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Register a new userFully Dynamic Maximal Independent Set with Polylogarithmic Update Time
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Soheil Behnezhad
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We present the first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph---which undergoes both edge insertions and deletions---in polylogarithmic time. Our algorithm is randomized and, per update, takes $O(log^2 Delta cdot log^2 n)$ expected time. Furthermore, the algorithm can be adjusted to have $O(log^2 Delta cdot log^4 n)$ worst-case update-time with high probability. Here, $n$ denotes the number of vertices and $Delta$ is the maximum degree in the graph. The MIS problem in fully dynamic graphs has attracted significant attention after a breakthrough result of Assadi, Onak, Schieber, and Solomon [STOC18] who presented an algorithm with $O(m^{3/4})$ update-time (and thus broke the natural $Omega(m)$ barrier) where $m$ denotes the number of edges in the graph. This result was improved in a series of subsequent papers, though, the update-time remained polynomial. In particular, the fastest algorithm prior to our work had $widetilde{O}(min{sqrt{n}, m^{1/3}})$ update-time [Assadi et al. SODA19]. Our algorithm maintains the lexicographically first MIS over a random order of the vertices. As a result, the same algorithm also maintains a 3-approximation of correlation clustering. We also show that a simpler variant of our algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time.
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Maintaining maximal independent set in dynamic graph is a fundamental open problem in graph theory and the first sublinear time deterministic algorithm was came up by Assadi, Onak, Schieber and Solomon(STOC18), which achieves $O(m^{3/4})$ amortized update time. We have two main contributions in this paper. We present a new simple deterministic algorithm with $O(m^{2/3}sqrt{log m})$ amortized update time, which improves the previous best result. And we also present the first randomized algorithm with expected $O(sqrt{m}log^{1.5}m)$ amortized time against an oblivious adversary.
The problem of (vertex) $(Delta+1)$-coloring a graph of maximum degree $Delta$ has been extremely well-studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized data structure for maintaining a $(Delta+1)$-coloring with $O(log Delta)$ expected amortized update time. In this paper, we present a $(Delta+1)$-coloring data structure that achieves a constant amortized update time and show that this time bound holds not only in expectation but also with high probability.
Maximal independent set (MIS), maximal matching (MM), and $(Delta+1)$-coloring in graphs of maximum degree $Delta$ are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for $(Delta+1)$-coloring that runs in $widetilde{O}(nsqrt{n})$ time, which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably admits a sublinear-time algorithm in general graphs. In this work, we dig deeper into the possibility of achieving sublinear-time algorithms for MIS and MM. The neighborhood independence number of a graph $G$, denoted by $beta(G)$, is the size of the largest independent set in the neighborhood of any vertex. We identify $beta(G)$ as the ``right parameter to measure the runtime of MIS and MM algorithms: Although graphs of bounded neighborhood independence may be very dense (clique is one example), we prove that carefully chosen variants of greedy algorithms for MIS and MM run in $O(nbeta(G))$ and $O(nlog{n}cdotbeta(G))$ time respectively on any $n$-vertex graph $G$. We complement this positive result by observing that a simple extension of the lower bound of Assadi et.al. implies that $Omega(nbeta(G))$ time is also necessary for any algorithm to either problem for all values of $beta(G)$ from $1$ to $Theta(n)$. We note that our algorithm for MIS is deterministic while for MM we use randomization which we prove is unavoidable: any deterministic algorithm for MM requires $Omega(n^2)$ time even for $beta(G) = 2$.
In the dynamic minimum set cover problem, a challenge is to minimize the update time while guaranteeing close to the optimal $min(O(log n), f)$ approximation factor. (Throughout, $m$, $n$, $f$, and $C$ are parameters denoting the maximum number of sets, number of elements, frequency, and the cost range.) In the high-frequency range, when $f=Omega(log n)$, this was achieved by a deterministic $O(log n)$-approximation algorithm with $O(f log n)$ amortized update time [Gupta et al. STOC17]. In the low-frequency range, the line of work by Gupta et al. [STOC17], Abboud et al. [STOC19], and Bhattacharya et al. [ICALP15, IPCO17, FOCS19] led to a deterministic $(1+epsilon)f$-approximation algorithm with $O(f log (Cn)/epsilon^2)$ amortized update time. In this paper we improve the latter update time and provide the first bounds that subsume (and sometimes improve) the state-of-the-art dynamic vertex cover algorithms. We obtain: 1. $(1+epsilon)f$-approximation ratio in $O(flog^2 (Cn)/epsilon^3)$ worst-case update time: No non-trivial worst-case update time was previously known for dynamic set cover. Our bound subsumes and improves by a logarithmic factor the $O(log^3 n/text{poly}(epsilon))$ worst-case update time for unweighted dynamic vertex cover (i.e., when $f=2$ and $C=1$) by Bhattacharya et al. [SODA17]. 2. $(1+epsilon)f$-approximation ratio in $Oleft((f^2/epsilon^3)+(f/epsilon^2) log Cright)$ amortized update time: This result improves the previous $O(f log (Cn)/epsilon^2)$ update time bound for most values of $f$ in the low-frequency range, i.e. whenever $f=o(log n)$. It is the first that is independent of $m$ and $n$. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [SODA19] for unweighted dynamic vertex cover (i.e., when $f = 2$ and $C = 1$).
We present fully dynamic approximation algorithms for the Maximum Independent Set problem on several types of geometric objects: intervals on the real line, arbitrary axis-aligned squares in the plane and axis-aligned $d$-dimensional hypercubes. It is known that a maximum independent set of a collection of $n$ intervals can be found in $O(nlog n)$ time, while it is already textsf{NP}-hard for a set of unit squares. Moreover, the problem is inapproximable on many important graph families, but admits a textsf{PTAS} for a set of arbitrary pseudo-disks. Therefore, a fundamental question in computational geometry is whether it is possible to maintain an approximate maximum independent set in a set of dynamic geometric objects, in truly sublinear time per insertion or deletion. In this work, we answer this question in the affirmative for intervals, squares and hypercubes. First, we show that for intervals a $(1+varepsilon)$-approximate maximum independent set can be maintained with logarithmic worst-case update time. This is achieved by maintaining a locally optimal solution using a constant number of constant-size exchanges per update. We then show how our interval structure can be used to design a data structure for maintaining an expected constant factor approximate maximum independent set of axis-aligned squares in the plane, with polylogarithmic amortized update time. Our approach generalizes to $d$-dimensional hypercubes, providing a $O(4^d)$-approximation with polylogarithmic update time. Those are the first approximation algorithms for any set of dynamic arbitrary size geometric objects; previous results required bounded size ratios to obtain polylogarithmic update time. Furthermore, it is known that our results for squares (and hypercubes) cannot be improved to a $(1+varepsilon)$-approximation with the same update time.
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