arXiv subjects

David G. Harris

Publications and source records attributed to David G. Harris.

At least 37 records · Page 2Linked to original sources

arXiv2023

Deterministic algorithms for the Lovasz Local Lemma: simpler, more general, and more parallel

The Lovász Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its simplest "symmetric" form, it asserts that whenever a bad-event has probability $p$ and affects at most $d$ bad-events, and $e p d < 1$, then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of Moser & Tardos (2010) gives nearly-automatic randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. We address three specific shortcomings of the prior deterministic algorithms. First, our algorithm applies to the LLL criterion of Shearer (1985); this is more powerful than alternate LLL criteria and also removes a number of nuisance parameters and leads to cleaner and more legible bounds. Second, we provide parallel algorithms with much greater flexibility in the functional form of of the bad-events. Third, we provide a derandomized version of the MT-distribution, that is, the distribution of the variables at the termination of the MT algorithm. We show applications to non-repetitive vertex coloring, independent transversals, strong coloring, and other problems. These give deterministic algorithms which essentially match the best previous randomized sequential and parallel algorithms.

cs.DS↗

arXiv2022

On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

Given a graph $G=(V,E)$ with arboricity $α$, we study the problem of decomposing the edges of $G$ into $(1+ε)α$ disjoint forests in the distributed LOCAL model. Barenboim and Elkin [PODC `08] gave a LOCAL algorithm that computes a $(2+ε)α$-forest decomposition using $O(\frac{\log n}ε)$ rounds. Ghaffari and Su [SODA `17] made further progress by computing a $(1+ε) α$-forest decomposition in $O(\frac{\log^3 n}{ε^4})$ rounds when $εα= Ω(\sqrt{α\log n})$, i.e. the limit of their algorithm is an $(α+ Ω(\sqrt{α\log n}))$-forest decomposition. This algorithm, based on a combinatorial construction of Alon, McDiarmid \& Reed [Combinatorica `92], in fact provides a decomposition of the graph into \emph{star-forests}, i.e. each forest is a collection of stars. Our main result in this paper is to reduce the threshold of $εα$ in $(1+ε)α$-forest decomposition and star-forest decomposition. This further answers the $10^{\text{th}}$ open question from Barenboim and Elkin's "Distributed Graph Algorithms" book. Moreover, it gives the first $(1+ε)α$-orientation algorithms with {\it linear dependencies} on $ε^{-1}$. At a high level, our results for forest-decomposition are based on a combination of network decomposition, load balancing, and a new structural result on local augmenting sequences. Our result for star-forest decomposition uses a more careful probabilistic analysis for the construction of Alon, McDiarmid, \& Reed; the bounds on star-arboricity here were not previously known, even non-constructively.

cs.DS↗

arXiv2022

Comparison of two convergence criteria for the variable-assignment Lopsided Lovasz Local Lemma

The Lopsided Lovasz Local Lemma (LLLL) is a cornerstone probabilistic tool for showing that it is possible to avoid a collection of "bad" events as long as their probabilities and interdependencies are sufficiently small. The strongest possible criterion in these terms is due to Shearer (1985), although it is technically difficult to apply to constructions in combinatorics. The original formulation of the LLLL was non-constructive; a seminal algorithm of Moser & Tardos (2010) gave an efficient algorithm for nearly all its applications, including to $k$-SAT instances where each variable appears in a bounded number of clauses. Harris (2015) later gave an alternate criterion for this algorithm to converge; unlike the LLL criterion or its variants, this criterion depends in a fundamental way on the decomposition of bad-events into variables. In this note, we show that the criterion given by Harris can be stronger in some cases even than Shearer's criterion. We construct $k$-SAT formulas with bounded variable occurrence, and show that the criterion of Harris is satisfied while the criterion of Shearer is violated. In fact, there is an exponentially growing gap between the bounds provable from any form of the LLLL and from the bound shown by Harris.

math.PR↗

arXiv2022

Some remarks on hypergraph matching and the Füredi-Kahn-Seymour conjecture

A classic conjecture of Füredi, Kahn and Seymour (1993) states that given any hypergraph with non-negative edge weights $w(e)$, there exists a matching $M$ such that $\sum_{e \in M} (|e|-1+1/|e|)\, w(e) \geq w^*$, where $w^*$ is the value of an optimum fractional matching. We show the conjecture is true for rank-3 hypergraphs, and is achieved by a natural iterated rounding algorithm. While the general conjecture remains open, we give several new improved bounds. In particular, we show that the iterated rounding algorithm gives $\sum_{e \in M} (|e|-δ(e))\, w(e) \geq w^*$, where $δ(e) = |e|/(|e|^2+|e|-1)$, improving upon the baseline guarantee of $\sum_{e \in M} |e|\,w(e) \geq w^*$.

math.CO↗

arXiv2021

Optimal Bounds for the $k$-cut Problem

In the $k$-cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into $k$ connected components. Algorithms of Karger \& Stein can solve this in roughly $O(n^{2k})$ time. On the other hand, lower bounds from conjectures about the $k$-clique problem imply that $Ω(n^{(1-o(1))k})$ time is likely needed. Recent results of Gupta, Lee \& Li have given new algorithms for general $k$-cut in $n^{1.98k + O(1)}$ time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed $k$-cut of weight $αλ_k$ with probability $Ω_k(n^{-αk})$, where $λ_k$ denotes the minimum $k$-cut weight. This also gives an extremal bound of $O_k(n^k)$ on the number of minimum $k$-cuts and an algorithm to compute $λ_k$ with roughly $n^k \mathrm{polylog}(n)$ runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight $k$-clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than $2 λ_k/k$, using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks -- and how the average degree evolves -- in the Karger process.

cs.DS↗

arXiv2021

Algorithms for weighted independent transversals and strong colouring

An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph with a given vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there remains a gap between the best bounds given by nonconstructive results, and those obtainable by efficient algorithms. Recently, Graf and Haxell (2018) described a new (deterministic) algorithm that asymptotically closes this gap, but there are limitations on its applicability. In this paper we develop a randomized version of this algorithm that is much more widely applicable, and demonstrate its use by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.

cs.DS↗

arXiv2020

Approximation algorithms for stochastic clustering

We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including clustering which is fairer and has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results.

cs.DS↗

arXiv2020

Oblivious resampling oracles and parallel algorithms for the Lopsided Lovasz Local Lemma

The Lovász Local Lemma (LLL) is a probabilistic tool which shows that, if a collection of "bad" events $\mathcal B$ in a probability space are not too likely and not too interdependent, then there is a positive probability that no bad-events in $\mathcal B$ occur. Moser & Tardos (2010) gave sequential and parallel algorithms which transformed most applications of the variable-assignment LLL into efficient algorithms. A framework of Harvey & Vondrák (2015) based on "resampling oracles" extended this to general sequential algorithms for other probability spaces satisfying the Lopsided Lovász Local Lemma (LLLL). We describe a new structural property which holds for all known resampling oracles, which we call "obliviousness." Essentially, it means that the interaction between two bad-events $B, B'$ depends only on the randomness used to resample $B$, and not the precise state within $B$ itself. This property has two major consequences. First, combined with a framework of Kolmogorov (2016), it is the key to achieving a unified parallel LLLL algorithm, which is faster than previous, problem-specific algorithms of Harris (2016) for the variable-assignment LLLL algorithm and of Harris \& Srinivasan (2014) for permutations. This gives the first RNC algorithms for rainbow perfect matchings and rainbow hamiltonian cycles of $K_n$. Second, this property allows us to build LLLL probability spaces out of relatively simple "atomic" events. This provides the first sequential resampling oracle for rainbow perfect matchings on the complete $s$-uniform hypergraph $K_n^{(s)}$, and the first commutative resampling oracle for hamiltonian cycles of $K_n$.

cs.DS↗

arXiv2020

Partial resampling to approximate covering integer programs

We consider column-sparse covering integer programs, a generalization of set cover, which have a long line of research of (randomized) approximation algorithms. We develop a new rounding scheme based on the Partial Resampling variant of the Lovász Local Lemma developed by Harris & Srinivasan (2019). This achieves an approximation ratio of $1 + \frac{\ln (Δ_1+1)}{a_{\min}} + O\Big( \log(1 + \sqrt{ \frac{\log (Δ_1+1)}{a_{\min}}} \Big)$, where $a_{\min}$ is the minimum covering constraint and $Δ_1$ is the maximum $\ell_1$-norm of any column of the covering matrix (whose entries are scaled to lie in $[0,1]$). When there are additional constraints on the variable sizes, we show an approximation ratio of $\ln Δ_0 + O(\log \log Δ_0)$ (where $Δ_0$ is the maximum number of non-zero entries in any column of the covering matrix). These results improve asymptotically, in several different ways, over results of Srinivasan (2006) and Kolliopoulos & Young (2005). We show nearly-matching inapproximability and integrality-gap lower bounds. We also show that the rounding process leads to negative correlation among the variables, which allows us to handle multi-criteria programs.

cs.DS↗

arXiv2020

Bounds and algorithms for graph trusses

The $k$-truss, introduced by Cohen (2005), is a graph where every edge is incident to at least $k$ triangles. This is a relaxation of the clique. It has proved to be a useful tool in identifying cohesive subnetworks in a variety of real-world graphs. Despite its simplicity and its utility, the combinatorial and algorithmic aspects of trusses have not been thoroughly explored. We provide nearly-tight bounds on the edge counts of $k$-trusses. We also give two improved algorithms for finding trusses in large-scale graphs. First, we present a simplified and faster algorithm, based on approach discussed in Wang & Cheng (2012). Second, we present a theoretical algorithm based on fast matrix multiplication; this converts a triangle-generation algorithm of Bjorklund et al. (2014) into a dynamic data structure.

math.CO↗

arXiv2020

Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank $r$. Our main result is a deterministic algorithm to generate a matching which is an $O(r)$-approximation to the maximum weight matching, running in $\tilde O(r \log Δ+ \log^2 Δ+ \log^* n)$ rounds. (Here, the $\tilde O()$ notations hides $\text{polyloglog } Δ$ and $\text{polylog } r$ factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). As a main application, we obtain nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a $(1+ε)$ approximation algorithm using $\tilde O(\log Δ/ ε^3 + \text{polylog}(1/ε, \log \log n))$ randomized time and $\tilde O(\log^2 Δ/ ε^4 + \log^*n / ε)$ deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for $(2 Δ- 1)$-edge-list coloring in $\tilde O(\log^2 Δ\log n)$ rounds deterministically or $\tilde O( (\log \log n)^3 )$ rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most $\lceil (1+ε) λ\rceil$ for a graph of arboricity $λ$; for fixed $ε$ this runs in $\tilde O(\log^6 n)$ rounds deterministically or $\tilde O(\log^3 n )$ rounds randomly.

cs.DS↗

arXiv2019

New bounds for the Moser-Tardos distribution

The Lovasz Local Lemma (LLL) is a probabilistic tool which has been used to show the existence of a variety of combinatorial structures with good "local" properties. The "LLL-distribution" can be used to show that the resulting structures have good global properties in expectation. The simplest, variable-based setting of the LLL was covered by the seminal algorithm of Moser & Tardos (2010). This has since been extended to other probability spaces including random permutations. One can similarly define an "MT-distribution" for these algorithms, that is, the distribution of the configuration they produce. Haeupler et al. (2011) showed bounds on the MT-distribution which essentially match the LLL-distribution for the variable-assignment setting; Harris & Srinivasan showed similar results for the permutation setting. In this work, we show new bounds on the MT-distribution which are significantly stronger than those known to hold for the LLL-distribution. In the variable-assignment setting, we show a tighter bound on the probability of a disjunctive event or singleton event. As a consequence, in $k$-SAT instances with bounded variable occurrence, the MT-distribution satisfies an $ε$-approximate independence condition asymptotically stronger than the LLL-distribution. We use this to show a nearly tight bound on the minimum implicate size of a CNF boolean formula. Another noteworthy application is constructing independent transversals which avoid a given subset of vertices; this provides a constructive analogue to a result of Rabern (2014). In the permutation LLL setting, we show a new type of bound which is similar to the cluster-expansion LLL criterion of Bissacot et al. (2011), but is stronger and takes advantage of the extra structure in permutations. We illustrate with improved bounds on weighted Latin transversals and partial Latin transversals.

math.CO↗

arXiv2019

Derandomizing the Lovasz Local Lemma via log-space statistical tests

The Lovász Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its simplest form, it asserts that whenever a bad-event has probability $p$ and affects at most $d$ other bad-events, and $e p (d+1) < 1$, then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of Moser & Tardos (2010) gives randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. Notably, prior deterministic LLL algorithms have required stringent conditions on $\mathcal B$; for example, they have required that events in $\mathcal B$ have low decision-tree complexity or depend on a small number of variables. For this reason, they can only be applied to small fraction of the numerous LLL applications in practice. We describe an alternate deterministic parallel (NC) algorithm for the LLL, based on a general derandomization method of Sivakumar (2002) using log-space statistical tests. The only requirement here is that bad-events should be computable via a finite automaton with $\text{poly}(d)$ states. This covers most LLL applications to graph theory and combinatorics. No auxiliary information about the bad-events, including any conditional probability calculations, are required. Additionally, the proof is a straightforward combination of general derandomization results and high-level analysis of the Moser-Tardos algorithm. We illustrate with applications to defective vertex coloring, domatic partition, and independent transversals.

cs.DS↗

arXiv2019

On Derandomizing Local Distributed Algorithms

The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we develop a generic and clean recipe for derandomizing LOCAL algorithms. We also exhibit how this simple recipe leads to significant improvements on a number of problem. Two main results are: - An improved distributed hypergraph maximal matching algorithm, improving on Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for edge-coloring, maximum matching approximation, and low out-degree edge orientation. The first gives an improved algorithm for Open Problem 11.4 of the book of Barenboim and Elkin, and the last gives the first positive resolution of their Open Problem 11.10. - An improved distributed algorithm for the Lovász Local Lemma, which gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads to improved distributed algorithms for problems such as defective coloring and $k$-SAT.

cs.DS↗

arXiv2019

The Moser-Tardos Framework with Partial Resampling

The resampling algorithm of Moser \& Tardos is a powerful approach to develop constructive versions of the Lovász Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random subset of the variables that define this event, rather than the entire set as in Moser & Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing etc. For instance, we settle a conjecture of Szabó & Tardos (2006) on graph transversals asymptotically, and obtain improved approximation ratios for a packet routing problem of Leighton, Maggs, & Rao (1994).

math.CO↗

arXiv2019

Derandomized concentration bounds for polynomials, and hypergraph maximal independent set

A parallel algorithm for maximal independent set (MIS) in hypergraphs has been a long-standing algorithmic challenge, dating back nearly 30 years to a survey of Karp & Ramachandran (1990). The best randomized parallel algorithm for hypergraphs of fixed rank $r$ was developed by Beame & Luby (1990) and Kelsen (1992), running in time roughly $(\log n)^{r!}$. We improve the randomized algorithm of Kelsen, reducing the runtime to roughly $(\log n)^{2^r}$ and simplifying the analysis through the use of more-modern concentration inequalities. We also give a method for derandomizing concentration bounds for low-degree polynomials, which are the key technical tool used to analyze that algorithm. This leads to a deterministic PRAM algorithm also running in $(\log n)^{2^{r+3}}$ time and $\text{poly}(m,n)$ processors. This is the first deterministic algorithm with sub-polynomial runtime for hypergraphs of rank $r > 3$. Our analysis can also apply when $r$ is slowly growing; using this in conjunction with a strategy of Bercea et al. (2015) gives a deterministic MIS algorithm running in time $\exp(O( \frac{\log (mn)}{\log \log (mn)}))$.

cs.DS↗

arXiv2019

Some results on chromatic number as a function of triangle count

A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak \& Tuza (1994), and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon et. al (1999) bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid, and has appeared in dozens of combinatorial constructions. In this paper, we show a similar type of result for $χ(G)$ as a function of the number of vertices $n$, the number of edges $m$, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem regarding fractional chromatic number and degeneracy in triangle-free graphs can resolve the small remaining gap in our bounds.

math.CO↗

arXiv2018

Edge-coloring linear hypergraphs with medium-sized edges

Motivated by the Erdős-Faber-Lovász (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper-edge sizes are bounded between $i$ and $C_{i,ε} \sqrt{n}$ inclusive, then there is a list edge coloring using $(1 + ε) \frac{n}{i - 1}$ colors. The dependence on $n$ in the upper bound is optimal (up to the value of $C_{i,ε}$).

math.CO↗