Complexity issues for the iterated h-preorders

SIGEST

In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: (1) We show that if $E: \mathbb{F}^n \mapsto \mathbb{F}^m$ is a linear LDC with two queries, then $m = \exp(\Omega(n))$. Previously this was known only for fields of size $\ll 2^n$ [O. Goldreich et al., Comput. Complexity, 15 (2006), pp. 263–296]. (2) We show that from every depth 3 arithmetic circuit ($\Sigma\Pi\Sigma$ circuit), ${\cal C}$, with a bounded (constant) top fan‐in that computes the zero polynomial, one can construct an LDC. More formally, assume that ${\cal C}$ is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by ${\cal C}$ and by r the rank of the linear functions appearing in ${\cal C}$. Then we can construct a linear LDC with two queries that encodes messages of length $r/{\operatorname{polylog}(d)}$ by codewords of length $O(d)$. (3) We prove a structural theorem for $\Sigma\Pi\Sigma$ circuits, with a bounded top fan‐in, that compute the zero polynomial. In particular we show that if such a circuit is simple, minimal, and of polynomial size, then its rank, r, is only polylogarithmic in the number of variables (a priori it could have been linear). (4) We give new PIT algorithms for $\Sigma\Pi\Sigma$ circuits with a bounded top fan‐in: (a) a deterministic algorithm that runs in quasipolynomial time, and (b) a randomized algorithm that runs in polynomial time and uses only a polylogarithmic number of random bits. Moreover, when the circuit is multilinear, our deterministic algorithm runs in polynomial time. Previously deterministic subexponential time algorithms for PIT in bounded depth circuits were known only for depth 2 circuits (in the black box model) [D. Grigoriev, M. Karpinski, and M. F. Singer, SIAM J. Comput., 19 (1990), pp. 1059–1063; M. Ben‐Or and P. Tiwari, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, ACM Press, New York, 1988, pp. 301–309; A. R. Klivans and D. Spielman, Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2001, pp. 216–223]. In particular, for the special case of depth 3 circuits with three multiplication gates our result resolves an open question asked by Klivans and Spielman.

We show how to simulate any BPP algorithm in polynomial time by using a weak random source of r bits and min-entropy $r^{\gamma}$ for any $\gamma >0$. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP [M. Saks, A. Srinivasan, and S. Zhou, Proc. 27th ACM Symp. on Theory of Computing, 1995, pp. 479--488] and a quasi-polynomial time simulation of BPP [A. Ta-Shma, Proc. 28th ACM Symp. on Theory of Computing, 1996, pp. 276--285]. Departing significantly from previous related works, we do not use extractors; instead, we use the OR-disperser of Saks, Srinivasan, and Zhou in combination with a tricky use of hitting sets borrowed from [Andreev, Clementi, and Rolim, J. ACM, 45 (1998), pp. 179--213].

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

On the core of the multicommodity flow game

This issue of SICOMP contains ten specially selected papers from STOC 2017, the Forty-ninth Annual ACM Symposium on the Theory of Computing, which was held June 19--23 in Montreal, Canada. The papers here were chosen to represent the range and quality of the STOC program. These papers have been revised and extended by their authors and subjected to the standard thorough reviewing process of SICOMP. The program committee for STOC 2017 consisted of Nina Balcan, Allan Borodin, Keren Censor-Hillel, Edith Cohen, Artur Czumaj, Yevgeniy Dodis, Andrew Drucker, Nick Harvey, Monika Henzinger, Russell Impagliazzo, Ken-ichi Kawarabayashi, Ravi Kumar, James R. Lee, Katrina Ligett, Aleksander Mądry, Cristopher Moore, Jelani Nelson, Eric Price, Amit Sahai, Jared Saia, Shubhangi Saraf, Alexander Sherstov, Mohit Singh, and Gábor Tardos. The program chair was Valerie King. Included in this issue are the following papers: ``Short Presburger Arithmetic Is Hard," by Danny Nguyen and Igor Pak, proves that the satisfiability of short sentences in Presburger arithmetic with $m+2$ alternating quantifiers is $\Sigma^{{P}}_m$-complete or $\Pi^{{P}}_m$-complete when the first quantifier is $\exists$ or $\forall$, respectively. ``An Efficient Reduction from Two-Source to Nonmalleable Extractors: Achieving Near-Logarithmic Min-Entropy," by Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma, gets an explicit bipartite Ramsey graph (or a twosource extractor) for sets of size 2$k$ for $k = O(\log n \log \log n)$, using the currently best explicit nonmalleable extractors. ``Holographic Algorithm with Matchgates Is Universal for Planar \#CSP over Boolean Domain," by Jin-Yi Cai and Zhiguo Fu, classifies all counting CSPs over Boolean variables into one of three categories: polynomial-time tractable, \#P-hard for general instances but solvable in polynomial time over planar graphs, and \#P-hard over planar graphs. ``Deciding Parity Games in Quasipolynomial Time," by Cristian S. Calude, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, and Frank Stephan, shows the parameterized parity game, with $n$ nodes and $m$ priorities, is in the class of fixed parameter tractable problems when parameterized over $m$. ``New Hardness Results for Routing on Disjoint Paths," by Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat, proves that node-disjoint paths is $2^{\Omega(\sqrt{\log n})}$-hard to approximate, unless all problems in NP have algorithms with running time $n^{O(\log n)}$. ``A Weighted Linear Matroid Parity Algorithm," by Satoru Iwata and Yusuke Kobayashi, presents a combinatorial, deterministic, strongly polynomial-time algorithm for the weighted linear matroid parity problem. ``Targeted Pseudorandom Generators, Simulation Advice Generators, and Derandomizing Logspace," by William M. Hoza and Chris Umans, shows that $\mathbf{BPL} \subseteq \bigcap_{\alpha > 0} {DSPACE}(\log^{1 + \alpha} n)$, assuming that for every derandomization result for log-space algorithms there is a pseudorandom generator strong enough to nearly recover the derandomization by iterating over all seeds and taking a majority vote. ``Approximating Rectangles by Juntas and Weakly Exponential Lower Bounds for LP Relaxations of CSPs," by Pravesh K. Kothari, Raghu Meka, and Prasad Raghavendra, shows that for CSPs, subexponential size LP relaxations are as powerful as $n^{\Omega(1)}$-rounds of the Sherali--Adams LP hierarchy. ``Equivocating Yao: Constant-Round Adaptively Secure Multiparty Computation in the Plain Model," by Ran Canetti, Oxana Poburinnaya, and Muthuramakrishnan Venkitasubramaniam, defines a new type of encryption and shows that Yao's garbling scheme, implemented with this encryption mechanism, is secure against adaptive adversaries. ``Geodesic Walks in Polytopes," by Yin Tat Lee and Santosh Vempala, introduces the geodesic walk for sampling Riemannian manifolds and applies it to the problem of generating uniform random points from the interior of polytopes in ${\mathbb{R}}^{n}$ specified by m inequalities; the resulting sampling algorithm for polytopes mixes in $O^{*}(mn^{\frac{3}{4}})$ steps. We thank the authors, the STOC 2017 program committee, the STOC 2017 external reviewers, and the SICOMP referees for all of their hard work. Andy Drucker, Ravi Kumar, Amit Sahai, Mohit Singh, Guest editors

Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich--Levin algorithm [O. Goldreich and L. Levin, “A Hard-Core Predicate for All One-Way Functions," in Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 1989, pp. 25--32] can be viewed as an algorithmic analogue of such a decomposition as it gives a way to efficiently find the linear phases associated with large Fourier coefficients. In the study of “quadratic Fourier analysis,” higher-degree analogues of such decompositions have been developed in which the pseudorandomness property is stronger but the structured part is correspondingly weaker. For example, it has been shown that it is possible to express a bounded function as a sum of a few quadratic phases plus a part that is small in the $U^3$ norm, defined by Gowers for the purpose of counting arithmetic progressions of length 4. We give a polynomial time algorithm for computing such a decomposition. A key part of the algorithm is a local self-correction procedure for Reed--Muller codes of order 2 (over $\F_2^n$) for a function at distance $1/2-\e$ from a codeword. Given a function $f:\F_2^n \to \{-1,1\}$ at fractional Hamming distance $1/2-\e$ from a quadratic phase (which is a codeword of the Reed--Muller code of order 2), we give an algorithm that runs in time polynomial in $n$ and finds a codeword at distance at most $1/2-\eta$ for $\eta = \eta(\e)$. This is an algorithmic analogue of Samorodnitsky's result [A. Samorodnitsky, “Low-Degree Tests at Large Distances,” in Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 2007, pp. 506--515], which gave a tester for the above problem. To our knowledge, it represents the first instance of a correction procedure for any class of codes, beyond the list-decoding radius. For this purpose, we give algorithmic versions of results from additive combinatorics used in Samorodnitsky's proof and a refined version of the inverse theorem for the Gowers $U^3$ norm over $\F_2^n$.

Let $C$ be a depth-3 circuit with $n$ variables, degree $d$, and top-fanin $k$ (called ${\Sigma\Pi\Sigma}(k,d,n)$ circuits) over base field ${\mathbb{F}}$. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests whether $C$ is identically zero. Klivans and Spielman [Proceedings of the 33rd Annual Symposium on Theory of Computing (STOC), 2001, pp. 216--223] observed that the problem is open even when $k$ is a constant. This case has been subjected to serious scrutiny over the past few years, starting from the work of Dvir and Shpilka [SIAM J. Comput., 36 (2007), pp. 1404--1434]. We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time ${\mbox{\rm poly}}(n)d^k$, regardless of the base field. The only field for which polynomial time algorithms were previously known is ${\mathbb{F}} = {\mathbb{Q}}$ [N. Kayal and S. Saraf, Proceedings of the 50th Annual Symposium on Foundations of Computer Science (FOCS), 2009, pp. 198--207; N. Saxena and C. Seshadhri, Proceedings of the 51st Annual Symposium on Foundations of Computer Science (FOCS), 2010, pp. 21--29]. This is the first blackbox algorithm for depth-3 circuits that does not use the rank-based approaches of Karnin and Shpilka [Proceedings of the 24th Annual Conference on Computational Complexity (CCC), 2009, pp. 274--285]. We prove an important tool for the study of depth-3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a ${\Sigma\Pi\Sigma}(k,d,n)$ circuit to $k$ variables but preserves the identity structure.

Toda (SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum–Shub–Smale real machines (Blum et al. in Bull. Am. Math. Soc. 21(1):1–46, 1989) was proved in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010). Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) is topological in nature; the properties of the topological join are used in a fundamental way. However, the constructions used in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) were semi-algebraic—they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence, we obtain a complex analogue of Toda’s theorem. The results of this paper, combined with those in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010), illustrate the central role of the Poincare polynomial in algorithmic algebraic geometry, as well as in computational complexity theory over the complex and real numbers: the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.

Given a matroidM with distinguished elemente, aport oracie with respect toe reports whether or not a given subset contains a circuit that containse. The first main result of this paper is an algorithm for computing ane-based ear decomposition (that is, an ear decomposition every circuit of which contains elemente) of a matroid using only a polynomial number of elementary operations and port oracle calls. In the case thatM is binary, the incidence vectors of the circuits in the ear decomposition form a matrix representation forM. Thus, this algorithm solves a problem in computational learning theory; it learns the class ofbinary matroid port (BMP) functions with membership queries in polynomial time. In this context, the algorithm generalizes results of Angluin, Hellerstein, and Karpinski [1], and Raghavan and Schach [17], who showed that certain subclasses of the BMP functions are learnable in polynomial time using membership queries. The second main result of this paper is an algorithm for testing independence of a given input set of the matroidM. This algorithm, which uses the ear decomposition algorithm as a subroutine, uses only a polynomial number of elementary operations and port oracle calls. The algorithm proves a constructive version of an early theorem of Lehman [13], which states that the port of a connected matroid uniquely determines the matroid.

Preference elicitation --- the process of asking queries to determine parties' preferences --- is a key part of many problems in electronic commerce. For example, a shopping agent needs to know a user's preferences in order to correctly act on her behalf, and preference elicitation can help an auctioneer in a combinatorial auction determine how to best allocate a given set of items to a given set of bidders. Unfortunately, in the worst case, preference elicitation can require an exponential number of queries even to determine an approximately optimal allocation. In this paper we study natural special cases of preferences for which elicitation can be done in polynomial time via value queries. The cases we consider all have the property that the preferences (or approximations to them) can be described in a polynomial number of bits, but the issue here is whether they can be elicited using the natural (limited) language of value queries. We make a connection to computational learning theory where the similar problem of exact learning with membership queries has a long history. In particular, we consider preferences that can be written as read-once formulas over a set of gates motivated by a shopping application, as well as a class of preferences we call Toolbox DNF, motivated by a type of combinatorial auction. We show that in each case, preference elicitation can be done in polynomial time. We also consider the computational problem of allocating items given the parties' preferences, and show that in certain cases it can be done in polynomial time and in other cases it is NP-complete. Given two bidders with Toolbox-DNF preferences, we show that allocation can be solved via network flow. If parties have read-once formula preferences, then allocation is NP-hard even with just two bidders, but if one of the two parties is additive (e.g., a shopping agent purchasing items individually and then bundling them to give to the user), the allocation problem is solvable in polynomial time.

To illustrate the "remarkable extent to which complexity theory operates by means of analogs from computability theory," Richard Karp created this conceptual map or jigsaw puzzle. To lay out the puzzle in the plane, he used a "graph planarity algorithm." The more distantly placed parts might not at first seem related, "but in the end, the theory of NP-completeness does bring them all together," Karp says. The upper right portion of the puzzle shows concepts related to combinatorial explosions and the notion of a "good" or "efficient" algorithm. In turn, "Complexity" connects these concepts to the upper left portion, which represents the concerns of early computability theorists. The traveling salesman problem is closer to the upper right corner because it is probably intractable. It therefore borders on "NP-completeness" and "Combinatorial explosion." To some extent, however, certain divisions blur. "Linear programming," for example, has an anomalous status—the most widely used algorithms for solving such problems in practice are not good in the theoretical sense, and those that are good in the theoretical sense are often not good in practice. One example is the ellipsoid method that was the object of so much attention six years ago. It ran in polynomial time, but the polynomial was of such a high degree that the method proved good in the technical sense, but ineffective in practice. "The reason is that our notion of polynomial-time algorithms doesn't exactly capture the notion of an intuitively efficient algorithm," Karp explains. "When you get up to n 5 or n 6 , then it's hard to justify saying that it is really efficient. So Edmonds's concept of a good algorithm isn't quite a perfect formal counterpart of good in the intuitive sense." Further, the simplex algorithm is good in every practical sense, Karp says, but not good according to the standard paradigm of complexity theory. The most recent addition to linear programming solutions, an algorithm devised by Narendra Karmarkar that some think challenges the simplex algorithm, is good in the technical sense and also appears to be quite effective in practice, says Karp. The good algorithm segment is adjacent to "Heuristics" because a heuristic algorithm may work well, but lack a theoretical pedigree. Some heuristic algorithms are always fast, but sometimes fail to give good solutions. Others always give an optimal solution, but are not guaranteed to be fast. The simplex algorithm is of the latter type. "Undecidability, " "Combinatorial explosion," and "Complexity" are on the same plane because they are analogs of one another; undecidability involves unbounded search, whereas combinatorial explosions are by definition very long but not unbounded searches. Complexity theory bridges the gap because, instead of asking whether a problem can be solved at all, it poses questions about the resources needed to solve a problem. The lower left region contains the segments Karp has been concerned with most recently and that contain open-ended questions. "Randomized algorithm," for example, is situated opposite "Probabilistic analysis" because both are alternatives to worst-case analyses of deterministic algorithms. Randomized algorithms might be able to solve problems in polynomial time that deterministic ones cannot and that could mean an extension of the notion of good algorithms. Perhaps through software designs for non-von Neumann machines, algorithms can be made more efficient in practice through parallelism. Finally, some parts of the puzzle are not yet defined. Says Karp, "They correspond to the unknown territory that remains to be explored in the future."

A famous result due to Ko and Friedman ( Theoretical Computer Science 20 ( 1982 ) 323–352) asserts that the problems of integration and maximisation of a univariate real function are computationally hard in a well-defined sense. Yet, both functionals are routinely computed at great speed in practice. We aim to resolve this apparent paradox by studying classes of functions which can be feasibly integrated and maximised, together with representations for these classes of functions which encode the information which is necessary to uniformly compute integral and maximum in polynomial time. The theoretical framework for this is the second-order complexity theory for operators in analysis which was introduced by Kawamura and Cook ( ACM Transactions on Computation Theory 4(2) ( 2012 ) 5). The representations we study are based on approximation by polynomials, piecewise polynomials, and rational functions. We compare these representations with respect to polytime reducibility. We show that the representation based on approximation by piecewise polynomials is polytime equivalent to the representation based on approximation by rational functions. With this representation, all terms in a certain language, which is expressive enough to contain the maximum and integral of most functions of practical interest, can be evaluated in polynomial time. By contrast, both the representation based on polynomial approximation and the standard representation based on function evaluation, which implicitly underlies the Ko-Friedman result, require exponential time to evaluate certain terms in this language. We confirm our theoretical results by an implementation in Haskell, which provides some evidence that second-order polynomial time computability is similarly closely tied with practical feasibility as its first-order counterpart.

We address the graph isomorphism problem and related fundamental complexity problems of computational group theory. The main results are these: A1. A polynomial time algorithm to test simplicity and find composition factors of a given permutation group (COMP). A2. A polynomial time algorithm to find elements of given prime order p in a permutation group of order divisible by p. A3. A polynomial time reduction of the problem of finding Sylow subgroups of permutation groups (SYLFIND) to finding the intersection of two cosets of permutation groups (INT). As a consequence, one can find Sylow subgroups of solvable groups and of groups with bounded nonabelian composition factors in polynomial time. A4. A polynomial time algorithm to solve SYLFIND for finite simple groups. A5. An ncd/log d algorithm for isomorphism (ISO) of graphs of valency less than d and a consequent improved moderately exponential general graph isomorphism test in exp(c√n log n) steps. A6. A moderately exponential, n,c√n algorithm for INT. Combined with A3, we obtain an nc√n algorithm for SYLFIND as well. All these problems have strong links to each other. ISO easily reduces to INT. A subcase of SYLFIND was solved in polynomial time and applied to bounded valence ISO in [Lul]. Now, SYLFIND is reduced to INT. Interesting special cases of SYLFIND belong to NP ∩ coNP and are not known to have subexponential solutions. All the results stated depend on the classification of finite simple groups. We note that no previous ISO test had no(d) worst case behavior for graphs of valency less than d. It appears that unless there is another radical breakthrough in ISO, independent of the previous one, the simple groups classification is an indispensable tool for further developments.

General Partitioning on Random Graphs