Closer Than Enemies

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Acyclic Partition has been introduced and proven to be NP -complete by Eppstein and Mumford [ 7 ] and, independently, by Guillemot et al. In both these papers it was used as a pivot problem for proving NP -completeness of other problems.

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For the sake of completeness we would like to revisit the NP -hardness proof, because we need explicit bounds on the size of the directed graph obtained in the reduction for further applications. We begin with an instance of the Set Splitting problem, which is known to be NP -complete [ 8 ]. This concludes the construction; it is easy to verify the claimed sizes of V and A.

Let us formally prove that the instances are equivalent. Therefore, both D [ V 1 ] and D [ V 2 ] are collections of isolated vertices and directed paths and V 1 , V 2 is a solution to the Acyclic Partition instance. In the other direction, let V 1 , V 2 be a solution to the instance of Acyclic Partition.

We construct the graph H as follows: Let us prove equivalence of the instances. Symmetrically, D [ V 2 ] has to be acyclic as well, which concludes the proof of V 1 , V 2 being a feasible solution. Note that the positive neighbours of the special vertex s form an interval, therefore the condition imposed on this vertex is satisfied.

Now consider a checker vertex c v , w. We need to prove that the condition imposed on a v is satisfied as well. The neighbours of v consist of:. Thus the only non-trivial check is whether the vertices from 2 lie before the vertices from 3.

Closer Than Enemies

We have verified that for all the vertices the conditions imposed on them are satisfied, so the instances are equivalent. As mentioned before, the question of finding the smallest dimension of the Euclidean space, into which the given graph can be embedded, clearly generalizes testing embeddability into a line. Therefore, we have the followingcorollary. It is NP-hard to decide the smallest dimension of the Euclidean space, into which a given signed graph can be embedded.

In this subsection we observe that the presented chain of reductions enables us also to establish a lower bound on the complexity of solving Line Cluster Embedding under ETH. Firstly, let us complete the chain of the reductions. Now, every set F x is split, as exactly one of the literals is true and one is false, whereas every set F C is split as well, as it contains a true literal, which belongs to X , and the special element s , which does not.

Every set F x is split by X ; therefore, exactly one literal of every variable belongs to X and exactly one does not. Note that by pipelining Lemmata 4.

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As the Exponential Time Hypothesis also excludes a possibility of having an algorithm for 3 -CNF-SAT with running time subexponential in the total number of variables and clauses of the formula [ 12 ], we obtain the following. To complete the picture of the complexity of Line Cluster Embedding , we show that a simple dynamic programming approach can give single-exponential time complexity.

This matches the lower bound obtained from under Exponential Time Hypothesis up to a constant in the base of the exponent. Before we proceed with the description of the algorithm, let us state a combinatorial observation that will be its main ingredient. One direction is obvious: Therefore, the condition imposed on v is satisfied for an arbitrary choice of v. Moreover, the algorithm can also output a feasible ordering of the vertices, if it exists. Observe that by Lemma 4.

Reachability of any sink from any source can be, however, tested in time linear in the size of the graph using a breadth-first search.

The search can also reconstruct the path in the same complexity, thus constructing the feasible solution. In this paper we addressed a number of problems raised by Kermarrec and Thraves in [ 14 ] for embeddability of a signed graph into a line. We refined their study of the case of a complete signed graph by showing relation with proper interval graphs.

Moreover, we have proven NP -hardness of the general case and shown an almost complete picture of its complexity. Although the general case of the problem appears to be hard, real-life social networks have a certain structure. Is it possible to develop faster, maybe even polynomial-time algorithms for classes of graphs reflecting this structure?

Publisher's Summary

We thank Sylvain Guillemot for pointing us to [ 7 ] and [ 9 ]. A preliminary version of this paper has been presented at MFCS [ 5 ]. Jakub Onufry Wojtaszczyk, Email: National Center for Biotechnology Information , U. Theory of Computing Systems.

Published online Jul 1. Jakub Onufry Wojtaszczyk Google Inc.

Author information Article notes Copyright and License information Disclaimer. Received Sep 26; Accepted May Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author s and the source are credited. Abstract Signed graphs, i. Signed graphs, Relationships, Embedding into metric space. Our Results We focus on the problem of embedding a signed graph into a line. Exponential Time Hypothesis [ 11 ]: Combinatorial Problem Statement In [ 14 ], Kermarrec and Thraves work with the metric definition of the problem: However, for the 1-dimensional case they have in essence proved the following result: Open in a separate window.

The Complete Signed Graph Case In their work, Kermarrec and Thraves [ 14 ] announced a polynomial-time algorithm solving the Line Cluster Embedding problem in the case where the input signed graph is complete. Proof We construct the graph H as follows: Lower Bound on the Complexity In this subsection we observe that the presented chain of reductions enables us also to establish a lower bound on the complexity of solving Line Cluster Embedding under ETH. Proof One direction is obvious: Conclusions In this paper we addressed a number of problems raised by Kermarrec and Thraves in [ 14 ] for embeddability of a signed graph into a line.

Contributor Information Marek Cygan, Email: Dynamics of social balance on networks. Graph classes with structured neighborhoods and algorithmic applications. Cartwright D, Harary F. Simple linear time recognition of unit interval graphs. Sitting closer to friends than enemies, revisited.

The Hunted The Enemy is Closer Than You Think by Charlie Higson - Paperback

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She enlists the help of her best friend Kimberly, who's also an attorney to assist her. Kimberly, who has been harbouring a secret for years soon Betrays her friend in a most unusual way. Their Friendship gives a whole new meaning to the phrase Log in to rate this item. You must be logged in to post a review. The book touched on a lot of topics - love, pain, friendship, careers, and intimate relationships. Karen did a good job of balancing the various character's stories, and then tying them all together. The two main characters, Peyton and Kimberly, who are best friends, are different like night and day but have a close bond!

Kimberly had some deep-rooted problems, though, and at times, I found her very unlikable because of her trifling behavior.

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