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Register a new userSolving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs
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Added by
Maximilian Katzmann
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time.
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Finding a minimum vertex cover in a network is a fundamental NP-complete graph problem. One way to deal with its computational hardness, is to trade the qualitative performance of an algorithm (allowing non-optimal outputs) for an improved running time. For the vertex cover problem, there is a gap between theory and practice when it comes to understanding this tradeoff. On the one hand, it is known that it is NP-hard to approximate a minimum vertex cover within a factor of $sqrt{2}$. On the other hand, a simple greedy algorithm yields close to optimal approximations in practice. A promising approach towards understanding this discrepancy is to recognize the differences between theoretical worst-case instances and real-world networks. Following this direction, we close the gap between theory and practice by providing an algorithm that efficiently computes nearly optimal vertex cover approximations on hyperbolic random graphs; a network model that closely resembles real-world networks in terms of degree distribution, clustering, and the small-world property. More precisely, our algorithm computes a $(1 + o(1))$-approximation, asymptotically almost surely, and has a running time of $mathcal{O}(m log(n))$. The proposed algorithm is an adaption of the successful greedy approach, enhanced with a procedure that improves on parts of the graph where greedy is not optimal. This makes it possible to introduce a parameter that can be used to tune the tradeoff between approximation performance and running time. Our empirical evaluation on real-world networks shows that this allows for improving over the near-optimal results of the greedy approach.
A bipartite graph $G=(A,B,E)$ is ${cal H}$-convex, for some family of graphs ${cal H}$, if there exists a graph $Hin {cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $bin B$ induces a connected subgraph of $H$. Many $mathsf{NP}$-complete problems, including problems such as Dominating Set, Feedback Vertex Set, Induced Matching and List $k$-Colouring, become polynomial-time solvable for ${mathcal H}$-convex graphs when ${mathcal H}$ is the set of paths. In this case, the class of ${mathcal H}$-convex graphs is known as the class of convex graphs. The underlying reason is that the class of convex graphs has bounded mim-width. We extend the latter result to families of ${mathcal H}$-convex graphs where (i) ${mathcal H}$ is the set of cycles, or (ii) ${mathcal H}$ is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least $3$. As a consequence, we can re-prove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of ${mathcal H}$-convex graphs is unbounded if ${mathcal H}$ is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least $3$. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of ${cal H}$-convex graphs.
Recent work has pinned down the existentially optimal size bounds for vertex fault-tolerant spanners: for any positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges resilient to $f$ vertex faults, and there are examples of input graphs on which this bound cannot be improved. However, these proofs work by analyzing the output spanner of a certain exponential-time greedy algorithm. In this work, we give the first algorithm that produces vertex fault tolerant spanners of optimal size and which runs in polynomial time. Specifically, we give a randomized algorithm which takes $widetilde{O}left( f^{1-1/k} n^{2+1/k} + mf^2right)$ time. We also derandomize our algorithm to give a deterministic algorithm with similar bounds. This reflects an exponential improvement in runtime over [Bodwin-Patel PODC 19], the only previously known algorithm for constructing optimal vertex fault-tolerant spanners.
We present a near-tight analysis of the average query complexity -- `a la Nguyen and Onak [FOCS08] -- of the randomized greedy maximal matching algorithm, improving over the bound of Yoshida, Yamamoto and Ito [STOC09]. For any $n$-vertex graph of average degree $bar{d}$, this leads to the following sublinear-time algorithms for estimating the size of maximum matching and minimum vertex cover, all of which are provably time-optimal up to logarithmic factors: $bullet$ A multiplicative $(2+epsilon)$-approximation in $widetilde{O}(n/epsilon^2)$ time using adjacency list queries. This (nearly) matches an $Omega(n)$ time lower bound for any multiplicative approximation and is, notably, the first $O(1)$-approximation that runs in $o(n^{1.5})$ time. $bullet$ A $(2, epsilon n)$-approximation in $widetilde{O}((bar{d} + 1)/epsilon^2)$ time using adjacency list queries. This (nearly) matches an $Omega(bar{d}+1)$ lower bound of Parnas and Ron [TCS07] which holds for any $(O(1), epsilon n)$-approximation, and improves over the bounds of [Yoshida et al. STOC09; Onak et al. SODA12] and [Kapralov et al. SODA20]: The former two take at least quadratic time in the degree which can be as large as $Omega(n^2)$ and the latter obtains a much larger approximation. $bullet$ A $(2, epsilon n)$-approximation in $widetilde{O}(n/epsilon^3)$ time using adjacency matrix queries. This (nearly) matches an $Omega(n)$ time lower bound in this model and improves over the $widetilde{O}(nsqrt{n})$-time $(2, epsilon n)$-approximate algorithm of [Chen, Kannan, and Khanna ICALP20]. It also turns out that any non-trivial multiplicative approximation in the adjacency matrix model requires $Omega(n^2)$ time, so the additive $epsilon n$ error is necessary too. As immediate corollaries, we get improved sublinear time estimators for (variants of) TSP and an improved AMPC algorithm for maximal matching.
We give an $n^{O(loglog n)}$-time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over ${pm 1}^n$. Even in the realizable setting, the previous fastest runtime was $n^{O(log n)}$, a consequence of a classic algorithm of Ehrenfeucht and Haussler. Our algorithm shares similarities with practical heuristics for learning decision trees, which we augment with additional ideas to circumvent known lower bounds against these heuristics. To analyze our algorithm, we prove a new structural result for decision trees that strengthens a theorem of ODonnell, Saks, Schramm, and Servedio. While the OSSS theorem says that every decision tree has an influential variable, we show how every decision tree can be pruned so that every variable in the resulting tree is influential.
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