Lecture Notes in Computer Science 10846

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Takeo KanadeCarnegie Mellon University, Pittsburgh, PA, USA

Josef KittlerUniversity of Surrey, Guildford, UK

Jon M. KleinbergCornell University, Ithaca, NY, USA

Friedemann MatternETH Zurich, Zurich, Switzerland

John C. MitchellStanford University, Stanford, CA, USA

Moni NaorWeizmann Institute of Science, Rehovot, Israel

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Bernhard SteffenTU Dortmund University, Dortmund, Germany

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Gerhard WeikumMax Planck Institute for Informatics, Saarbrücken, Germany

More information about this series at http://www.springer.com/series/7407

Fedor V. Fomin • Vladimir V. Podolskii (Eds.)

Computer Science –

Theory and Applications13th International Computer Science Symposium in Russia, CSR 2018Moscow, Russia, June 6–10, 2018Proceedings

123

EditorsFedor V. FominDepartment of InformaticsUniversity of BergenBergenNorway

Vladimir V. PodolskiiSteklov Mathematical InstituteMoscowRussia

ISSN 0302-9743 ISSN 1611-3349 (electronic)Lecture Notes in Computer ScienceISBN 978-3-319-90529-7 ISBN 978-3-319-90530-3 (eBook)https://doi.org/10.1007/978-3-319-90530-3

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Preface

The 13th International Computer Science Symposium in Russia (CSR 2018) was heldduring June 6–10, 2018, in Moscow, Russia. The symposium was organized by theHigher School of Economics. It was the 13th event in the CSR series of regularinternational meetings, following St. Petersburg (2006), Ekaterinburg (2007), Moscow(2008), Novosibirsk (2009), Kazan (2010), St. Petersburg (2011), Nizhny Novgorod(2012), Ekaterinburg (2013), Moscow (2014), Listvyanka (2015), St. Petersburg(2011), and Kazan (2017). CSR covers a wide range of areas in theoretical computerscience and its applications.

The opening lecture was given by Noga Alon (Tel Aviv University and Princeton).Seven other invited plenary lectures were given by Markus Bläser (Saarland Univer-sity), Vladimir Gurvich (Rutgers University), Alexander Kulikov (St. PetersburgDepartment of Steklov Institute of Mathematics), Kurt Mehlhorn (Max-Planck-Institutfür Informatik), Michael Saks (Rutgers University), Rahul Santhanam (University ofOxford), and László A. Végh (London School of Economics).

This volume contains the accepted papers and those sent by the invited speakers. Wereceived 42 submissions in total, and out of these the Program Committee selected 24papers for presentation at the symposium and for publication in the proceedings. Eachsubmission was reviewed by at least three Program Committee members. The ProgramCommittee also selected the winners of the two Yandex Best Paper Awards.

The Best Paper Award: Jayakrishnan Madathil, Saket Saurabh, andMeirav Zehavi, “Max-Cut Above Spanning Tree Is Fixed Parameter Tractable”

The Best Student Paper Award: Alexander Kozachinskiy, “RecognizingRead-Once Functions from Depth-Three Formulas”

Many people and organizations contributed to the smooth running and the successof CSR 2016. In particular our thanks go to:

– All authors who submitted their current research to CSR– All invited speakers who agreed to give a talk at the conference– Our reviewers and additional reviewers, whose expertise flowed into the decision

process– The members of the Program Committee, who graciously gave their time and

energy– The members of the local Organizing Committee, who made the conference

possible– The EasyChair conference management system for hosting the evaluation process– The Higher School of Economics for hosting the conference– Yandex for supporting the conference and providing Best Paper Awards– The European Association for Theoretical Computer Science (EATCS) for the

scientific support of the conference

March 2018 Fedor V. FominVladimir Podolskii

Organization

Program Committee

Maxim Babenko Moscow State University, RussiaMikołaj Bojńaczyk Warsaw University, PolandHolger Dell Saarland University and Cluster of Excellence, MMCI,

GermanyEdith Elkind University of Oxford, UKFedor Fomin University of Bergen, NorwayFabrizio Grandoni IDSIA, University of Lugano, SwitzerlandDmitry Itsykson Steklov Institute of Mathematics at St. Petersburg, RussiaMikko Koivisto University of Helsinki, FinlandAntonina Kolokolova Memorial University of Newfoundland, CanadaAlexander Kulikov Steklov Mathematical Institute at St. Petersburg, RussiaJakob Nordström KTH Royal Institute of Technology, SwedenAlexander Okhotin St. Petersburg State University, RussiaVladimir Podolskii Steklov Mathematical Institute, RussiaIlya Razenshteyn Microsoft Research Redmond, USASaket Saurabh The Institute of Mathematical Sciences, Chennai, IndiaAlexander Shen LIRMM CNRS and University Montpellier 2, France, on

leave from IITP RAS, MoscowArseny Shur Ural Federal University, RussiaDirk Oliver Theis University of Tartu, EstoniaRené van Bevern Novosibirsk State University, RussiaMeirav Zehavi Ben-Gurion University, Israel

Additional Reviewers

Aganezov, SergeyAghighi, MeysamAkhmedov, MaximBliznets, IvanBoiret, AdrienDe Oliveira Oliveira,

Mateusde Rezende, Susanna F.Dey, PalashGerasimov, AlexanderGlinskih, LudmilaGolovach, Petr

Golovnev, AlexanderGuillon, BrunoGálvez, WaldoGöös, MikaHermelin, DannyIskhakov, TimurJohannsen, JanKalenkova, AnnaKarpov, NikolaiKarthik, C. S.Kaski, PetteriKnop, Alexander

Kolesnichenko, IgnatKorhonen, Janne H.Kosolobov, DmitryKuske, DietrichKutrib, MartinKuznetsov, StepanLasota, SławomirM. S., RamanujanMukhopadhyay, SagnikNasre, MeghanaNichterlein, AndréNikulin, Yury

Oparin, VsevolodPettie, SethPyatkin, ArtemRadhakrishnan, JaikumarRajan, VarunRaskin, MikhailReinhardt, KlausRisse, KilianRobere, Robert

Romashchenko, AndreiRoth, MarcSalomaa, KaiSavchenko, RuslanSchaeffer, LukeScharf, NadjaScquizzato, MicheleSegev, DannySeidl, Helmut

Shapoval, AlexanderSokolov, DmitrySperanski, Stanislav O.Stephan, FrankSuchy, OndrejSwernofsky, JosephTörmä, IlkkaVyalyi, Mikhail

VIII Organization

Abstract of Invited Talks

Constructive and Non-constructiveCombinatorics

Noga Alon1,2

1 Princeton University, Princeton, NJ 08544, USA2 Tel Aviv University, Tel Aviv, 69978, Israel

nalon@math.princeton.edu

Purely combinatorial proofs of combinatorial statements usually provide efficientprocedures for solving the corresponding algorithmic problems, even if they deal withNP-hard invariants. One representative classical example out of many is Dirac’sTheorem that asserts that any simple graph with n� 3 vertices and minimum degree atleast n/2 contains a Hamilton cycle. The proof supplies a simple polynomial timealgorithm for finding a Hamilton cycle in any such graph, although the problem ofdeciding whether or not a given input graph contains a Hamilton cycle is NP-hard.Similar examples include Turán’s Theorem, Vizing’s Theorem or the Four ColorTheorem.

Modern combinatorics often applies more sophisticated, non-combinatorial tools,including probabilistic, topological or algebraic techniques. Probabilistic proofs usuallysupply efficient (deterministic or randomized) algorithms, and in many cases the ran-domized algorithms can be derandomized and converted into deterministic ones. See[5, 8, 10] for some prominent examples obtained during the last decade. In contrast,proofs based on topological and algebraic reasoning are often non-constructive andprovide no efficient solutions for the corresponding algorithmic problems. Finding suchsolutions is an intriguing challenge. In the lecture I will describe several old and newexamples of this type. A representative example is the following, known as the Cycleand Triangles question. While it may look somewhat special, I have chosen it as it canbe solved either by applying topological techniques or by using algebraic tools, and yetthere is neither a known combinatorial solution nor a known algorithmic one.

Du, Hsu and Wang [6] conjectured that if a graph on 3n vertices is the edge disjointunion of a Hamilton cycle of length 3n and n vertex disjoint triangles then its inde-pendence number is n. Erdős conjectured that in fact any such graph is 3 colorable.Using the algebraic approach in [4], Fleischner and Stiebitz [7] proved this conjecturein a stronger form - any such graph is in fact 3-list-colorable, namely, for everyassignment of a list of 3 colors for each vertex, there is a proper coloring assigning toeach vertex a color from its list.

As proved in a recent paper [1] the original conjecture, in a slightly stronger form,can be derived quickly from a result of Schrijver about critical subgraphs of theKneser graph [11]. Strengthening Lovász theorem he proved, using the Borsuk-Ulam

Research supported in part by an ISF grant and by a GIF grant.

Theorem, that in any coloring of the independent sets of vertices of size k in a cycle oflength n by less than n� 2kþ 2 colors there are two disjoint independent sets havingthe same color. This supplies a short proof of the following statement: LetC3n ¼ ðV ;EÞ be cycle of length 3n and let V ¼ A1 [A2 [ . . .[An be a partition of itsvertex set into n pairwise disjoint sets, each of size 3. Then there exist two disjointindependent sets in the cycle, each containing one point from each Ai.

Here is a proof. Define a coloring of the independent sets of size n in C3n asfollows. If S is such an independent set and there is an index i so that jS\Aij � 2, colorS by the smallest such i. Otherwise, color S by the color nþ 1. By the above result ofSchrijver there are two disjoint independent sets S1; S2 with the same color. This colorcannot be any i� n, since if this is the case then

jðS1 [ S2Þ \Aij ¼ jS1 \Aij þ jS2 \Aij � 2þ 2 ¼ 4[ 3 ¼ jAij;which is impossible. Thus S1 and S2 are both colored nþ 1, meaning that each of themcontains exactly one element of each Ai. h

Several extensions are proved in [1]. The Fleischner-Stiebitz theorem implies thatthe representing set in the Du-Hsu-Huang conjecture can be required to contain anygiven vertex. This can also be deduced from the topological version of Hall’s Theo-rem of Aharoni and Haxell, as shown in [2]. None of the above proofs supplies anefficient algorithm for finding the desired independent set.

In the lecture I will describe several additional non-constructive proofs of combi-natorial statements proved by applying the Borsuk-Ulam theorem, as well as additionalalgebraic proofs that supply no efficient algorithms. Certain results of this type can befound in [3, 9], and I will mention a few more recent examples. I will also discuss thereasons that suggest that the derivation of constructive proofs may be hard.

References

1. Aharoni, R., et al.: Fair representation by independent sets. In: Loebl, M., Nešetřil, J.,Thomas, R. (eds.) A Journey Through Discrete Mathematics, pp. 31–58. Springer, Cham(2017)

2. Aharoni, R., Holzman, R., Howard, D., Sprüsell, P.: Cooperative colorings and systems ofindependent representatives. Electron. J. Combin. 22(2), 14 (2015). Paper 2.27

3. Alon, N.: Discrete mathematics: methods and challenges. In: Proceedings of the Interna-tional Congress of Mathematicians (ICM), Beijing 2002, China, pp. 119–135. HigherEducation Press (2003)

4. Alon, N., Tarsi, M.: Chromatic numbers and orientations of graphs. Combinatorica 12,125–134 (1992)

5. Bansal, N.: Constructive algorithms for discrepancy minimization. In: Proceedings of 2010IEEE FOCS, pp. 3–10 (2010)

6. Du, D.Z., Hsu, D.F., Hwang, F.K.: The Hamiltonian property of consecutive-d digraphs, ingraph-theoretic models in computer science, II (Las Cruces, NM, 1988. 1990). Math.Comput. Model. 17, 61–63 (1993)

7. Fleischner, H., Stiebitz, M.: A solution to a colouring problem of P. Erdős. Discrete Math.101, 39–48 (1992)

XII N. Alon

8. Lovett, S., Meka, R.: Constructive discrepancy minimization by walking on the edges. In:Proceedings of 2012 IEEE FOCS, pp. 61–67 (2012)

9. Matoušek, J.: Using the Borsuk-Ulam theorem. In: Lectures on Topological Methods inCombinatorics and Geometry, Springer, Heidelberg (2003)

10. Moser, R.A., Tardos, G.: A constructive proof of the general Lovász local lemma. J. ACM57, 15 (2010). Art. 11

11. Schrijver, A.: Vertex-critical subgraphs of Kneser graphs. Nieuw Arch. Wisk. 26, 454–461(1978)

Constructive and Non-constructive Combinatorics XIII

Physarum Solves Non-negative UndirectedLinear Programs

Ruben Becker1, Vincenzo Bonifaci2, Andreas Karrenbauer1,Pavel Kolev1, and Kurt Mehlhorn1

1 Max Planck Institute for Informatics, Saarland Informatics Campus,Saarbrücken, Germany

2 Institute for the Analysis of Systems and Informatics,National Research Council of Italy (IASI-CNR), Rome, Italy

Physarum polycephalum is a slime mold that apparently is able to solve shortest pathproblems. Nakagaki, Yamada, and Tóth [1] report about the following experiment;see Fig. 1. They built a maze, covered it by pieces of Physarum (the slime can be cutinto pieces which will reunite if brought into vicinity), and then fed the slime withoatmeal at two locations. After a few hours the slime retracted to a path that followsthe shortest path in the maze connecting the food sources. The authors report thatthey repeated the experiment with different mazes; in all experiments, Physarumretracted to the shortest path.

The paper [3] proposes a mathematical model for the behavior of the slime andargues extensively that the model is adequate. Physarum is modeled as an electricalnetwork with time varying resistors. We have a simple undirected graph G ¼ ðN;EÞwith distinguished nodes s0 and s1 modeling the food sources. Each edge e 2 E has apositive length ce and a positive capacity xeðtÞ; ce is fixed, but xeðtÞ is a function oftime. The resistance reðtÞ of e is reðtÞ ¼ ce=xeðtÞ. In the electrical network defined bythese resistances, a current of value 1 is forced from s0 to s1. For an (arbitrarilyoriented) edge e ¼ ðu; vÞ, let qeðtÞ be the resulting current over e. Then, the capacity ofe evolves according to the differential equation

Fig. 1. The experiment in [1] (reprinted from there): (a) shows the maze uniformly covered byPhysarum; yellow color indicates presence of Physarum. Food (oatmeal) is provided at the locationslabeled AG. After a while the mold retracts to the shortest path connecting the food sources as shownin (b) and (c). (d) shows the underlying abstract graph. The video [2] shows the experiment.

_xeðtÞ ¼ jqeðtÞj � xeðtÞ; ð1Þ

where _xe is the derivative of xe with respect to time. In equilibrium ( _xe ¼ 0 for all e), theflow through any edge is equal to its capacity. In non-equilibrium, the capacity grows(shrinks) if the absolute value of the flow is larger (smaller) than the capacity. In thesequel, we will mostly drop the argument t as is customary in the treatment ofdynamical systems. It is well-known that the electrical flow q is the feasible flowminimizing energy dissipation

Pe req

2e (Thomson’s principle).

Miyaji and Ohnishi were the first to analyze convergence for special graphs (par-allel links and planar graphs with source and sink on the same face) in [4]. In [5]convergence was proven for all graphs. We state the result from [5] for the special casethat the shortest path is unique.

Theorem 1 ([5]). Assume c[ 0 and that the undirected shortest path P� from s0 to s1w.r.t. the cost vector c is unique. Assume xð0Þ[ 0. Then x(t) in (1) converges to P�.Namely, xeðtÞ ! 1 for e 2 P� and xe ! 0 for e 62 P� as t ! 1.

[5] also proves an analogous result for the undirected transportation problem; [6]simplified the argument under additional assumptions. The paper [7] studies a moregeneral dynamics and proves convergence for parallel links.

In this paper1, we extend this result to non-negative undirected linear programs

minfcTx : Af ¼ b; jf j � xg; ð2Þ

where A 2 IRn�m, b 2 IRn, x 2 IRm, c 2 IRm� 0, and the absolute values are taken

componentwise. Undirected LPs can model a wide range of problems, e.g. optimizationproblems such as shortest path and min-cost flow in undirected graphs, and the BasisPursuit problem in signal processing [9].

We use n for the number of rows of A and m for the number of columns, since thisnotation is appropriate when A is the node-edge-incidence matrix of a graph. A vectorf is feasible if Af ¼ b. We assume that the system Af ¼ b has a feasible solution andthat there is no non-zero f in the kernel of A with cefe ¼ 0 for all e. A vector f lies in thekernel of A if Af ¼ 0. The vector q in (1) is now the minimum energy feasible solution

qðtÞ ¼ argminf2IRm

Xe:xe 6¼0

cexeðtÞ f

2e : Af ¼ b ^ fe ¼ 0 whenever xe ¼ 0

( ): ð3Þ

We remark that q is unique. If A is the incidence matrix of a graph (the columncorresponding to an edge e has one entry þ 1, one entry �1 and all other entries areequal to zero), (2) is a transshipment problem with flow sources and sinks encoded by ademand vector b. The condition that there is no solution in the kernel of A with cefe ¼ 0for all e states that every cycle contains at least one edge of positive cost. In that setting,

1 The full paper [8] is available on the arxiv.

Physarum Solves Non-negative Undirected Linear Programs XV

q(t) as defined by (3) coincides with the electrical flow induced by resistors of valuece=xeðtÞ.Theorem 2. Let c� 0 satisfy cT jf j[ 0 for every nonzero f in the kernel of A. Let x� bean optimum solution of (2) and let XI be the set of optimum solutions. Assumexð0Þ[ 0. The following holds for the dynamics (1) with q as in (3):

(i) The solution x(t) exists for all t� 0.(ii) The cost cTxðtÞ converges to cTx� as t goes to infinity.(iii) The vector x(t) converges to XI.(iv) For all e with ce [ 0, xeðtÞ � jqeðtÞj converges to zero as t goes to infinity.2 If x�

is unique, x(t) and q(t) converge to x� as t goes to infinity.

Item (i) was previously shown in [10] for the case of a strictly positive cost vector.The result in [10] is stated for the cost vector c ¼ 1. The case of a general positive costvector reduces to this special case by rescaling the solution vector x. We stress that thedynamics (1) is biologically-grounded. It was proposed to model a biological systemand not as an optimization method. Nevertheless, it can solve a large class ofnon-negative LPs. Table 1 summarizes our first main result and puts it into context.

References

1. Nakagaki, T., Yamada, H., Tóth, A.: Maze-solving by an amoeboid organism. Nature 407,470 (2000)

2. http://people.mpi-inf.mpg.de/*mehlhorn/ftp/SlimeAusschnitt.webm3. Tero, A., Kobayashi, R., Nakagaki, T.: A mathematical model for adaptive transport network

in path finding by true slime mold. J. Theor. Biol. 553–564 (2007)4. Miyaji, T., Ohnishi, I.: Physarum can solve the shortest path problem on Riemannian surface

mathematically rigourously. Int. J. Pure Appl. Math. 47, 353–369 (2008)5. Bonifaci, V., Mehlhorn, K., Varma, G.: Physarum can compute shortest paths. J. Theor.

Biol. 309, 121–133 (2012)

Table 1. Convergence results for the continuous undirected Physarum dynamics (1).

Reference Problem Existence ofsolution

Convergenceto OPT

Comments

[4] Undirected shortestpath

Yes Yes Parallel edges, planargraphs

[5] Undirected shortestpath

Yes Yes All graphs

[10] Undirected positiveLP

Yes No c� 0

Our Result UndirectedNonnegative LP

Yes Yes (1) c� 0

(2) 8v 2 kerðAÞ : cT jvj[ 0

XVI R. Becker et al.

2 We conjecture that this also holds for the edges e with ce = 0.

6. Bonifaci, V.: Physarum can compute shortest paths: a short proof. Inf. Process. Lett. 113,4–7 (2013)

7. Bonifaci, V.: A revised model of network transport optimization in Physarum Polycephalum(2015)

8. Becker, R., Bonifaci, V., Karrenbauer, A., Kolev, P., Mehlhorn, K.: Two results on slimemold computations. Technical report (2017). https://arxiv.org/abs/1707.06631

9. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit.SIAM J. Sci. Comput. 20, 33–61 (1998)

10. Straszak, D., Vishnoi, N.K.: IRLS and slime mold: equivalence and convergence (2016).CoRR, abs/1601.02712

Physarum Solves Non-negative Undirected Linear Programs XVII

Limitations of Algebraic Lower Bound Proofs

Markus Bläser

Saarland Universitymblaeser@cs.uni-saarland.de

Abstract. Algebraic natural proofs were recently introduced by Forbes et al.[FSV17] and independently by Grochow et al. [GKSS17]. Assume we are givensome polynomial of which we think that it is hard to compute, say the per-manent pern 2 K½X1;1; . . .;Xn;n� of size n� n. pern is a multilinear polynomial ofdegree n in n2 variables. The space of all such polynomials has dimensionn2

n

� �. For such a polynomial f, let cf denote its coefficient vector, which has

length n2

n

� �. An algebraic proof that the permanent is hard is a polynomial

P that vanishes on all polynomials of low complexity but not on the permanent,that is, Pðcf Þ ¼ 0 for all f as above of low complexity but PðcpernÞ 6¼ 0. What isthe complexity of such a P? If such a P has high complexity, then this meansthat proving a circuit lower bound for pern is hard.

For Boolean circuit complexity, Razborov and Rudich [RR97] introducedso-called natural proofs. The objects they consider are truth tables of Booleanfunctions (instead of coefficient vectors). A natural proof P is a predicate on theset of truth tables that has two properties: The first one is largeness, that is, P istrue for a sufficiently large fraction of all Boolean functions. The second one isconstructivity, that is, P is computable by small circuits. Razborov and Rudichsfamous barrier result states that natural proofs can only yield superpolynomialbounds if certain pseudorandom generators do not exist.In the algebraic setting, largeness comes for free, the zero set of any nonzero

polynomial is small in the sense that almost all inputs do not lie in the zero set.We can now ask the question whether there is some sort of barrier in thealgebraic world, too, or could it be that there is a polynomial P that is easy tocompute, vanishes on all coefficient vectors of polynomials of low complexity,but PðcpernÞ 6¼ 0Unfortunately, there is no known analogous theory of pseudorandomness in

the algebraic setting. Therefore, Forbes et al. use a concept called succincthitting sets instead. This assumption is related to polynomial identity testing, butit is currently not clear how plausible this assumption is. Forbes et al. are onlyable to construct succinct hitting sets against rather weak models of arithmeticcircuits.Let S�K½X1; . . .;Xn� be a set of polynomials.H�Kn is called a hitting set for S,

if for all p 2 S, there is an x 2 H such that pðxÞ 6¼ 0. If P is a polynomial thatvanishes on all cofficient vectors cf of polynomials f of low complexity and P hasitself low complexity, then this can be interpreted as follows: Polynomials of lowcomplexity do not have simple hitting sets. This is the main idea behind theconcept of succinct hitting sets.

There is one further complication: If a polynomial vanishes on a particular set,it also vanishes on the Zariski closure of this set. So an algebraic proof againstsome class S will vanish on polynomials f that are not contained in S, but arecontained in the closure S. Polynomials in the border S n S have higher com-plexity than polynomials in S (otherwise, they would be in S), yet they cannot bedistinguished by an algebraic proof from polynomials in S, independently of anybarrier. Therefore, to study algebraic proofs properly, one needs to look at Zariskiclosed classes of polynomials.In the setting above, the proof polynomial P has much more variables than the

polynomials f. There are also settings were these numbers are polynomiallyrelated, tensors are such an example. One can think of a tensor as a“three-dimensional matrix” t ¼ ðth;i;jÞ 2 K‘�m�n :¼ K‘ Km Kn. A rank-onetensor is a tensor of the form u v w with u 2 K‘, v 2 Km and w 2 Kn. Therank R(t) of t is the smallest number r of rank-one tensors s1; . . .; sr such thatt ¼ s1 þ s2 þ . . .þ sr . Let Sr ¼ fs 2 K‘ Km Kn j RðsÞ� rg be the set of alltensors of rank at most r. An algebraic proof that RðtÞ[ r is a polynomial P in‘mn variables such that P vanishes on Sr and PðtÞ 6¼ 0. However, the set Sr is notZariski-closed. That is, it is not the vanishing set of a set of polynomials. So welook at the Zariski closure Xr of Sr instead. These tensors are called the tensors ofborder rank � r. As seen above, the appropriate quantity to study when con-sidering algebraic proofs is the border rank.Given a tensor t and a bound b, it is NP-hard to decide whether RðTÞ� b as

shown by Håstad [Hås90], however, it is not known whether this holds for theborder rank. We define a similar quantity, which we call (border) completionrank, and proof that completion rank and even border completion rank are NP-hard to compute. Next we construct a small family of tensors (small means thatthey come from a closed, even low dimensional set) such that not all of thesetensors can have algebraic proofs of polynomial size against the set of all tensorsof completion rank � r for some appropriately chosen r. This means that there isa tensor t such that any polynomial P with PðtÞ 6¼ 0 that vanishes on all tensor ofcompletion rank r has superpolynomial circuit complexity. This result if ofcourse conditional, but it is based on the widely believed assumption thatcoNP 6�9BPP. One can view this as a meta-result: Proving lower bounds viaalgebraic proofs is difficult. At least, if we want to represent the proof by analgebraic circuit.Even the geometric complexity approach initiated by Mulmuley and Sohoni

[MS01] eventually produces an algebraic proof. However, it is produced fromsome intermediate representation, which can be more compact. We provide onesuch example: We show, of course conditionally, that matrices with nonzeropermanent cannot have small algebraic proofs. However, geometric complexitytheory provides very short proofs for them.(Parts of the presentation are joint work with Christian Ikenmeyer, Gorav

Jindal, and Vladimir Lysikov)

Limitations of Algebraic Lower Bound Proofs XIX

References

[FSV17] Forbes, M.A., Shpilka, A., Volk, B.L.: Succinct hitting sets and barriers to provingalgebraic circuits lower bounds. In: Hatami, H., McKenzie, P., King, H. (eds.)Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Com-puting, STOC 2017, Montreal, QC, Canada, 19–23 June 2017, pp. 653–664. ACM(2017)

[GKSS17] Grochow, J.A., Kumar, M. Saks, M.E., Saraf, S.: Towards an algebraic natural proofsbarrier via polynomial identity testing (2017). CoRR, abs/1701.01717

[Hås90] Håstad, J.: Tensor rank is NP-complete. J. Algorithms 11(4), 644–654 (1990)[MS01] Mulmuley, K., Sohoni, M.A..: Geometric complexity theory I: an approach to the P

vs. NP and related problems. SIAM J. Comput. 31(2), 496–526 (2001)[RR97] Razborov, A.A., Rudich, S.: Natural proofs. J. Comput. Syst. Sci. 55(1), 24–35

(1997)

XX M. Bläser

Complexity of Generation

Vladimir Gurvich1,2

1 RUTCOR & RBS, Rutgers University, 100 Rockafeller Road,Piscataway NJ 08854, USA

2 National Research Institute Higher School of Economics (HSE),Moscow, Russia

vladimir.gurvich@gmail.com

Abstract. In this talk I summarize the results obtained in 1999–2008 byLeonid Khachiyan, Endre Boros, Konrad Borys, Khaled Elbassiony, KazuhisaMakino, and myself, on complexity of generation algorithms. These algorithmscan be partitioned into three groups: supergraph, flash-light (backtrack), anddual-bounded generation. We will call a problem tractable if it can be solved bya polynomial (nconst) or quasi-polynomial (npolylogðnÞ) time algorithm. Moregenerally, for any positive non-decreasing function g ¼ gðnÞ, generating can beperformed in total or incremental time g, or with g-delay. Most of the poly-nomial delay algorithms are provided by the flash-light (backtrack) method. Asfor the incremental algorithms, generating the next object is equivalent with justverifying its existence, which is a standard decision problem. Thus, incrementalgeneration, in contrast to the delay one, may be NP-hard or NP-complete. Forexample, we show that generating all vertices of a polyhedron, given by itsfacets, is NP-complete (while the complexity status is still open in case of thepolytopes, that is, bounded polyhedra). This problem is reduced to generating allnegative cycles of a weighted digraph, which is NP-complete (for graphs, too).Generating all minimal transversals to a hypergraph, so-called dualization, playsan important role. For this problem an incremental quasi-polynomial algorithm(but no polynomial one) is known. We outline several wide classes of generationproblems that can be reduced to dualization and, thus, solved in incrementalquasi-polynomial time. We survey algorithms and complexity bounds for theabove and many other generation problems.

This work was supported in part by the Russian Academic Excellence Project ‘5-100’.

Lower Bounds for Unrestricted BooleanCircuits: Open Problems

Alexander S. Kulikov

St. Petersburg Department of Steklov Institute of Mathematics,St. Petersburg, Russia

Abstract. To prove that P 6¼ NP, it suffices to prove a superpolynomial lowerbound on Boolean circuit complexity of a function from NP. Currently, we arenot even close to achieving this goal: we do not know how to prove a 4n lowerbound. What is more depressing is that there are almost no techniques forproving circuit lower bounds.

In this note, we briefly review various approaches that could potentially leadto stronger linear or superlinear lower bounds for unrestricted Boolean circuits(i.e., circuits with no restriction on depth, fan-out, or basis).

Online Labeling: Algorithms, Lower Boundsand Open Questions

Michael Saks

Department of Mathematics, Rutgers University, New Brunswick,NJ 08854, USA

saks@math.rutgers.edu

Abstract. The online labeling problem (also known as the file maintenanceproblem), is a natural algorithmic problem that has arisen as a buidling block fordata structures. A stream of distinct integer items is to be assigned labels onlinefrom a label set f1; . . .;mg so that the order of the labels respects the naturalorder of the items. Maintaining order on the labels may require relabeling items.The algorithm pays 1 each time an item is labeled or relabeled and the goalof the algorithm is to minimize the total cost.

We survey upper and lower bounds and open problems in both the deter-ministic and randomized setting.

Reading MCSP Through SAT

Rahul Santhanam

Department of Computer Science, University of Oxford, UKrahul.santhanam@cs.ox.ac.uk

Abstract. The Minimum Circuit Size Problem (MCSP) and the Circuit Satis-fiability Problem (SAT) are fundamental problems in theoretical computer sci-ence. These problems are in a sense dual to each other - while MCSP asks if aBoolean function given by its truth table has small circuits, SAT and its variantsask about properties of the Boolean function corresponding to a given circuit.SAT has featured in some of the most important and influential results incomplexity theory over the past few decades, including the Cook-Levin theo-rem, the PCP theorem and Williams’ connection between SAT algorithms andcircuit lower bounds. MCSP, however, remains mysterious. This lecture willdescribe a research program aiming for a deeper understanding of MCSP guidedby a comparative analysis with SAT, emphasizing the following phenomena:

1. Explicit constructions: While it is easy to compute explicitly positive andnegative instances of SAT of any given length, the corresponding questionfor MCSP is intimately tied to long sought-after circuit lower bounds.

2. NP-hardness: While the classical Cook-Levin theorem establishes theNP-hardness of SAT, the NP-hardness of MCSP remains open. There aresome NP-hardness results for variants of MCSP where the circuit class isstrongly restricted. On the other hand, there is a recent line of work ruling outNP-hardness under restricted reductions or deriving hard-to-prove com-plexity consequences from NP-hardness under standard reductions. In sum,there is no strong evidence for or against NP-completeness.

3. Search to decision reductions: The complexity of deciding SAT ispolynomial-time equivalent to the complexity of finding a satisfyingassignment for a satisfiable circuit, using the downward self-reducibility ofSAT. MCSP does not appear to have a similar downward self-reducibilityproperty. Recent work gives an approximate search-to-decision reduction forMCSP using an intriguing connection with learning theory.

4. Average-case complexity: SAT for k-CNFs appears hard empirically for highenough clause density under a natural distribution where clauses are pickeduniformly and independently at random. This motivates Feige’s hypothesis,which is known to imply strong hardness of approximation results for variousnatural NP problems. The complexity of MCSP under the uniform distri-bution on inputs has been studied under the guise of the “natural proofs” ofRazborov and Rudich, which have strong connections to cryptography andproof complexity. Though SAT is not known to reduce to MCSP in the worstcase, a recent result shows that the MKTP problem, a close cousin of MCSP,is hard on average under the uniform distribution if Feige’s hypothesis holds.

Keywords: Minimum circuit size problem • Satisfiability • NP-hardnessLearning • Natural proofs

References

1. Carmosino, M., Impagliazzo, R., Kabanets, V., Kolokolova, A.: Learning algorithms fromnatural proofs. In: Proceedings of 31st Conference on Computational Complexity (CCC),pp. 10:1–10:24 (2016)

2. Feige, U.: Relations between average case complexity and approximation complexity. In:Proceedings of 34th Annual ACM Symposium on Theory of Computing (STOC),pp. 534–543 (2002)

3. Hirahara, S., Oliveira, I. C., Santhanam, R.: NP-hardness of minimum circuit size problem forOR-AND-MOD circuits. Electronic colloquium on computational complexity (ECCC),pp. 25–30 (2018)

4. Hirahara, S., Santhanam, R.: On the average-case complexity of MCSP and its variants. In:Proceedings of 32nd Computational Complexity Conference (CCC), pp. 7:1–7:20 (2017)

5. Kabanets, V., Cai, J.-Y.: Circuit minimization problem. In: Proceedings of 32nd Annual ACMSymposium on Theory of Computing (STOC), pp. 73–79 (2000)

6. Razborov, A., Rudich, S.: Natural proofs. J. Compt. Syst. Sci. 55(1), 24–35 (1997)7. Trakhtenbrot, B.: A survey of Russian approaches to perebor algorithms. IEEE Ann. Hist.

Comput. 6(4), 384–400 (1984)8. Williams, R.: Improving exhaustive search implies superpolynomial lower bounds.

SIAM J. Comput. 42(3), 1218–1244 (2013)

Reading MCSP Through SAT XXV

Recent Developments on the AsymmetricTraveling Salesman Problem

László A. Végh

London School of Economics and Political Science, London, WC2A 2AE, UKl.vegh@lse.ac.uk

http://personal.lse.ac.uk/veghl

Abstract. The talk presents an overview of recent developments on theapproximability of the Asymmetric Traveling Salesman Problem

Keywords: Traveling salesman problem Approximation algorithms

In the traveling salesman problem (TSP), the input is given by n cities and theirpairwise distances, and the goal is to find a shortest tour visiting every city andreturning to the starting city. It is one of the best known optimization problems, withvariants studied since the 19th century. The problem is NP-complete; if no assumptionsare made on the distance function, it cannot be approximated within any constantfactor. It is therefore common to assume that the distances satisfy the triangleinequality, or equivalently, that the traveler is allowed to visit some cities more thanonce.

In this talk, we focus on approximation algorithms for TSP. A classical algorithmby Christofides from 1976 [5] gives a 3

2-approximation algorithm if the distancefunction is assumed to be symmetric. Forty years on, this simple algorithm is still thebest known approximation algorithm for general symmetric costs. Improved guaranteeshave been given in recent work for the special setting of unweighted shortest pathmetrics, [9, 13–15].

In contrast to the tight guarantees for the symmetric case, our understanding of themore general asymmetric traveling salesman problem (ATSP) is far from complete.A classical LP relaxation was obtained by Held and Karp in the 1970s [10, 11]. Thishas been used as the lower bound in all approximation algorithms (both symmetric andasymmetric). Whereas the best lower bound on the integrality gap is 2 in the asym-metric case [4], even finding a constant factor approximation guarantee has remainedopen until very recently.

An elegant approximation algorithm for ATSP was given by Frieze, Galbiati andMaffioli [7], with approximation guarantee of log2ðnÞ. This algorithm constructs asequence of cycle covers, gradually growing the connected components of the solution.A series of papers [3, 6, 12] have improved the approximation factor by constantfactors, but finding an oðlognÞ approximation guarantee remained open for a longertime period.

This was first achieved in the breakthrough result Asadpour et al. [2], who obtainedan Oðlog n = log log nÞ-approximation factor for ATSP. They introduced a new andinfluential approach, making a connection between the approximability of ATSP andthe existence of thin trees, a problem studied in the context of graph theory. In par-ticular, they showed that finding a tree of constant “thinness” would imply a constantfactor approximation for ATSP. Following this approach, Oveis Gharan and Saberigave a constant factor approximation for graphs with bounded genus [8]. Morerecently, Anari and Oveis Gharan have obtained an upper bound Oðpoly log log nÞ [1]on the integrality gap of the Held-Karp relaxation. This, however, does not provide anefficient approximation algorithm of the same guarantee, since their arguments arenon-constructive.

An ATSP solution must satisfy three properties simultaneously: connectivity,Eulerian degree constraints, and integrality. For any two among these three require-ments, a minimum cost solution can be found efficiently. Indeed, connected andEulerian (but not necessarily integral) vectors are exactly the feasible solutions to theHeld-Karp relaxation; connected and integral (but not necessary Eulerian) edge sets arethe spanning trees; and Eulerian and integral (but not necessary connected) edge setsare the cycle covers. At a very high level, all ATSP approximation algorithms start byeither relaxing the Eulerian constraints, or relaxing connectivity.

The thin tree approach start by relaxing the Eulerian constraint, starting with aspecial spanning tree. The thinness property can be then used to fix the violations of theEulerian constraints. In contrast, the Frieze, Galbiati and Maffioli [7] algorithm start byrelaxing connectivity: they construct a sequence of cycle covers whose union becomesconnected.

Svensson introduced a new approach via relaxing connectivity, by giving areduction from ATSP to a problem called Local-Connectivity ATSP [16]. An “a-light”solution to this problem implies an OðaÞ approximation to ATSP. The paper [16] usedthis approach for giving a ð27þ eÞ-approximation algorithm for the special case ofnode-weighted metrics, where Local-Connectivity ATSP is relatively easy to imple-ment. We have generalized this to graphs with at most two different edge weights insubsequent work [17], however, this already required substantial technical effort.

In our recent paper [18], we have built upon and generalized both of these results toobtain the first constant-factor approximation algorithm for ATSP for arbitrary metrics.The constant factor in the paper is 5500; however, this is not optimized for the sake ofsimplicity in the presentation.

In contrast to the two edge weights result [17], we do not try to tackleLocal-Connectivity ATSP directly in arbitrary weighted graphs. Instead, we introduce aseries of natural reduction steps to reduce the problem of approximating ATSP ingeneral to that of approximating ATSP on special, structured instances called verte-brate pairs. These instances enjoy properties that make them amenable forLocal-Connectivity ATSP.

All these reductions build on classical techniques from mathematical optimizationand from graph theory. We start by applying the uncrossing technique for the optimalsolution to the dual of the Held-Karp LP to reduce general ATSP instances tolaminarly-weighted ATSP instances. These enjoy a special weight structure defined bya laminar family of vertex subsets.

Recent Developments on the Asymmetric Traveling Salesman Problem XXVII

The subsequent steps define a natural contraction operation and, using a recursivealgorithm, reduce the problem to irreducible instances. The very same property thatmakes these instances difficult for the reduction in turn enables us to construct abackbone, a special subtour that visits most vertices in the instance. In the final step, wetake advantage of the backbone for implementing Local-Connectivity ATSP.

We believe that, by further optimizing these techniques, the integrality gap of theLP relaxation can be upper-bounded by the hundreds. In order to achieve an upperbound closer to the current lower bound 2, even to say 50 is likely to require somesubstantial new ideas.

References

1. Anari, N., Gharan, S.O.: Effective-resistance-reducing flows, spectrally thin trees, andasymmetric TSP. In: IEEE 56th Annual Symposium on Foundations of Computer Science(FOCS), pp. 20–39 (2015)

2. Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem. In: Proceedingsof the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010,pp. 379–389 (2010)

3. Bläser, M.: A new approximation algorithm for the asymmetric TSP with triangle inequality.ACM Trans. Algorithms 4(4) (2008)

4. Charikar, M., Goemans, M.X., Karloff, H.J.: On the integrality ratio for the asymmetrictraveling salesman problem. Math. Oper. Res. 31(2), 245–252 (2006)

5. Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem.Technical report, Graduate School of Industrial Administration, CMU (1976)

6. Feige, U., Singh, M.: Improved approximation ratios for traveling salesperson tours andpaths in directed graphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.)APPROX/RANDOM -2007. LNCS, vol. 4627, pp. 104–118. Springer, Heidelberg (2007).https://doi.org/10.1007/978-3-540-74208-1_8

7. Frieze, A.M., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithmsfor the asymmetric traveling salesman problem. Networks 12(1), 23–39 (1982)

8. Gharan, S.O., Saberi, A.: The asymmetric traveling salesman problem on graphs withbounded genus. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium onDiscrete Algorithms, SODA 2011, pp. 967–975 (2011)

9. Gharan, S.O., Saberi, A., Singh, M.: A randomized rounding approach to the travelingsalesman problem. In: IEEE 52nd Annual Symposium on Foundations of Computer Science,FOCS 2011, pp. 550–559 (2011)

10 Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Oper.Res. 18(6), 1138–1162 (1970)

11. Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees: Part II.Math. Program. 1(1), 6–25 (1971)

12. Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.: Approximation algorithms forasymmetric TSP by decomposing directed regular multigraphs. J. ACM 52(4), 602–626(2005)

13. Mömke, T., Svensson, O.: Removing and adding edges for the traveling salesman problem.J. ACM 63(1), 2:1–2:28 (2016)

XXVIII L. A. Végh

14. Mucha, M.: 13/9-approximation for graphic TSP. In: 29th International Symposium onTheoretical Aspects of Computer Science, STACS 2012, pp. 30–41 (2012)

15. Sebő, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2for the path version, and 4/3 for two-edge-connected subgraphs. Combinatorica 34(5),597–629 (2014)

16. Svensson, O.: Approximating ATSP by relaxing connectivity. In: FOCS 2015: Proceedingsof the 56th Annual IEEE Symposium on Foundations of Computer Science (2015).http://arxiv.org/abs/1502.02051

17. Svensson, O., Tarnawski, J., Végh, L.A.: Constant factor approximation for ATSP with twoedge weights. In: Louveaux, Q., Skutella, M. (eds.) Integer Programming and CombinatorialOptimization. IPCO 2016. LNCS, vol. 9682, pp. 226–237. Springer, Cham (2016)

18. Svensson, O., Tarnawski, J., Végh, L.A.: A constant-factor approximation algorithm for theasymmetric traveling salesman problem. In: STOC (2018, forthcoming)

Recent Developments on the Asymmetric Traveling Salesman Problem XXIX

Contents

Complexity of Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Vladimir Gurvich

Lower Bounds for Unrestricted Boolean Circuits: Open Problems . . . . . . . . . 15Alexander S. Kulikov

Online Labeling: Algorithms, Lower Bounds and Open Questions. . . . . . . . . 23Michael Saks

Maintaining Chordal Graphs Dynamically: Improved Upperand Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Niranka Banerjee, Venkatesh Raman, and Srinivasa Rao Satti

Distributed Symmetry-Breaking Algorithms for Congested Cliques . . . . . . . . 41Leonid Barenboim and Victor Khazanov

The Clever Shopper Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Laurent Bulteau, Danny Hermelin, Anthony Labarre,and Stéphane Vialette

A Tight Lower Bound for Steiner Orientation . . . . . . . . . . . . . . . . . . . . . . . 65Rajesh Chitnis and Andreas Emil Feldmann

Can We Create Large k-Cores by Adding Few Edges? . . . . . . . . . . . . . . . . 78Rajesh Chitnis and Nimrod Talmon

Periodicity in Data Streams with Wildcards . . . . . . . . . . . . . . . . . . . . . . . . 90Funda Ergün, Elena Grigorescu, Erfan Sadeqi Azer, and Samson Zhou

Maximum Colorful Cycles in Vertex-Colored Graphs . . . . . . . . . . . . . . . . . 106Giuseppe F. Italiano, Yannis Manoussakis, Nguyen Kim Thang,and Hong Phong Pham

Grammar-Based Compression of Unranked Trees . . . . . . . . . . . . . . . . . . . . 118Adrià Gascón, Markus Lohrey, Sebastian Maneth, Carl Philipp Reh,and Kurt Sieber

Complement for Two-Way Alternating Automata . . . . . . . . . . . . . . . . . . . . 132Viliam Geffert

Closure Under Reversal of Languages over Infinite Alphabets . . . . . . . . . . . 145Daniel Genkin, Michael Kaminski, and Liat Peterfreund

Structural Parameterizations of Dominating Set Variants . . . . . . . . . . . . . . . 157Dishant Goyal, Ashwin Jacob, Kaushtubh Kumar, Diptapriyo Majumdar,and Venkatesh Raman

Complexity and Inapproximability Results for Parallel Task Schedulingand Strip Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Sören Henning, Klaus Jansen, Malin Rau, and Lars Schmarje

Operations on Boolean and Alternating Finite Automata . . . . . . . . . . . . . . . 181Michal Hospodár, Galina Jirásková, and Ivana Krajňáková

Conflict Free Version of Covering Problems on Graphs: Classicaland Parameterized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Pallavi Jain, Lawqueen Kanesh, and Pranabendu Misra

Quadratically Tight Relations for Randomized Query Complexity . . . . . . . . . 207Rahul Jain, Hartmut Klauck, Srijita Kundu, Troy Lee, Miklos Santha,Swagato Sanyal, and Jevgēnijs Vihrovs

On Vertex Coloring Without Monochromatic Triangles . . . . . . . . . . . . . . . . 220Michał Karpiński and Krzysztof Piecuch

Recognizing Read-Once Functions from Depth-Three Formulas . . . . . . . . . . 232Alexander Kozachinskiy

MAX-CUT ABOVE SPANNING TREE is Fixed-Parameter Tractable. . . . . . . . . . . . 244Jayakrishnan Madathil, Saket Saurabh, and Meirav Zehavi

Slopes of 3-Dimensional Subshifts of Finite Type . . . . . . . . . . . . . . . . . . . . 257Etienne Moutot and Pascal Vanier

Facility Location on Planar Graphs with Unreliable Links . . . . . . . . . . . . . . 269N. S. Narayanaswamy, Meghana Nasre, and R. Vijayaragunathan

On the Decision Trees with Symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . 282Artur Riazanov

On Emptiness and Membership Problems for Set Automata . . . . . . . . . . . . . 295A. Rubtsov and M. Vyalyi

On Strong NP-Completeness of Rational Problems . . . . . . . . . . . . . . . . . . . 308Dominik Wojtczak

A New Algorithm for Finding Closest Pair of Vectors (Extended Abstract) . . . 321Ning Xie, Shuai Xu, and Yekun Xu

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

XXXII Contents