On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter

Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this paper we consider partially dynamic versions of three classic problems in graph theory. Based on popular conjectures we show that: -- No algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for incremental or decremental maximum cardinality bipartite matching. This significantly improves on the $O(m^{1/2-\varepsilon})$ bound for sparse graphs of Henzinger et al. [STOC'15] and $O(n^{1/3-\varepsilon})$ bound of Kopelowitz, Pettie and Porat. Our linear bound also appears more natural. In addition, the result we present separates the node-addition model from the edge insertion model, as an algorithm with total update time $O(m\sqrt{n})$ exists for the former by Bosek et al. [FOCS'14]. -- No algorithm with amortized update time $O(m^{1-\varepsilon})$ exists for incremental or decremental maximum flow in directed and weighted sparse graphs. No such lower bound was known for partially dynamic maximum flow previously. Furthermore no algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for directed and unweighted graphs or undirected and weighted graphs. -- No algorithm with amortized update time $O(n^{1/2 - \varepsilon})$ exists for incremental or decremental $(4/3-\varepsilon')$-approximating the diameter of an unweighted graph. We also show a slightly stronger bound if node additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit

Fast and Compact Exact Distance Oracle for Planar Graphs

For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an $O(n^{5/3})$-space distance oracle which answers exact distance queries in $O(\log n)$ time for $n$-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space $O(n^{2 - \epsilon})$ for some constant $\epsilon > 0$) either required query time polynomial in $n$ or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any $S \ge n^{3/2}$, we show how to obtain an $S$-space distance oracle that answers queries in time $O((n^{5/2}/ S^{3/2}) \log n)$. This is a polynomial improvement over the previous planar distance oracles with $o(n^{1/4})$ query time

Finding Even Cycles Faster via Capped k-Walks

In this paper, we consider the problem of finding a cycle of length $2k$ (a $C_{2k}$) in an undirected graph $G$ with $n$ nodes and $m$ edges for constant $k\ge2$. A classic result by Bondy and Simonovits [J.Comb.Th.'74] implies that if $m \ge100k n^{1+1/k}$, then $G$ contains a $C_{2k}$, further implying that one needs to consider only graphs with $m = O(n^{1+1/k})$. Previously the best known algorithms were an $O(n^2)$ algorithm due to Yuster and Zwick [J.Disc.Math'97] as well as a $O(m^{2-(1+\lceil k/2\rceil^{-1})/(k+1)})$ algorithm by Alon et al. [Algorithmica'97]. We present an algorithm that uses $O(m^{2k/(k+1)})$ time and finds a $C_{2k}$ if one exists. This bound is $O(n^2)$ exactly when $m=\Theta(n^{1+1/k})$. For $4$-cycles our new bound coincides with Alon et al., while for every $k>2$ our bound yields a polynomial improvement in $m$. Yuster and Zwick noted that it is "plausible to conjecture that $O(n^2)$ is the best possible bound in terms of $n$". We show "conditional optimality": if this hypothesis holds then our $O(m^{2k/(k+1)})$ algorithm is tight as well. Furthermore, a folklore reduction implies that no combinatorial algorithm can determine if a graph contains a $6$-cycle in time $O(m^{3/2-\epsilon})$ for any $\epsilon>0$ under the widely believed combinatorial BMM conjecture. Coupled with our main result, this gives tight bounds for finding $6$-cycles combinatorially and also separates the complexity of finding $4$- and $6$-cycles giving evidence that the exponent of $m$ in the running time should indeed increase with $k$. The key ingredient in our algorithm is a new notion of capped $k$-walks, which are walks of length $k$ that visit only nodes according to a fixed ordering. Our main technical contribution is an involved analysis proving several properties of such walks which may be of independent interest.Comment: To appear at STOC'1

Fast Similarity Sketching

We consider the Similarity Sketching problem: Given a universe $[u]= \{0,\ldots,u-1\}$ we want a random function $S$ mapping subsets $A\subseteq [u]$ into vectors $S(A)$ of size $t$, such that similarity is preserved. More precisely: Given sets $A,B\subseteq [u]$, define $X_i=[S(A)[i]= S(B)[i]]$ and $X=\sum_{i\in [t]}X_i$. We want to have $E[X]=t\cdot J(A,B)$, where $J(A,B)=|A\cap B|/|A\cup B|$ and furthermore to have strong concentration guarantees (i.e. Chernoff-style bounds) for $X$. This is a fundamental problem which has found numerous applications in data mining, large-scale classification, computer vision, similarity search, etc. via the classic MinHash algorithm. The vectors $S(A)$ are also called sketches. The seminal $t\times$MinHash algorithm uses $t$ random hash functions $h_1,\ldots, h_t$, and stores $\left(\min_{a\in A}h_1(A),\ldots, \min_{a\in A}h_t(A)\right)$ as the sketch of $A$. The main drawback of MinHash is, however, its $O(t\cdot |A|)$ running time, and finding a sketch with similar properties and faster running time has been the subject of several papers. Addressing this, Li et al. [NIPS'12] introduced one permutation hashing (OPH), which creates a sketch of size $t$ in $O(t + |A|)$ time, but with the drawback that possibly some of the $t$ entries are "empty" when $|A| = O(t)$. One could argue that sketching is not necessary in this case, however the desire in most applications is to have one sketching procedure that works for sets of all sizes. Therefore, filling out these empty entries is the subject of several follow-up papers initiated by Shrivastava and Li [ICML'14]. However, these "densification" schemes fail to provide good concentration bounds exactly in the case $|A| = O(t)$, where they are needed. (continued...

Practical Hash Functions for Similarity Estimation and Dimensionality Reduction

Hashing is a basic tool for dimensionality reduction employed in several aspects of machine learning. However, the perfomance analysis is often carried out under the abstract assumption that a truly random unit cost hash function is used, without concern for which concrete hash function is employed. The concrete hash function may work fine on sufficiently random input. The question is if it can be trusted in the real world when faced with more structured input. In this paper we focus on two prominent applications of hashing, namely similarity estimation with the one permutation hashing (OPH) scheme of Li et al. [NIPS'12] and feature hashing (FH) of Weinberger et al. [ICML'09], both of which have found numerous applications, i.e. in approximate near-neighbour search with LSH and large-scale classification with SVM. We consider mixed tabulation hashing of Dahlgaard et al.[FOCS'15] which was proved to perform like a truly random hash function in many applications, including OPH. Here we first show improved concentration bounds for FH with truly random hashing and then argue that mixed tabulation performs similar for sparse input. Our main contribution, however, is an experimental comparison of different hashing schemes when used inside FH, OPH, and LSH. We find that mixed tabulation hashing is almost as fast as the multiply-mod-prime scheme ax+b mod p. Mutiply-mod-prime is guaranteed to work well on sufficiently random data, but we demonstrate that in the above applications, it can lead to bias and poor concentration on both real-world and synthetic data. We also compare with the popular MurmurHash3, which has no proven guarantees. Mixed tabulation and MurmurHash3 both perform similar to truly random hashing in our experiments. However, mixed tabulation is 40% faster than MurmurHash3, and it has the proven guarantee of good performance on all possible input.Comment: Preliminary version of this paper will appear at NIPS 201

Testing the Effect of a Short Cheap Talk Script in Choice Experiments

The application of stated preference methods rests on the assumption that respondents act rationally and that their demand for the non-market good on the hypothetical market is equal to what their real demand would be. Previous studies have shown that this is not the case and this gap is known as hypothetical bias. The present paper attempts to frame the description of the hypothetical market so as to induce more “true market behaviour” in the respondents by including a short Cheap Talk script. The script informs respondents that in similar studies using stated preference methods, people have a tendency to overestimate how much they are willing to pay compared to their actual (true) willingness to pay. Applying a two-split sample approach to a Choice Experiment study focusing on preferences for reducing visual disamenities from offshore wind farms, the Cheap Talk script is found to be a preference mover, but does not affect preferences significantly. Significant effects are found when relating the effect of the Cheap Talk script to the cost levels of the alternatives, in that female respondents are found to choose higher cost alternatives less frequently when presented with the Cheap Talk script, while male respondents are not affected.Cheap Talk, Stated Preferences, Choice Experiment, Hypothetical Bias, Gender

Sublinear Distance Labeling

A distance labeling scheme labels the $n$ nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A $D$-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least $D$ from each other. In this paper we consider distance labeling schemes for the classical case of unweighted graphs with both directed and undirected edges. We present a $O(\frac{n}{D}\log^2 D)$ bit $D$-preserving distance labeling scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of $\Omega(\frac{n}{D})$. With our $D$-preserving distance labeling scheme as a building block, we additionally achieve the following results: 1. We present the first distance labeling scheme of size $o(n)$ for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from distance labeling in general graphs which require $\Omega(n)$ bits, Moon [Proc. of Glasgow Math. Association 1965]. 2. For approximate $r$-additive labeling schemes, that return distances within an additive error of $r$ we show a scheme of size $O\left ( \frac{n}{r} \cdot\frac{\operatorname{polylog} (r\log n)}{\log n} \right )$ for $r \ge 2$. This improves on the current best bound of $O\left(\frac{n}{r}\right)$ by Alstrup et. al. [SODA 2016] for sub-polynomial $r$, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for $r=2$.Comment: A preliminary version of this paper appeared at ESA'1

Povezanost med inovativno usmerjenimi dejavnostmi trajnostnega razvoja ter učinkovitostjo in uspešnostjo organizacije

Background and Purpose: The purpose of this paper is to empirically analyse the effects of sustainability-oriented innovation practices on the overall organizational performance. Further, this paper also aims to advance understanding of the measurement of corporate sustainability practices with the focus on innovation dimensions. Design/Methodology/Approach: The study uses data obtained from a survey of 116 organizations encompassing both the manufacturing and service industries in Slovenia. Descriptive statistics were used in order to determine the level of sustainability-oriented innovation practices deployment. Exploratory factor analysis was applied to extract the underlying factors and to provide a basis for assessing their reliability and validity. In addition, regression analysis was used to quantify the effect of sustainability practices on the organizational performance. Results: Data analysis result showed that sustainability-oriented innovation practices are significantly associated with organizational performance. Therefore, empirical evidence from this research confirmed the premise that building innovation competencies and integrating innovation activities in organization’s processes lead to performance benefits. This contributes to the debate about the potential for organizations to be sustainable and competitive. Conclusion: The presented research on corporate sustainability provides important theoretical and practical insights on which the deployment of sustainability-oriented innovation practices are conducive to fostering a broader set of performance benefits. As such, managers should increase organizations’ capacity for innovation which can be beneficial in terms of performance implications and achieving sustainability goalsOzadje in namen: Poglavitni namen članka je predstaviti empirično raziskavo o vplivu dejavnosti trajnostnega razvoja, ki so osredotočene na inovativnost, na celokupno učinkovitost in uspešnost organizacije. Namen članka je prav tako izboljšati razumevanje operacionalizacije spremenljivk dejavnosti trajnostnega razvoja, s poudarkom na dimenzijahinovativnosti. Zasnova in metodologija: Pričujoči članek temelji na anketni raziskavi, na osnovi katere smo pridobili 116 uporabnih odgovorov s strani slovenskih proizvodnih in storitvenih organizacij. S pomočjo faktorske analize smo preverili konvergentno veljavnost merjenega konstrukta in s tem oblikovali posamezne dimenzije na inventivnost osredotočenih dejavnosti trajnostnega razvoja organizacije. Vplive na inventivnost osredotočenih dejavnosti trajnostnega razvoja na celokupno učinkovitost in uspešnost organizacije smo proučevali s pomočjo regresijske analize. Rezultati: – Rezultati raziskave so pokazali, da na inventivnost osredotočene dejavnosti trajnostnega razvoja pozitivno in statistično značilno vplivajo na celokupno učinkovitost in uspešnost organizacije. Rezultati raziskave so torej potrdili predpostavko, da razvoj kompetenc na področju inovativnosti in vključitev le-teh v procese organizacije, prinese številne koristi za organizacijo. Na ta način članek prispeva k razpravi o priložnostih in možnostih organizacije, da razvijakonkurenčne prednosti in hkrati prispeva k trajnostnemu razvoju. Zaključek: Raziskava doprinaša pomembne teoretične in praktične vpoglede na področju trajnostnega razvojaorganizacije in prikazuje pomembnost izvajanja na inventivnost osredotočenih dejavnosti trajnostnega razvoja z vidika doseganja učinkovitosti in uspešnosti organizacije. Izsledki raziskave poudarjajo vlogo vodstva, ki mora spodbujati povečevanje sposobnosti organizacije na področju inovativnosti, saj organizacije lahko posledično izboljšujejo različne segmente učinkovitosti in uspešnosti ter hkrati dosegajo cilje trajnostnega razvoja