGC while trying to prove hardness results, and it has received immense attention since. It is in fact equivalent to a stronger version of UGC but distinct from Strong UGC.

Definition 7.

Given a -regular graph , for every let us define

Moreover, for every we define

Conjecture 8.

[RS10] For every there exists a such that given a graph it is NP-Hard to distinguish between the following two cases:

We reduce the Maximum Edge Biclique (MEB) problem to finding the largest WeakCont as defined in the paper by Bernstein et al. [BDD18], called Weak Contraction with tolerance function .

MEB was proven to be hard to approximate under SSEH by Manurangsi [Man17]. It was also proven to be hard to approximate under the Strong UGC by Bhangale et al [BGH16].

2.4 Biclique

Problem 9.

Maximum Edge Biclique (MEB): given a bipartite graph G, find a complete bipartite subgraph of G with maximum number of edges.

Problem 10.

Maximum Balanced Biclique (MBB): given a bipartite graph G, find a balanced complete bipartite subgraph of G with maximum number of vertices.

Theorem 11.

[Man17] Assuming SSEH, there is no polynomial time algorithm that approximates MEB or MBB to within factor of the optimum for every , unless .

Concretely, they prove this by showing the following lemma.

Lemma 12.

[Man17] Assume SSEH. Then given a bipartite graph with , for every it is NP-hard to distinguish between the following two cases:

  • (Completeness) G contains as a subgraph.

  • (Soundness) G does not contain as a subgraph. Here denotes the complete bipartite graph in which each side contains t vertices.

This lemma works for MBB but also implies a similar result for MEB. Since the paper does not explicitly state this reduction, we state it here for exposition.

Lemma 13.

Assume SSEH. Then given a bipartite graph with , for every it is NP-hard to distinguish between the following two cases:

  • (Completeness) G contains a biclique having edges..

  • (Soundness) G does not any biclique containing edges.

Proof.

Given input graph for MEB, we construct

(tensor product). Let us divide the vertex set of

into 4 parts. . The only edges in this graph are between and and hence this graph is also bipartite, of size . Let .

  • (Completeness) A in corresponds to a in . Therefore, any biclique in with edges corresponds to a in . Let .

  • (Soundness) Given a in , we know that for all and , by definition . Hence we have a biclique with edges in . If there is no in for , there can be no biclique of size in G.

Theorem 11 follows from gap amplification via randomized graph product which is discussed in Appendix B of the full version of [Man17]. This is also where we assume NP BPP. Without this assumption (assuming only PNP), both [Man17] and [BGH16] show that the problem cannot be approximated within any constant factor, under SSEH and Strong UGC respectively.

3 Reduction

First let us see a simple property of WeakCont on graphs with unit edge lengths.

Lemma 14.

For any path in , if two disjoint edges then .

Proof.

By way of contradiction let . One of reduces by 2 after contraction, depending on the relative position of . ∎

Our results are based on a simple reduction gadget, which we describe first.

Given an input bipartite graph we create an instance as follows. We create two copies each of and , let us call them . The subset of vertices induces . We also add edges of the form and for all to . An illustration in Fig 1.

Figure 1: A representative drawing of

The following lemma will help us prove that finding the largest WeakCont is hard.

Lemma 15.

Assigning weights 1 to all edges in , any Weak Contraction of size strictly greater than 1 must contract a biclique in .

Proof.

There are two kinds of edges we can contract. One are the “matching” edges of the form or . The other edges correspond to those in .

First, we will see that if a “matching” edge belongs to , it must be that , or in other words that it is the only edge contracted. Without loss of generality let contracted edge be .

  • First let us show one cannot contract any disjoint edge in the graph induced by . By way of contradiction is contracted. Any other edge in the graph can be expressed as a part of a path including these edges. Hence by Lemma 14, the entire graph must be contracted. But that means is not a WeakCont.

  • Now say some edge was contracted. Then must also be contracted, else decreases by 2. Let us take some other edge induced by . There is a path from to . Similarly from to . Now the path satisfies the condition for Lemma 14. Hence, . But we have seen that once such an edge is contracted it leads to a contradiction.

  • Now the only edges left to contract are the matching edges themselves. But if we contract any other matching edge, we can take a path including this edge and . Now we are forced to contract an edge in which we have seen leads to a contradiction previously.

So now we can focus on the graph induced by . Let be vertices such that their neighbours are endpoints of some edge in . Let these edges be and . Let us assume they are not involved in a biclique, which means does not exist. Since the graph is connected, there exists a path from to . Let the path from to be and the path from to be . Let us consider . Since we have contracted more than one edge in this path, we must contract the entire path. But since we know contracting matching edges leads to a contradiction, this is not possible. Therefore we must contract a biclique. ∎

Theorem 16.

Assuming SSEH OR assuming Strong UGC, no polynomial algorithm can approximate WeakCont within a factor of for every , even for bipartite graphs with unit edge lengths, unless NP BPP.

Proof.

For every

  • (Completeness) If we have a biclique with edges in , since induce in , we have a biclique of the same size in .

  • (Soundness) If we have no biclique of size in , there is no biclique of that size in . By Lemma 15, the largest Weak Contraction contracts less than edges.

Corollary 17.

Assuming SSEH OR assuming Strong UGC, no polynomial algorithm can approximate WeakCont within a factor of for every , even for bipartite graphs with unit edge lengths, unless NP BPP.

Proof.

In our construction if we replace the edge weights with , everything follows similarly. ∎

4 Acknowledgements

The author thanks Bundit Laekhanukit for his mentorship and useful discussion on the problem, especially for suggesting the inapproximability results for the Maximum Biclique problem. In addition, the author would like to thank him for funding through his 1000-talents award by the Chinese government. This work was partially done while the author was visiting the Institute for Theoretical Computer Science at the Shanghai University of Finance and Economics (ITCS@SUFE), and the author would like to thank ITCS@SUFE for the office space and all the support.

References

  • [BDD18] Aaron Bernstein, Karl Däubel, Yann Disser, Max Klimm, Torsten Mütze, and Frieder Smolny. Distance-Preserving Graph Contractions. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of Leibniz International Proceedings in Informatics (LIPIcs), pages 51:1–51:14, Dagstuhl, Germany, 2018. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.
  • [BGH16] Amey Bhangale, Rajiv Gandhi, Mohammad Taghi Hajiaghayi, Rohit Khandekar, and Guy Kortsarz. Bicovering: Covering Edges With Two Small Subsets of Vertices. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 55 of Leibniz International Proceedings in Informatics (LIPIcs), pages 6:1–6:12, Dagstuhl, Germany, 2016. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.
  • [BK09] Nikhil Bansal and Subhash Khot. Optimal long code test with one free bit. In Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’09, pages 453–462, Washington, DC, USA, 2009. IEEE Computer Society.
  • [Man17] Pasin Manurangsi. Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 79:1–79:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.
  • [RS10] Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In

    Proceedings of the Forty-second ACM Symposium on Theory of Computing

    , STOC ’10, pages 755–764, New York, NY, USA, 2010. ACM.