17,661 research outputs found

    Robustness: a New Form of Heredity Motivated by Dynamic Networks

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    We investigate a special case of hereditary property in graphs, referred to as {\em robustness}. A property (or structure) is called robust in a graph GG if it is inherited by all the connected spanning subgraphs of GG. We motivate this definition using two different settings of dynamic networks. The first corresponds to networks of low dynamicity, where some links may be permanently removed so long as the network remains connected. The second corresponds to highly-dynamic networks, where communication links appear and disappear arbitrarily often, subject only to the requirement that the entities are temporally connected in a recurrent fashion ({\it i.e.} they can always reach each other through temporal paths). Each context induces a different interpretation of the notion of robustness. We start by motivating the definition and discussing the two interpretations, after what we consider the notion independently from its interpretation, taking as our focus the robustness of {\em maximal independent sets} (MIS). A graph may or may not admit a robust MIS. We characterize the set of graphs \forallMIS in which {\em all} MISs are robust. Then, we turn our attention to the graphs that {\em admit} a robust MIS (\existsMIS). This class has a more complex structure; we give a partial characterization in terms of elementary graph properties, then a complete characterization by means of a (polynomial time) decision algorithm that accepts if and only if a robust MIS exists. This algorithm can be adapted to construct such a solution if one exists

    Families with infants: a general approach to solve hard partition problems

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    We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with O((2ε(d))n)O^*((2-\varepsilon(d))^n) time and exponential space for graphs of average degree dd. This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in O((2ε(d))n)O^*((2-\varepsilon(d))^n) time and polynomial space for graphs of average degree dd. The previously known results of this kind is a polyspace algorithm by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~dd we present an algorithm with running time O((2ε(d))n/2)O^*((2-\varepsilon(d))^{n/2}) and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at http://arxiv.org/abs/1410.220

    Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

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    We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit C(x1,,xn)C(x_1,\ldots,x_n) of size ss and degree dd over a field F{\mathbb F}, and any inputs a1,,aKFna_1,\ldots,a_K \in {\mathbb F}^n, \bullet the Prover sends the Verifier the values C(a1),,C(aK)FC(a_1), \ldots, C(a_K) \in {\mathbb F} and a proof of O~(Kd)\tilde{O}(K \cdot d) length, and \bullet the Verifier tosses poly(log(dKF/ε))\textrm{poly}(\log(dK|{\mathbb F}|/\varepsilon)) coins and can check the proof in about O~(K(n+d)+s)\tilde{O}(K \cdot(n + d) + s) time, with probability of error less than ε\varepsilon. For small degree dd, this "Merlin-Arthur" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in cnc^{n} time (for various c<2c < 2) for the Permanent, #\#Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of 00-11 linear programs. In general, the value of any polynomial in Valiant's class VP{\sf VP} can be certified faster than "exhaustive summation" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. We also give a three-round (AMA) proof system for quantified Boolean formulas running in 22n/3+o(n)2^{2n/3+o(n)} time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in nk/2+O(1)n^{k/2+O(1)}-time for counting kk-cliques in graphs. We point to some potential future directions for refuting the Nondeterministic Strong ETH.Comment: 17 page

    On the Feasibility of Maintenance Algorithms in Dynamic Graphs

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    Near ubiquitous mobile computing has led to intense interest in dynamic graph theory. This provides a new and challenging setting for algorithmics and complexity theory. For any graph-based problem, the rapid evolution of a (possibly disconnected) graph over time naturally leads to the important complexity question: is it better to calculate a new solution from scratch or to adapt the known solution on the prior graph to quickly provide a solution of guaranteed quality for the changed graph? In this paper, we demonstrate that the former is the best approach in some cases, but that there are cases where the latter is feasible. We prove that, under certain conditions, hard problems cannot even be approximated in any reasonable complexity bound --- i.e., even with a large amount of time, having a solution to a very similar graph does not help in computing a solution to the current graph. To achieve this, we formalize the idea as a maintenance algorithm. Using r-Regular Subgraph as the primary example we show that W[1]-hardness for the parameterized approximation problem implies the non-existence of a maintenance algorithm for the given approximation ratio. Conversely we show that Vertex Cover, which is fixed-parameter tractable, has a 2-approximate maintenance algorithm. The implications of NP-hardness and NPO-hardness are also explored
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