arXiv subjects

David G. Harris

Publications and source records attributed to David G. Harris.

51 records · Page 3Linked to original sources

arXiv2018

The Combinatorial Data Fusion Problem in Conflicted-supervised Learning

The best merge problem in industrial data science generates instances where disparate data sources place incompatible relational structures on the same set $V$ of objects. Graph vertex labelling data may include (1) missing or erroneous labels,(2) assertions that two vertices carry the same (unspecified) label, and (3) denying some subset of vertices from carrying the same label. Conflicted-supervised learning applies to cases where no labelling scheme satisfies (1), (2), and (3). Our rigorous formulation starts from a connected weighted graph $(V, E)$, and an independence system $\mathcal{S}$ on $V$, characterized by its circuits, called forbidden sets. Global incompatibility is expressed by the fact $V \notin \mathcal{S}$. Combinatorial data fusion seeks a subset $E_1 \subset E$ of maximum edge weight so that no vertex component of the subgraph $(V, E_1)$ contains any forbidden set. Multicut and multiway cut are special cases where all forbidden sets have cardinality two. The general case exhibits unintuitive properties, shown in counterexamples. The first in a series of papers concentrates on cases where $(V, E)$ is a tree, and presents an algorithm on general graphs, in which the combinatorial data fusion problem is transferred to the Gomory-Hu tree, where it is solved using greedy set cover. Experimental results are given.

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arXiv2018

Deterministic parallel algorithms for fooling polylogarithmic juntas and the Lovasz Local Lemma

Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low (almost-) independence. A series of papers, beginning with Luby (1993) and continuing with Berger & Rompel (1991) and Chari et al. (2000), showed that these techniques can be combined to give deterministic parallel algorithms for combinatorial optimization problems involving sums of $w$-juntas. We improve these algorithms through derandomized variable partitioning and a new code construction for fooling Fourier characters over $GF(2)$. This reduces the processor complexity to essentially independent of $w$ while the running time is reduced from exponential in $w$ to linear in $w$. As a key subroutine, we give a new algorithm to generate a probability space which can fool a given set of neighborhoods. Schulman (1992) gave an NC algorithm to do so for neighborhoods of size $w \leq O(\log n)$. Our new algorithm is NC1, with essentially optimal time and processor complexity, when $w = O(\log n)$; it remains NC up to $w = \text{polylog}(n)$. This answers an open problem of Schulman. One major application of these algorithms is an NC algorithm for the Lovász Local Lemma. Previous NC algorithms, including the seminal algorithm of Moser & Tardos (2010) and the work of Chandrasekaran et. al (2013), required that (essentially) the bad-events could span only $O(\log n)$ variables; we relax this to $\text{polylog}(n)$ variables. We use this for an $\text{NC}^2$ algorithm for defective vertex coloring, which works for arbitrary degree graphs.

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arXiv2018

Deterministic parallel algorithms for bilinear objective functions

Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low independence. A series of papers, beginning with work by Luby (1988), showed that in many cases these techniques can be combined to give deterministic parallel (NC) algorithms for a variety of combinatorial optimization problems, with low time- and processor-complexity. We extend and generalize a technique of Luby for efficiently handling bilinear objective functions. One noteworthy application is an NC algorithm for maximal independent set. On a graph $G$ with $m$ edges and $n$ vertices, this takes $\tilde O(\log^2 n)$ time and $(m + n) n^{o(1)}$ processors, nearly matching the best randomized parallel algorithms. Other applications include reduced processor counts for algorithms of Berger (1997) for maximum acyclic subgraph and Gale-Berlekamp switching games. This bilinear factorization also gives better algorithms for problems involving discrepancy. An important application of this is to automata-fooling probability spaces, which are the basis of a notable derandomization technique of Sivakumar (2002). Our method leads to large reduction in processor complexity for a number of derandomization algorithms based on automata-fooling, including set discrepancy and the Johnson-Lindenstrauss Lemma.

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arXiv2018

Distributed $(Δ+1)$-Coloring in Sublogarithmic Rounds

We give a new randomized distributed algorithm for $(Δ+1)$-coloring in the LOCAL model, running in $O(\sqrt{\log Δ})+ 2^{O(\sqrt{\log \log n})}$ rounds in a graph of maximum degree~$Δ$. This implies that the $(Δ+1)$-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of $Ω\left( \min \left( \sqrt{\frac{\log n}{\log \log n}}, \frac{\log Δ}{\log \log Δ} \right) \right)$ by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains $Δ+1$ colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts.

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arXiv2017

Tight bounds and conjectures for the isolation lemma

Given a hypergraph $H$ and a weight function $w: V \rightarrow \{1, \dots, M\}$ on its vertices, we say that $w$ is isolating if there is exactly one edge of minimum weight $w(e) = \sum_{i \in e} w(i)$. The Isolation Lemma is a combinatorial principle introduced in Mulmuley et. al (1987) which gives a lower bound on the number of isolating weight functions. Mulmuley used this as the basis of a parallel algorithm for finding perfect graph matchings. It has a number of other applications to parallel algorithms and to reductions of general search problems to unique search problems (in which there are one or zero solutions). The original bound given by Mulmuley et al. was recently improved by Ta-Shma (2015). In this paper, we show improved lower bounds on the number of isolating weight functions, and we conjecture that the extremal case is when $H$ consists of $n$ singleton edges. When $M \gg n$ our improved bound matches this extremal case asymptotically. We are able to show that this conjecture holds in a number of special cases: when $H$ is a linear hypergraph or is 1-degenerate, or when $M = 2$. We also show that it holds asymptotically when $M \gg n \gg 1$.

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arXiv2017

A Lottery Model for Center-type Problems With Outliers

In this paper, we give tight approximation algorithms for the $k$-center and matroid center problems with outliers. Unfairness arises naturally in this setting: certain clients could always be considered as outliers. To address this issue, we introduce a lottery model in which each client $j$ is allowed to submit a parameter $p_j \in [0,1]$ and we look for a random solution that covers every client $j$ with probability at least $p_j$. Our techniques include a randomized rounding procedure to round a point inside a matroid intersection polytope to a basis plus at most one extra item such that all marginal probabilities are preserved and such that a certain linear function of the variables does not decrease in the process with probability one.

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arXiv2017

Parallel algorithms and concentration bounds for the Lovasz Local Lemma via witness DAGs

The Lovász Local Lemma (LLL) is a cornerstone principle in the probabilistic method of combinatorics, and a seminal algorithm of Moser & Tardos (2010) provides an efficient randomized algorithm to implement it. This can be parallelized to give an algorithm that uses polynomially many processors and runs in $O(\log^3 n)$ time on an EREW PRAM, stemming from $O(\log n)$ adaptive computations of a maximal independent set (MIS). Chung et al. (2014) developed faster local and parallel algorithms, potentially running in time $O(\log^2 n)$, but these algorithms require more stringent conditions than the LLL. We give a new parallel algorithm that works under essentially the same conditions as the original algorithm of Moser & Tardos but uses only a single MIS computation, thus running in $O(\log^2 n)$ time on an EREW PRAM. This can be derandomized to give an NC algorithm running in time $O(\log^2 n)$ as well, speeding up a previous NC LLL algorithm of Chandrasekaran et al. (2013). We also provide improved and tighter bounds on the run-times of the sequential and parallel resampling-based algorithms originally developed by Moser & Tardos. These apply to any problem instance in which the tighter Shearer LLL criterion is satisfied.

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arXiv2017

Fairness in Resource Allocation and Slowed-down Dependent Rounding

We consider an issue of much current concern: could fairness, an issue that is already difficult to guarantee, worsen when algorithms run much of our lives? We consider this in the context of resource-allocation problems, we show that algorithms can guarantee certain types of fairness in a verifiable way. Our conceptual contribution is a simple approach to fairness in this context, which only requires that all users trust some public lottery. Our technical contributions are in ways to address the $k$-center and knapsack-center problems that arise in this context: we develop a novel dependent-rounding technique that, via the new ingredients of "slowing down" and additional randomization, guarantees stronger correlation properties than known before.

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arXiv2017

Improved bounds and algorithms for graph cuts and network reliability

Karger (SIAM Journal on Computing, 1999) developed the first fully-polynomial approximation scheme to estimate the probability that a graph $G$ becomes disconnected, given that its edges are removed independently with probability $p$. This algorithm runs in $n^{5+o(1)} ε^{-3}$ time to obtain an estimate within relative error $ε$. We improve this run-time through algorithmic and graph-theoretic advances. First, there is a certain key sub-problem encountered by Karger, for which a generic estimation procedure is employed, we show that this has a special structure for which a much more efficient algorithm can be used. Second, we show better bounds on the number of edge cuts which are likely to fail. Here, Karger's analysis uses a variety of bounds for various graph parameters, we show that these bounds cannot be simultaneously tight. We describe a new graph parameter, which simultaneously influences all the bounds used by Karger, and obtain much tighter estimates of the cut structure of $G$. These techniques allow us to improve the runtime to $n^{3+o(1)} ε^{-2}$, our results also rigorously prove certain experimental observations of Karger & Tai (Proc. ACM-SIAM Symposium on Discrete Algorithms, 1997). Our rigorous proofs are motivated by certain non-rigorous differential-equation approximations which, however, provably track the worst-case trajectories of the relevant parameters. A key driver of Karger's approach (and other cut-related results) is a bound on the number of small cuts: we improve these estimates when the min-cut size is "small" and odd, augmenting, in part, a result of Bixby (Bulletin of the AMS, 1974).

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arXiv2017

Algorithmic and enumerative aspects of the Moser-Tardos distribution

Moser & Tardos have developed a powerful algorithmic approach (henceforth "MT") to the Lovasz Local Lemma (LLL); the basic operation done in MT and its variants is a search for "bad" events in a current configuration. In the initial stage of MT, the variables are set independently. We examine the distributions on these variables which arise during intermediate stages of MT. We show that these configurations have a more or less "random" form, building further on the "MT-distribution" concept of Haeupler et al. in understanding the (intermediate and) output distribution of MT. This has a variety of algorithmic applications; the most important is that bad events can be found relatively quickly, improving upon MT across the complexity spectrum: it makes some polynomial-time algorithms sub-linear (e.g., for Latin transversals, which are of basic combinatorial interest), gives lower-degree polynomial run-times in some settings, transforms certain super-polynomial-time algorithms into polynomial-time ones, and leads to Las Vegas algorithms for some coloring problems for which only Monte Carlo algorithms were known. We show that in certain conditions when the LLL condition is violated, a variant of the MT algorithm can still produce a distribution which avoids most of the bad events. We show in some cases this MT variant can run faster than the original MT algorithm itself, and develop the first-known criterion for the case of the asymmetric LLL. This can be used to find partial Latin transversals -- improving upon earlier bounds of Stein (1975) -- among other applications. We furthermore give applications in enumeration, showing that most applications (where we aim for all or most of the bad events to be avoided) have many more solutions than known before by proving that the MT-distribution has "large" min-entropy and hence that its support-size is large.

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arXiv2017

Tighter inapproximability for set cover

Set Cover is a classic NP-hard problem; as shown by Slavík (1997) the greedy algorithm gives an approximation ratio of $\ln n - \ln \ln n + Θ(1)$. A series of works by Lund \& Yannakakis (1994), Feige (1998), Moshkovitz (2015) have shown that, under the assumption $P \neq NP$, it is impossible to obtain a polynomial-time approximation ratio with approximation ratio $(1 - α) \ln n$, for any constant $α> 0$. In this note, we show that under the Exponential Time Hypothesis (a stronger complexity-theoretic assumptions than $P \neq NP$), there are no polynomial-time algorithms achieving approximation ratio $\ln n - C \ln \ln n$, where $C$ is some universal constant. Thus, the greedy algorithm achieves an essentially optimal approximation ratio (up to the coefficient of $\ln \ln n$).

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arXiv2016

A constructive algorithm for the LLL on permutations

While there has been significant progress on algorithmic aspects of the Lovász Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations. The breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. We resolve this by developing a randomized polynomial-time algorithm for such applications. A noteworthy application is for Latin transversals: the best-known general result here (Bissacot et al., improving on Erdős and Spencer), states that any $n \times n$ matrix in which each entry appears at most $(27/256)n$ times, has a Latin transversal. We present the first polynomial-time algorithm to construct such a transversal. We also develop RNC algorithms for Latin transversals, rainbow Hamiltonian cycles, strong chromatic number, and hypergraph packing. In addition to efficiently finding a configuration which avoids bad-events, the algorithm of Moser & Tardos has many powerful extensions and properties. These include a well-characterized distribution on the output distribution, parallel algorithms, and a partial resampling variant. We show that our algorithm has nearly all of the same useful properties as the Moser-Tardos algorithm, and present a comparison of this aspect with recent works on the LLL in general probability spaces.

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arXiv2016

Lopsidependency in the Moser-Tardos framework: Beyond the Lopsided Lovasz Local Lemma

The Lopsided Lovász Local Lemma (LLLL) is a powerful probabilistic principle which has been used in a variety of combinatorial constructions. While originally a general statement about probability spaces, it has recently been transformed into a variety of polynomial-time algorithms. The resampling algorithm of Moser & Tardos (2010) is the most well-known example of this. A variety of criteria have been shown for the LLLL; the strongest possible criterion was shown by Shearer, and other criteria which are easier to use computationally have been shown by Bissacot et al (2011), Pegden (2014), Kolipaka & Szegedy (2011), and Kolipaka, Szegedy, Xu (2012). We show a new criterion for the Moser-Tardos algorithm to converge. This criterion is stronger than the LLLL criterion; this is possible because it does not apply in the same generality as the original LLLL; yet, it is strong enough to cover many applications of the LLLL in combinatorics. We show a variety of new bounds and algorithms. A noteworthy application is for $k$-SAT, with bounded occurrences of variables. As shown in Gebauer, Szábo, and Tardos (2011), a $k$-SAT instance in which every variable appears $L \leq \frac{2^{k+1}}{e (k+1)}$ times, is satisfiable. Although this bound is asymptotically tight (in $k$), we improve it to $L \leq \frac{2^{k+1} (1 - 1/k)^k}{k-1} - \frac{2}{k}$ which can be significantly stronger when $k$ is small. We introduce a new parallel algorithm for the LLLL. While Moser & Tardos described a simple parallel algorithm for the Lovász Local Lemma, and described a simple sequential algorithm for a form of the Lopsided Lemma, they were not able to combine the two. Our new algorithm applies in nearly all settings in which the sequential algorithm works --- this includes settings covered by our new stronger LLLL criterion.

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arXiv2015

Distinct volume subsets

Suppose that $a$ and $d$ are positive integers with $a \geq 2$. Let $h_{a,d}(n)$ be the largest integer $t$ such that any set of $n$ points in $\mathbb{R}^d$ contains a subset of $t$ points for which all the non-zero volumes of the ${t \choose a}$ subsets of order $a$ are distinct. Beginning with Erdős in 1957, the function $h_{2,d}(n)$ has been closely studied and is known to be at least a power of $n$. We improve the best known bound for $h_{2,d}(n)$ and show that $h_{a,d}(n)$ is at least a power of $n$ for all $a$ and $d$.

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arXiv2014

On Computing Maximal Independent Sets of Hypergraphs in Parallel

Whether or not the problem of finding maximal independent sets (MIS) in hypergraphs is in (R)NC is one of the fundamental problems in the theory of parallel computing. Unlike the well-understood case of MIS in graphs, for the hypergraph problem, our knowledge is quite limited despite considerable work. It is known that the problem is in \emph{RNC} when the edges of the hypergraph have constant size. For general hypergraphs with $n$ vertices and $m$ edges, the fastest previously known algorithm works in time $O(\sqrt{n})$ with $\text{poly}(m,n)$ processors. In this paper we give an EREW PRAM algorithm that works in time $n^{o(1)}$ with $\text{poly}(m,n)$ processors on general hypergraphs satisfying $m \leq n^{\frac{\log^{(2)}n}{8(\log^{(3)}n)^2}}$, where $\log^{(2)}n = \log\log n$ and $\log^{(3)}n = \log\log\log n$. Our algorithm is based on a sampling idea that reduces the dimension of the hypergraph and employs the algorithm for constant dimension hypergraphs as a subroutine.

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