# Theory of Computing Blog Aggregator

My daughters, now in college, never knew a time when they couldn't communicate with anyone instantaneously. Molly, now 18, takes pride having the same birth year as Google. They have never in their memory seen a true technological change that so dramatically affects the world they live in. But they are about to.

The 50's and 60's saw a transportation revolution. The Interstate highway system made local and national car and truck travel feasible. The jet engine allowed short trips to faraway places. The shipping container made transporting a good, made anywhere in the world, cheaper than producing it.

We could have national and worldwide academic conferences. China became a superpower by shipping us low-cost goods. Dock worker jobs, the kind held by Archie Bunker, have morphed and shrunk, even as the number of imported goods has grown.

In the 90's we had a communications revolution. The cell phone kept us connected all the time. The home computer and the Internet gave us immediate access to the world's information and Google helped us sort through it.

It fundamentally changed how we interacted. No longer did we need to make plans in advance. Eventually we would have little need for encyclopedias, almanacs, maps, or physical media for music, photos and movies. Not to mention new types of companies and the transformation of how businesses could work with their employees and contractors spread across the world.

That brings us to today. We are at the brink, if it hasn't already started, of an intelligence revolution, the combination of big data, machine learning and automation. The initial obvious transformation will come from autonomous cars, which will change not only how we get from point A to point B but how we plan roads, businesses and where we live. Beyond that work itself will change as we will continue to automate an ever growing number of white collar jobs. As with every transformation, the world we live in will change in unforeseen ways, hopefully more good than bad.

I really look forward to watching these changes through my daughter's eyes, to see how this new society directly affects them as they start their working lives. And how their children will one day see an old-fashioned automobile with relics like foot pedals and a steering wheel and be shocked that their parents once drove cars themselves.

by Lance Fortnow (noreply@blogger.com) at October 27, 2016 11:29 AM UTC

### A Proposed Algorithm for Minimum Vertex Cover Problem and its Testing

Authors: Gang Hu
Abstract: The paper presents an algorithm for minimum vertex cover problem, which is an NP-Complete problem. The algorithm computes a minimum vertex cover of each input simple graph. Tested by the attached MATLAB programs, Stage 1 of the algorithm is applicable to, i.e., yields a proved minimum vertex cover for, about 99.99% of the tested 610,000 graphs of order 16 and 99.67% of the tested 1,200 graphs of order 32, and Stage 2 of the algorithm is applicable to all of the above tested graphs. All of the tested graphs are randomly generated graphs of random "edge density" or in other words, random probability of each edge. It is proved that Stage 1 and Stage 2 of the algorithm run in $O(n^{5+logn})$ and $O(n^{3(5+logn)/2})$ time respectively, where $n$ is the order of input graph. Because there is no theoretical proof yet that Stage 2 is applicable to all graphs, further stages of the algorithm are proposed, which are in a general form that is consistent with Stages 1 and 2.

### On the Exact Amount of Missing Information that makes Finding Possible Winners Hard

Authors: Palash Dey, Neeldhara Misra
Abstract: We consider election scenarios with incomplete information, a situation that arises often in practice. There are several models of incomplete information and accordingly, different notions of outcomes of such elections. In one well-studied model of incompleteness, the votes are given by partial orders over the candidates. In this context we can frame the problem of finding a possible winner, which involves determining whether a given candidate wins in at least one completion of a given set of partial votes for a specific voting rule.

The possible winner problem is well-known to be NP-complete in general, and it is in fact known to be NP-complete for several voting rules where the number of undetermined pairs in every vote is bounded only by some constant. In this paper, we address the question of determining precisely the smallest number of undetermined pairs for which the possible winner problem remains NP-complete. In particular, we find the exact values of $t$ for which the possible winner problem transitions to being NP-complete from being in P, where $t$ is the maximum number of undetermined pairs in every vote. We demonstrate tight results for a broad subclass of scoring rules which includes all the commonly used scoring rules (such as plurality, veto, Borda, $k$-approval, and so on), Copeland$^\alpha$ for every $\alpha\in[0,1]$, maximin, and Bucklin voting rules. A somewhat surprising aspect of our results is that for many of these rules, the possible winner problem turns out to be hard even if every vote has at most one undetermined pair of candidates.

### The geometry of rank decompositions of matrix multiplication I: 2x2 matrices

Authors: Luca Chiantini, Christian Ikenmeyer, J. M. Landsberg, Giorgio Ottaviani
Abstract: This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper we: establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix multiplication decompositions, give a geometric proof of the theorem of Burichenko's theorem establishing the symmetry group of Strassen's algorithm, and present two particularly nice subfamilies in the Strassen family of decompositions.

### Multiplayer parallel repetition for expander games

Authors: Irit Dinur, Prahladh Harsha, Rakesh Venkat, Henry Yuen
Abstract: We investigate the value of parallel repetition of one-round games with any number of players $k\ge 2$. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games.

Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are connected in a certain sense.

We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.

### On-line algorithms for multiplication and division in real and complex numeration systems

Authors: Marta Brzicova, Christiane Frougny, Edita Pelantova, Milena Svobodova
Abstract: A positional numeration system is given by a base and by a set of digits. The base is a real or complex number $\beta$ such that $|\beta|>1$, and the digit set $\A$ is a finite set of digits including $0$. Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi and Ercegovac, is a mode of computation where operands and results flow through arithmetic units in a digit serial manner, starting with the most significant digit.

In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that $\beta$ is any real or complex number, and digits are real or complex. We then define the so-called OL Property, and show that if $(\beta, \A)$ has the OL Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base $\beta$ and a digit set $\A$ of contiguous integers, the system $(\beta, \A)$ has the OL Property if $# \A > |\beta|$. For a complex base $\beta$ and symmetric digit set $\A$ of contiguous integers, the system $(\beta, \A)$ has the OL Property if $# \A > \beta\overline{\beta} + |\beta + \overline{\beta}|$. Provided that addition and subtraction are realizable in parallel in the system $(\beta, \A)$ and that preprocessing of the denominator is possible, our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base $\beta=\frac{3+\sqrt{5}}{2}$ with digits $\A=\{-1,0,1\}$; base $\beta=2i$ with digits $\A = \{-2,-1, 0,1,2\}$; and base $\beta = -\frac{3}{2} + i \frac{\sqrt{3}}{2} = -1 + \omega$, where $\omega = \exp{\frac{2i\pi}{3}}$, with digits $\A = \{0, \pm 1, \pm \omega, \pm \omega^2 \}$.

### Optimal In-Place Suffix Sorting

Authors: Zhize Li, Jian Li, Hongwei Huo
Abstract: Suffix array is a fundamental data structure for many applications that involve string searching and data compression. Designing time/space-efficient suffix array construction algorithms has attracted significant attentions and considerable advances have been made in the last 20 years. We obtain the suffix array construction algorithms that are optimal both in time and space for both integer alphabets and general alphabets. Concretely, we make the following contributions:

1. For integer alphabets, we obtain the first algorithm which takes linear time and uses only $O(1)$ workspace (the workspace is the space needed beyond the input string and the output suffix array). The input string may be modified during the execution of the algorithm, but should be restored upon termination of the algorithm. Our algorithm is easy to implement. Our C implementation of the algorithm requires only 8 Bytes of workspace.

2. We strengthen the first result by providing the first linear time in-place algorithm for read-only integer alphabets (i.e., we cannot modify the input string $T$). This settles the open problem posed by Franceschini and Muthukrishnan in ICALP 2007 \cite{franceschini2007place}.

3. For read-only general alphabets (i.e., only comparisons are allowed), we present an in-place $O(n\log n)$ time algorithm, recovering the result obtained by Franceschini and Muthukrishnan \cite{franceschini2007place}.

### LP Rounding and Combinatorial Algorithms for Minimizing Active and Busy Time

Authors: Jessica Chang, Samir Khuller, Koyel Mukherjee
Abstract: We consider fundamental scheduling problems motivated by energy issues. In this framework, we are given a set of jobs, each with a release time, deadline and required processing length. The jobs need to be scheduled on a machine so that at most g jobs are active at any given time. The duration for which a machine is active (i.e., "on") is referred to as its active time. The goal is to find a feasible schedule for all jobs, minimizing the total active time. When preemption is allowed at integer time points, we show that a minimal feasible schedule already yields a 3-approximation (and this bound is tight) and we further improve this to a 2-approximation via LP rounding techniques. Our second contribution is for the non-preemptive version of this problem. However, since even asking if a feasible schedule on one machine exists is NP-hard, we allow for an unbounded number of virtual machines, each having capacity of g. This problem is known as the busy time problem in the literature and a 4-approximation is known for this problem. We develop a new combinatorial algorithm that gives a 3-approximation. Furthermore, we consider the preemptive busy time problem, giving a simple and exact greedy algorithm when unbounded parallelism is allowed, i.e., g is unbounded. For arbitrary g, this yields an algorithm that is 2-approximate.

### Efficient Pattern Matching in Elastic-Degenerate Strings

Authors: Costas Iliopoulos, Ritu Kundu, Solon Pissis
Abstract: In this paper, we extend the notion of gapped strings to elastic-degenerate strings. An elastic-degenerate string can been seen as an ordered collection of k > 1 seeds (substrings/subpatterns) interleaved by elastic-degenerate symbols such that each elastic-degenerate symbol corresponds to a set of two or more variable length strings. Here, we present an algorithm for solving the pattern matching problem with (solid) pattern and elastic-degenerate text, running in O(N+{\alpha}{\gamma}nm) time; where m is the length of the given pattern; n and N are the length and total size of the given elastic-degenerate text, respectively; {\alpha} and {\gamma} are small constants, respectively representing the maximum number of strings in any elastic-degenerate symbol of the text and the largest number of elastic-degenerate symbols spanned by any occurrence of the pattern in the text. The space used by the algorithm is linear in the size of the input for a constant number of elastic-degenerate symbols in the text; {\alpha} and {\gamma} are so small in real applications that the algorithm is expected to work very efficiently in practice.

### Almost Optimal Streaming Algorithms for Coverage Problems

Authors: Mohammadhossein Bateni, Hossein Esfandiari, Vahab Mirrokni
Abstract: Maximum coverage and minimum set cover problems --collectively called coverage problems-- have been studied extensively in streaming models. However, previous research not only achieve sub-optimal approximation factors and space complexities, but also study a restricted set arrival model which makes an explicit or implicit assumption on oracle access to the sets, ignoring the complexity of reading and storing the whole set at once. In this paper, we address the above shortcomings, and present algorithms with improved approximation factor and improved space complexity, and prove that our results are almost tight. Moreover, unlike most of previous work, our results hold on a more general edge arrival model. More specifically, we present (almost) optimal approximation algorithms for maximum coverage and minimum set cover problems in the streaming model with an (almost) optimal space complexity of $\tilde{O}(n)$, i.e., the space is {\em independent of the size of the sets or the size of the ground set of elements}. These results not only improve over the best known algorithms for the set arrival model, but also are the first such algorithms for the more powerful {\em edge arrival} model. In order to achieve the above results, we introduce a new general sketching technique for coverage functions: This sketching scheme can be applied to convert an $\alpha$-approximation algorithm for a coverage problem to a $(1-\eps)\alpha$-approximation algorithm for the same problem in streaming, or RAM models. We show the significance of our sketching technique by ruling out the possibility of solving coverage problems via accessing (as a black box) a $(1 \pm \eps)$-approximate oracle (e.g., a sketch function) that estimates the coverage function on any subfamily of the sets.

### TR16-161 | Robust sensitivity | Shachar Lovett, Jiapeng Zhang

from ECCC papers

The sensitivity conjecture is one of the central open problems in boolean complexity. A recent work of Gopalan et al. [CCC 2016] conjectured a robust analog of the sensitivity conjecture, which relates the decay of the Fourier mass of a boolean function to moments of its sensitivity. We prove this robust analog in this work.

### TR16-160 | Multiplayer parallel repetition for expander games | Henry Yuen, Irit Dinur, Prahladh Harsha, Rakesh Venkat

from ECCC papers

We investigate the value of parallel repetition of one-round games with any number of players $k\ge 2$. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games. Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are connected in a certain sense. We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.

### Rapid Mixing of Hypergraph Independent Set

Authors: Jonathan Hermon, Allan Sly, Yumeng Zhang
Abstract: We prove that the the mixing time of the Glauber dynamics for sampling independent sets on $n$-vertex $k$-uniform hypergraphs is $O(n\log n)$ when the maximum degree $\Delta$ satisfies $\Delta \leq c 2^{k/2}$, improving on the previous bound [BDK06] of $\Delta \leq k-2$. This result brings the algorithmic bound to within a constant factor of the hardness bound of [BGG+16] which showed that it is NP-hard to approximately count independent sets on hypergraphs when $\Delta \geq 5 \cdot 2^{k/2}$.

### A pumping lemma for non-cooperative self-assembly

Authors: Pierre-Étienne Meunier, Damien Regnault
Abstract: We prove the computational weakness of a model of tile assembly that has so far resisted many attempts of formal analysis or positive constructions. Specifically, we prove that, in Winfree's abstract Tile Assembly Model, when restricted to use only noncooperative bindings, any long enough path that can grow in all terminal assemblies is pumpable, meaning that this path can be extended into an infinite, ultimately periodic path. This result can be seen as a geometric generalization of the pumping lemma of finite state automata, and closes the question of what can be computed deterministically in this model. Moreover, this question has motivated the development of a new method called visible glues. We believe that this method can also be used to tackle other long-standing problems in computational geometry, in relation for instance with self-avoiding paths. Tile assembly (including non-cooperative tile assembly) was originally introduced by Winfree and Rothemund in STOC 2000 to understand how to program shapes. The non-cooperative variant, also known as temperature 1 tile assembly, is the model where tiles are allowed to bind as soon as they match on one side, whereas in cooperative tile assembly, some tiles need to match on several sides in order to bind. In this work, we prove that only very simple shapes can indeed be programmed, whereas exactly one known result (SODA 2014) showed a restriction on the assemblies general non-cooperative self-assembly could achieve, without any implication on its computational expressiveness. With non-square tiles (like polyominos, SODA 2015), other recent works have shown that the model quickly becomes computationally powerful.

### Subexponential parameterized algorithms for graphs of polynomial growth

Authors: Dániel Marx, Marcin Pilipczuk
Abstract: We show that for a number of parameterized problems for which only $2^{O(k)} n^{O(1)}$ time algorithms are known on general graphs, subexponential parameterized algorithms with running time $2^{O(k^{1-\frac{1}{1+\delta}} \log^2 k)} n^{O(1)}$ are possible for graphs of polynomial growth with growth rate (degree) $\delta$, that is, if we assume that every ball of radius $r$ contains only $O(r^\delta)$ vertices. The algorithms use the technique of low-treewidth pattern covering, introduced by Fomin et al. [FOCS 2016] for planar graphs; here we show how this strategy can be made to work for graphs with polynomial growth.

Formally, we prove that, given a graph $G$ of polynomial growth with growth rate $\delta$ and an integer $k$, one can in randomized polynomial time find a subset $A \subseteq V(G)$ such that on one hand the treewidth of $G[A]$ is $O(k^{1-\frac{1}{1+\delta}} \log k)$, and on the other hand for every set $X \subseteq V(G)$ of size at most $k$, the probability that $X \subseteq A$ is $2^{-O(k^{1-\frac{1}{1+\delta}} \log^2 k)}$. Together with standard dynamic programming techniques on graphs of bounded treewidth, this statement gives subexponential parameterized algorithms for a number of subgraph search problems, such as Long Path or Steiner Tree, in graphs of polynomial growth.

We complement the algorithm with an almost tight lower bound for Long Path: unless the Exponential Time Hypothesis fails, no parameterized algorithm with running time $2^{k^{1-\frac{1}{\delta}-\varepsilon}}n^{O(1)}$ is possible for any $\varepsilon > 0$ and an integer $\delta \geq 3$.

### Online Submodular Maximization with Free Disposal: Randomization Beats 0.25 for Partition Matroids

Authors: T-H. Hubert Chan, Zhiyi Huang, Shaofeng H.-C. Jiang, Ning Kang, Zhihao Gavin Tang
Abstract: We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access.

For $k$-uniform matroids, we give a deterministic algorithm with competitive ratio at least $0.2959$, and the ratio approaches $\frac{1}{\alpha_\infty} \approx 0.3178$ as $k$ approaches infinity, improving the previous best ratio of $0.25$ by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015) and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased.

Further, we prove that no deterministic monotone algorithm can be strictly better than $0.25$-competitive even for partition matroids, the most modest generalization of $k$-uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio $\frac{1}{\alpha_\infty} \approx 0.3178$.

### Hardness of approximation for strip packing

Authors: Anna Adamaszek, Tomasz Kociumaka, Marcin Pilipczuk, Michał Pilipczuk
Abstract: Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, e.g. in scheduling and stock-cutting, and has been studied extensively.

When the dimensions of objects are allowed to be exponential in the total input size, it is known that the problem cannot be approximated within a factor better than $3/2$, unless $\mathrm{P}=\mathrm{NP}$. However, there was no corresponding lower bound for polynomially bounded input data. In fact, Nadiradze and Wiese [SODA 2016] have recently proposed a $(1.4 + \epsilon)$ approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values in the input data are allowed. Their result has subsequently been improved to a $(4/3 + \epsilon)$ approximation by two independent research groups [FSTTCS 2016, arXiv:1610.04430]. This raises a question whether strip packing with polynomially bounded input data admits a quasi-polynomial time approximation scheme, as is the case for related two-dimensional packing problems like maximum independent set of rectangles or two-dimensional knapsack.

In this paper we answer this question in negative by proving that it is NP-hard to approximate strip packing within a factor better than $12/11$, even when admitting only polynomially bounded input data. In particular, this shows that the strip packing problem admits no quasi-polynomial time approximation scheme, unless $\mathrm{NP} \subseteq \mathrm{DTIME}(2^{\mathrm{polylog}(n)})$.

### Categorical Complexity

Authors: Saugata Basu, M. Umut Isik
Abstract: We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest, such as finite sets, Boolean functions, topological spaces, vector spaces, graded algebras, schemes and modules. We show that on one hand categorical complexity recovers in several settings classical notions of non-uniform computational complexity (such as circuit complexity), while on the other hand it has features which make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes.

### Derandomization for k-submodular maximization

Authors: Hiroki Oshima
Abstract: Submodularity is one of the most important property of combinatorial optimization, and $k$-submodularity is a generalization of submodularity. Maximization of $k$-submodular function is NP-hard, and approximation algorithms are studied. For monotone $k$-submodular function, [Iwata, Tanigawa, and Yoshida 2016] gave $k/(2k-1)$-approximation algorithm. In this paper, we give a deterministic algorithm by derandomizing that algorithm. Derandomization scheme is from [Buchbinder and Feldman 2016]. Our algorithm is $k/(2k-1)$-approximation and polynomial-time algorithm.

### Bias-Aware Sketches

Authors: Jiecao Chen, Qin Zhang
Abstract: Count-Sketch \cite{CCFC02} and Count-Median \cite{CM04} are two widely used sketching algorithms for processing large-scale distributed and streaming datasets, such as finding frequent elements, computing frequency moments, performing point queries, etc. % Their popularity is largely due to the fact that linear sketches can be naturally composed in the distributed model and be efficiently updated in the streaming model. Moreover, both the sketching phase and the recovery phase can be parallelized. The errors of Count-Sketch and Count-Median are expressed in terms of the sum of coordinates of the input vector excluding those largest ones, or, the mass on the tail of the vector. Thus, the precondition for these algorithms to perform well is that the mass on the tail is small, which is, however, not always the case -- in many real-world datasets the coordinates of the input vector have a non-zero bias, which will generate a large mass on the tail.

In this paper we propose linear sketches that are {\em bias-aware}. They can be used as substitutes to Count-Sketch and Count-Median, and achieve strictly better error guarantees. We also demonstrate their practicality by an extensive experimental evaluation on both real and synthetic datasets.

### The Markov Memory for Generating Rare Events

Authors: C. Aghamohammadi, J. P. Crutchfield
Abstract: We classify the rare events of structured, memoryful stochastic processes and use this to analyze sequential and parallel generators for these events. Given a stochastic process, we introduce a method to construct a new process whose typical realizations are a given process' rare events. This leads to an expression for the minimum memory required to generate rare events. We then show that the recently discovered classical-quantum ambiguity of simplicity also occurs when comparing the structure of process fluctuations.

### Online and Random-order Load Balancing Simultaneously

Authors: Marco Molinaro
Abstract: We consider the problem of online load balancing under lp-norms: sequential jobs need to be assigned to one of the machines and the goal is to minimize the lp-norm of the machine loads. This generalizes the classical problem of scheduling for makespan minimization (case l_infty) and has been thoroughly studied. However, despite the recent push for beyond worst-case analyses, no such results are known for this problem. In this paper we provide algorithms with simultaneous guarantees for the worst-case model as well as for the random-order (i.e. secretary) model, where an arbitrary set of jobs comes in random order. First, we show that the greedy algorithm (with restart), known to have optimal O(p) worst-case guarantee, also has a (typically) improved random-order guarantee. However, the behavior of this algorithm in the random-order model degrades with p. We then propose algorithm SIMULTANEOUSLB that has simultaneously optimal guarantees (within constants) in both worst-case and random-order models. In particular, the random-order guarantee of SIMULTANEOUSLB improves as p increases.

One of the main components is a new algorithm with improved regret for Online Linear Optimization (OLO) over the non-negative vectors in the lq ball. Interestingly, this OLO algorithm is also used to prove a purely probabilistic inequality that controls the correlations arising in the random-order model, a common source of difficulty for the analysis. Another important component used in both SIMULTANEOUSLB and our OLO algorithm is a smoothing of the lp-norm that may be of independent interest. This smoothness property allows us to see algorithm SIMULTANEOUSLB as essentially a greedy one in the worst-case model and as a primal-dual one in the random-order model, which is instrumental for its simultaneous guarantees.

### My 5-minute quantum computing talk at the White House

from Scott Aaronson

(OK, technically it was in the Eisenhower Executive Office Building, which is not exactly the White House itself, but is adjacent to the West Wing in the White House complex.  And President Obama wasn’t there—maybe, like Justin Trudeau, he already knows everything about quantum computing?  But lots of people from the Office of Science and Technology Policy were!  And some of us talked with Valerie Jarrett, Obama’s adviser, when she passed us on her way to the West Wing.

The occasion was a Quantum Information Science policy workshop that OSTP held, and which the White House explicitly gave us permission to discuss on social media.  Indeed, John Preskill already tweeted photos from the event.  Besides me and Preskill, others in attendance included Umesh Vazirani, Seth Lloyd, Yaoyun Shi, Rob Schoelkopf, Krysta Svore, Hartmut Neven, Stephen Jordan…

I don’t know whether this is the first time that the polynomial hierarchy, or the notion of variation distance, were ever invoked in a speech at the White House.  But in any case, I was proud to receive a box of Hershey Kisses bearing the presidential seal.  I thought of not eating them, but then I got hungry, and realized that I can simply refill the box later if desired.

For regular readers of Shtetl-Optimized, my talk won’t have all that much that’s new, but in any case it’s short.

Incidentally, during the workshop, a guy from OSTP told me that, when he and others at the White House were asked to prepare materials about quantum computing, posts on Shtetl-Optimized (such as Shor I’ll Do It) were a huge help.  Honored though I was to have “served my country,” I winced, thinking about all the puerile doofosities I might’ve self-censored had I had any idea who might read them.  I didn’t dare ask whether anyone at the White House also reads the comment sections!

Thanks so much to all the other participants and to the organizers for a great workshop.  –SA)

Quantum Supremacy

by Scott Aaronson (UT Austin)

October 18, 2016

Thank you; it’s great to be here.  There are lots of directions that excite me enormously right now in quantum computing theory, which is what I work on.  For example, there’s the use of quantum computing to get new insight into classical computation, into condensed matter physics, and recently, even into the black hole information problem.

But since I have five minutes, I wanted to talk here about one particular direction—one that, like nothing else that I know of, bridges theory and experiment in the service of what we hope will be a spectacular result in the near future.  This direction is what’s known as “Quantum Supremacy”—John [Preskill], did you help popularize that term?  [John nods yes]—although some people have been backing away from the term recently, because of the campaign of one of the possible future occupants of this here complex.

But what quantum supremacy means to me, is demonstrating a quantum speedup for some task as confidently as possible.  Notice that I didn’t say a useful task!  I like to say that for me, the #1 application of quantum computing—more than codebreaking, machine learning, or even quantum simulation—is just disproving the people who say quantum computing is impossible!  So, quantum supremacy targets that application.

What is important for quantum supremacy is that we solve a clearly defined problem, with some relationship between inputs and outputs that’s independent of whatever hardware we’re using to solve the problem.  That’s part of why it doesn’t cut it to point to some complicated, hard-to-simulate molecule and say “aha!  quantum supremacy!”

One discovery, which I and others stumbled on 7 or 8 years ago, is that quantum supremacy seems to become much easier to demonstrate if we switch from problems with a single valid output to sampling problems: that is, problems of sampling exactly or approximately from some specified probability distribution.

Doing this has two advantages.  First, we no longer need a full, fault-tolerant quantum computer—in fact, very rudimentary types of quantum hardware appear to suffice.  Second, we can design sampling problems for which we can arguably be more confident that they really are hard for a classical computer, than we are that (say) factoring is classically hard.  I like to say that a fast classical factoring algorithm might collapse the world’s electronic commerce, but as far as we know, it wouldn’t collapse the polynomial hierarchy!  But with sampling problems, at least with exact sampling, we can often show the latter implication, which is about the best evidence you can possibly get for such a problem being hard in the present state of mathematics.

One example of these sampling tasks that we think are classically hard is BosonSampling, which Alex Arkhipov and I proposed in 2011.  BosonSampling uses a bunch of identical photons that are sent through a network of beamsplitters, then measured to count the number of photons in each output mode.  Over the past few years, this proposal has been experimentally demonstrated by quantum optics groups around the world, with the current record being a 6-photon demonstration by the O’Brien group in Bristol, UK.  A second example is the IQP (“Instantaneous Quantum Polynomial-Time”) or Commuting Hamiltonians model of Bremner, Jozsa, and Shepherd.

A third example—no doubt the simplest—is just to sample from the output distribution of a random quantum circuit, let’s say on a 2D square lattice of qubits with nearest-neighbor interactions.  Notably, this last task is one that the Martinis group at Google is working toward achieving right now, with 40-50 qubits.  They say that they’ll achieve it in as little as one or two years, which translated from experimental jargon, means maybe five years?  But not infinity years.

The challenges on the experimental side are clear: get enough qubits with long enough coherence times to achieve this.  But there are also some huge theoretical challenges remaining.

A first is, can we still solve classically hard sampling problems even in the presence of realistic experimental imperfections?  Arkhipov and I already thought about that problem—in particular, about sampling from a distribution that’s merely close in variation distance to the BosonSampling one—and got results that admittedly weren’t as satisfactory as the results for exact sampling.  But I’m delighted to say that, just within the last month or two, there have been some excellent new papers on the arXiv that tackle exactly this question, with both positive and negative results.

A second theoretical challenge is, how do we verify the results of a quantum supremacy experiment?  Note that, as far as we know today, verification could itself require classical exponential time.  But that’s not the showstopper that some people think, since we could target the “sweet spot” of 40-50 qubits, where classical verification is difficult (and in particular, clearly “costlier” than running the experiment itself), but also far from impossible with cluster computing resources.

If I have any policy advice, it’s this: recognize that a clear demonstration of quantum supremacy is at least as big a deal as (say) the discovery of the Higgs boson.  After this scientific milestone is achieved, I predict that the whole discussion of commercial applications of quantum computing will shift to a new plane, much like the Manhattan Project shifted to a new plane after Fermi built his pile under the Chicago stadium in 1942.  In other words: at this point, the most “applied” thing to do might be to set applications aside temporarily, and just achieve this quantum supremacy milestone—i.e., build the quantum computing Fermi pile—and thereby show the world that quantum computing speedups are a reality.  Thank you.

by Scott at October 25, 2016 10:50 PM UTC

### Are there any heuristic-free NP complete problems?

Are there any NP complete problems with no infinite subset of instances $\Phi$ such that membership in $\Phi$ can be decided in polynomial time, and for all $x \in \Phi$, $x$ can be solved in polynomial time? (Assuming $P \neq NP$)

by Phylliida at October 25, 2016 08:54 PM UTC

### Target Set Selection in Dense Graph Classes

Authors: Pavel Dvořák, Dušan Knop, Tomáš Toufar
Abstract: In this paper we study the Target Set Selection problem, a fundamental problem in computational social choice, from a parameterized complexity perspective. Here for a given graph and a threshold for each vertex the task is to find a set of active vertices that activates whole graph. A vertex becomes active if the number of activated vertices in its neighborhood is at least its threshold.

We give two parameterized algorithms for a special case where each vertex has threshold set to half of its neighbors (the so called Majority Target Set Selection problem) for parameterizations by neighborhood diversity and twin cover number of the input graph. From the opposite side we give hardness proof for the Majority Target Set Selection problem when parameterized by (restriction of) the modular-width - a natural generalization of both previous structural parameters. Finally, we give hardness proof for the Target Set Selection problem parameterized by the neighborhood diversity when there is no restriction on the thresholds.

### Dynamic Complexity of the Dyck Reachability

Authors: Patricia Bouyer, Vincent Jugé
Abstract: Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of Dyck reachability problems in directed and undirected graphs, where updates may add or delete edges. We show a strong dichotomy between such problems, based on the size of the Dyck alphabet. Some of them are P-complete (under a strong notion of reduction) while the others lie either in DynFO or in NL.

### An Attempt to Design a Better Algorithm for the Uncapacitated Facility Location Problem

Authors: Haotian Jiang
Abstract: The uncapacitated facility location has always been an important problem due to its connection to operational research and infrastructure planning. Byrka obtained an algorithm that is parametrized by $\gamma$ and proved that it is optimal when $\gamma>1.6774$. He also proved that the algorithm achieved an approximation ratio of 1.50. A later work by Shi Li achieved an approximation factor of 1.488. In this research, we studied these algorithms and several related works. Although we didn't improve upon the algorithm of Shi Li, our work did provide some insight into the problem. We also reframed the problem as a vector game, which provided a framework to design balanced algorithms for this problem.

### On Solving Non-preemptive Mixed-criticality Match-up Scheduling Problem with Two and Three Criticality Levels

Authors: Antonin Novak, Premysl Sucha, Zdenek Hanzalek
Abstract: In this paper, we study an NP-hard problem of a single machine scheduling minimizing the makespan, where the mixed-critical tasks with an uncertain processing time are scheduled. We show the derivation of F-shaped tasks from the probability distribution function of the processing time, then we study the structure of problems with two and three criticality levels for which we propose efficient exact algorithms and we present computational experiments for instances with up to 200 tasks. Moreover, we show that the considered problem is approximable within a constant multiplicative factor.

### Molecular solutions for the Maximum K-colourable Sub graph Problem in Adleman-Lipton model

Authors: Akbar Moazzam, Babak Dalvand
Abstract: Adleman showed that deoxyribonucleic acid DNA strands could be employed towards calculating solutions to an instance of the Hamiltonian path problem . Lipton also demonstrated that Adleman techniques could be used to solve the Satisfiability problem. In this paper, we use Adleman Lipton model for developing a DNA algorithm to solve Maximum k-colourable Sub graph problem. In spite of the NP-hardness of Maximum k-colourable Sub graph problem our DNA procedures is done in a polynomial time.

### Power of one non-clean qubit

Authors: Tomoyuki Morimae, Keisuke Fujii, Harumichi Nishimura
Abstract: The one-clean qubit model (or the DQC1 model) is a restricted model of quantum computing where only a single qubit of the initial state is pure and others are maximally mixed. Although the model is not universal, it can efficiently solve several problems whose classical efficient solutions are not known. Furthermore, it was recently shown that if the one-clean qubit model is classically efficiently simulated, the polynomial hierarchy collapses to the second level. A disadvantage of the one-clean qubit model is, however, that the clean qubit is too clean: for example, in realistic NMR experiments, polarizations are not enough high to have the perfectly pure qubit. In this paper, we consider a more realistic one-clean qubit model, where the clean qubit is not clean, but depolarized. We first show that, for any polarization, a multiplicative-error calculation of the output probability distribution of the model is possible in a classical polynomial time if we take an appropriately large multiplicative error. The result is in a strong contrast to that of the ideal one-clean qubit model where the classical efficient multiplicative-error calculation (or even the sampling) with the same amount of error causes the collapse of the polynomial hierarchy. We next show that, for any polarization lower-bounded by an inverse polynomial, a classical efficient sampling (in terms of a sufficiently small multiplicative error or an exponentially-small additive error) of the output probability distribution of the model is impossible unless BQP is contained in the second level of the polynomial hierarchy, which suggests the hardness of the classical efficient simulation of the one non-clean qubit model.

### Output-sensitive Complexity of Multiobjective Combinatorial Optimization

Authors: Fritz Bökler, Matthias Ehrgott, Christopher Morris, Petra Mutzel
Abstract: We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, i.e., its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of MOCO problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions.

### The Security of Hardware-Based Omega(n^2) Cryptographic One-Way Functions: Beyond Satisfiability and P=NP

Authors: Javier A. Arroyo-Figueroa
Abstract: We present a class of hardware-based cryptographic one-way functions that, in practice, would be hard to invert even if P=NP and linear-time satisfiability algorithms exist. Such functions use a hardware-based component with omega(n^2) size circuits, and omega(n^2) run time.

### Not All Multi-Valued Partial CFL Functions Are Refined by Single-Valued Functions

Authors: Tomoyuki Yamakami
Abstract: Multi-valued partial CFL functions are computed by one-way nondeterministic pushdown automata equipped with write-only output tapes. We give an answer to a fundamental question, raised by Konstantinidis, Santean, and Yu [Act. Inform. 43 (2007) 395-417], of whether all multi-valued partial CFL functions can be refined by single-valued partial CFL functions. We negatively solve this question by presenting a special multi-valued partial CFL function as an example function and by proving that no refinement of this particular function becomes a single-valued partial CFL function. This contrasts an early result of Kobayashi [Inform. Control 15 (1969) 95-109] that multi-valued partial NFA functions are always refined by single-valued NFA functions, where NFA functions are computed by nondeterministic finite automata with output tapes. Our example function turns out to be unambiguously 2-valued, and thus we obtain a stronger separation result, in which no refinement of unambiguously 2-valued partial CFL functions can be single-valued. For the proof, we first introduce a new concept of colored automata having no output tapes but having "colors," which can simulate pushdown automata with constant-space output tapes. We then conduct an extensive combinatorial analysis on the behaviors of transition records of stack contents (called stack histories) of colored automata.

### Bounded embeddings of graphs in the plane

Abstract: A drawing in the plane ($\mathbb{R}^2$) of a graph $G=(V,E)$ equipped with a function $\gamma: V \rightarrow \mathbb{N}$ is \emph{$x$-bounded} if (i) $x(u) <x(v)$ whenever $\gamma(u)<\gamma(v)$ and (ii) $\gamma(u)\leq\gamma(w)\leq \gamma(v)$, where $uv\in E$ and $\gamma(u)\leq \gamma(v)$, whenever $x(w)\in x(uv)$, where $x(.)$ denotes the projection to the $x$-axis. We prove a characterization of isotopy classes of graph embeddings in the plane containing an $x$-bounded embedding. Then we present an efficient algorithm, that relies on our result, for testing the existence of an $x$-bounded embedding if the given graph is a tree or generalized $\Theta$-graph. This partially answers a question raised recently by Angelini et al. and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable if each connected component of the underlying abstract graph is a tree.

### Local Maxima and Improved Exact Algorithm for MAX-2-SAT

Authors: M. B. Hastings
Abstract: Given a MAX-2-SAT instance, we define a local maximum to be an assignment such that changing any single variable reduces the number of satisfied clauses. We consider the question of the number of local maxima that an instance of MAX-2-SAT can have. We give upper bounds in both the sparse and nonsparse case, where the sparse case means that there is a bound $d$ on the average number of clauses involving any given variable. The bounds in the nonsparse case are tight up to polylogarithmic factors, while in the sparse case the bounds are tight up to a multiplicative factor in $d$ for large $d$. Additionally, we generalize to the question of assignments which are maxima up to changing $k> 1$ variables simultaneously; in this case, we give explicit constructions with large (in a sense explained below) numbers of such maxima in the sparse case. The basic idea of the upper bound proof is to consider a random assignment to some subset of the variables and determine the probability that some fraction of the remaining variables can be fixed without considering interactions between them. The bounded results hold in the case of weighted MAX-2-SAT as well. Using this technique and combining with ideas from Ref. 6, we find an algorithm for weighted MAX-2-SAT which is faster for large $d$ than previous algorithms which use polynomial space; this algorithm does require an additional bounds on maximum weights and degree.

### p-Causality: Identifying Spatiotemporal Causal Pathways for Air Pollutants with Urban Big Data

Authors: Julie Yixuan Zhu, Chao Zhang, Shi Zhi, Victor O. K. Li, Jiawei Han, Yu Zheng
Abstract: Many countries are suffering from severe air pollution. Understanding how different air pollutants accumulate and propagate is critical to making relevant public policies. In this paper, we use urban big data (air quality data and meteorological data) to identify the \emph{spatiotemporal (ST) causal pathways} for air pollutants. This problem is challenging because: (1) there are numerous noisy and low-pollution periods in the raw air quality data, which may lead to unreliable causality analysis, (2) for large-scale data in the ST space, the computational complexity of constructing a causal structure is very high, and (3) the \emph{ST causal pathways} are complex due to the interactions of multiple pollutants and the influence of environmental factors. Therefore, we present \emph{p-Causality}, a novel pattern-aided causality analysis approach that combines the strengths of \emph{pattern mining} and \emph{Bayesian learning} to efficiently and faithfully identify the \emph{ST causal pathways}. First, \emph{Pattern mining} helps suppress the noise by capturing frequent evolving patterns (FEPs) of each monitoring sensor, and greatly reduce the complexity by selecting the pattern-matched sensors as "causers". Then, \emph{Bayesian learning} carefully encodes the local and ST causal relations with a Gaussian Bayesian network (GBN)-based graphical model, which also integrates environmental influences to minimize biases in the final results. We evaluate our approach with three real-world data sets containing 982 air quality sensors, in three regions of China from 01-Jun-2013 to 19-Dec-2015. Results show that our approach outperforms the traditional causal structure learning methods in time efficiency, inference accuracy and interpretability.

### P_3-Games on Chordal Bipartite Graphs

Authors: Wing-Kai Hon, Ton Kloks, Fu-Hong Liu, Hsiang-Hsuan Liu, Tao-Ming Wang, Yue-Li Wang
Abstract: Let G=(V,E) be a connected graph. A set U subseteq V is convex if G[U] is connected and all vertices of V\U have at most one neighbor in U. Let sigma(W) denote the unique smallest convex set that contains W subseteq V. Two players play the following game. Consider a convex set U and call it the `playground.' Initially, U = emptyset. When U=V, the player to move loses the game. Otherwise, that player chooses a vertex x in V\U which is at distance at most two from U. The effect of the move is that the playground U changes into sigma(U cup {x}) and the opponent is presented with this new playground.

A graph is chordal bipartite if it is bipartite and has no induced cycle of length more than four. In this paper we show that, when G is chordal bipartite, there is a polynomial-time algorithm that computes the Grundy number of the P_3-game played on G. This implies that there is an efficient algorithm to decide whether the first player has a winning strategy.

### A Noisy-Influence Regularity Lemma for Boolean Functions

Authors: Chris Jones
Abstract: We present a regularity lemma for Boolean functions $f:\{-1,1\}^n \to \{-1,1\}$ based on noisy influence, a measure of how locally correlated $f$ is with each input bit. We provide an application of the regularity lemma to weaken the conditions on the Majority is Stablest Theorem. We also prove a "homogenized" version stating that there is a set of input bits so that most restrictions of $f$ on those bits have small noisy influences. These results were sketched out by [OSTW10], but never published. With their permission, we present the full details here.

### The K Shortest Paths Problem with Application to Routing

Authors: David Burstein, Leigh Metcalf
Abstract: We present a simple algorithm for explicitly computing all k shortest paths bounded by length L from a fixed source to a target in O(m + kL) and O(mlogm + kL) time for unweighted and weighted directed graphs with m edges respectively. For many graphs, this outperforms existing algorithms by exploiting the fact that real world networks have short average path length. Consequently, we would like to adapt our almost shortest paths algorithm to find an efficient solution to the almost short- est simple paths, where we exclude paths that visit any node more than once. To this end, we consider realizations from the Chung-Lu random graph model as the Chung-Lu random graph model is not only amenable to analysis, but also emulates many of the properties frequently observed in real world networks including the small world phenomenon and degree heterogeneity. We provide theoretical and numeric evidence regarding the efficiency of utilizing our almost shortest paths algorithm to find al- most shortest simple paths for Chung-Lu random graphs for a wide range of parameters. Finally, we consider a special application of our almost shortest paths algorithm to study internet routing (withdrawals) in the Autonomous System graph.

### Exaggeration is one thing but this is....

This website is about the history of math and lists famous mathematicians. The ones from the 20th century are biased towards logic, but you should go there yourself and see who you think they left out.

There entry on Paul Cohen is... odd. Its here. I quote from it:

-------------------------------------
His findings were as revolutionary as Gödel’s own. Since that time, mathematicians have built up two different mathematical worlds, one in which the continuum hypothesis applies and one in which it does not, and modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.
-------------------------------------

When was the last time you had to put into the premise of a theorem CH or NOT-CH?
I did once here in an expository work about a problem in combinatorics that was ind of set theory since it was equivalent to CH. Before you get too excited it was a problem in infinite combinatorics having to do with coloring the reals.

I suspect that at least 99% of my readers have never had to insert a  note in a paper about if they were assuming CH or NOT-CH. If you have I'd love to hear about it in the comments. And I suspect
you are a set theorist.

Paul Cohen's work was very important--- here we have an open problem in math that will always be open (thats one interpretation). And there will sometimes be other problems that are ind of ZFC or equiv to CH or something like that. But it does not affect the typical mathematician working on a typical problem.

I have a sense that its bad to exaggerate like this. One reason would be that if the reader finds out
the truth he or she will be disillusioned. But somehow, that doesn't seem to apply here. So I leave it to the reader to comment:  Is it  bad to exaggerate Paul Cohen's (or anyone's) accomplishments? And if so,
then why?

by GASARCH (noreply@blogger.com) at October 24, 2016 10:20 PM UTC