Some Results on Average-Case Hardness Within the Polynomial Hierarchy

During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what before-the-fact seemed to be) lower complexity levels. For example, for each k ≥ 2 it. is now known that if one query to σ p k is as powerful as two queries to σ p k (i.e., P Σ[1] = P Σ[2] ), then PH = σ p k . This article surveys the history, the results, and the method---the so-called easy-hard technique---of the just-mentioned nine-paper progression:1. J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing , 17(6):1263--1282, 1988. Erratum appears in the same journal. 20(2):404.2. K. Wagner. Number-of-query hierarchies. Technical Report 158, Institut für Mathematik. Universität Augsburg, Augsburg, Germany, October 1987.3. K. Wagner. Number-of-query hierarchies. Technical Report 4. Institut für Informatik. Universität Würzburg, Würzburg, Germany, February 1989.4. R. Chang and J. Kadin. The boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM Journal on Computing . 25(2):340--354. 1996.5. R. Beigel. R. Chang. and M. Ogiwara. A relationship between difference hierarchies and relativized polynomial hierarchies. Mathematical Systems Theory . 26(3):293--310, 1993.6. E. Hemaspaandra, L. Hemaspaandra. and H. Hempel. An upward separation in the polynomial hierarchy. Technical Report Math/Inf/96/15, Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany, June 1996.7. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. A downward collapse within the polynomial hierarchy. SIAM Journal on Computing , 28(2):383--393, 1999.8. H. Buhrman and L. Fortnow. Two queries. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity . 13--19. IEEE Computer Society Press, June 1998.9. E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Translating equality downwards. Technical Report TR-657. Department of Computer Science, University of Rochester, Rochester, NY. April 1997.

Judgment aggregation is a collective decision making framework where the opinions of a group of agents is combined into a collective opinion. This can be done using many different judgment aggregation procedures. We study the computational complexity of computing the group opinion for several of the most prominent judgment aggregation procedures. In particular, we show that the complexity of this winner determination problem for analogues of the Kemeny rule, the Slater rule and the Young rule lies at the Θp2-level of the Polynomial Hierarchy (PH). Moreover, we show that the problem has a complexity at the Δp2-level of the PH for the analogue of Tideman's procedure with a fixed tie-breaking rule, and at the Σp2-level of the PH for the analogue of Tideman's procedure without a fixed tie-breaking rule.

This paper is an introduction to the computational complexity theory. I believe that the standard courses in complexity theory make some things much more important than they really are, while things which I find extremely interesting are marginalized. Thus, it can be seen as my subjective view on what is important and interesting in computational complexity, and what is not. We state the basic facts from computational complexity theory using the blackboard computations (one-dimensional cellular automata) as the standard computation model; we argue the benefits of such approach. This article is somewhat based on my series of talks from Warsztaty Logiczne 2014 in Białka Tatrzańska, although these talks covered more topics, such as NL=coNL, and links between BPP and the polynomial hierarchy.

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a characterization of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class. To test this notion, we pose some hypotheses related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses may be over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2Ω(n), then there exist inapproximable-on-average optimization problems of a particularly elegant form. All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardnessamplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.

We study computational complexity theory and define a class of optimization problems called OptP (Optimization Polynomial Time), and we show that TRAVELLING SALESPERSON, KNAPSACK and 0-1 INTEGER LINEAR PROGRAMMING are complete for OptP. OptP is a natural generalization of NP (Nondeterministic Polynomial Time), but while NP only considers problems at the level of their yes/no question, the value of an OptP function is the optimal value of the problem. This approach enables us to show a deeper level of structure in these problems than is possible in NP. OptP is a subset of FPSAT, the class of functions computable in polynomial time with an oracle for NP. Our central result is that any FPSAT function decomposes into an OptP function followed by polynomial-time computation. The significance of this result is that it quantifies how much NP-completeness is in a problem, i.e., the number of NP queries it takes to compute the function. It also allows us to unify the classes NP, DP and DELTA-2 in a natural way. For example, we prove that an OptP-completeness result implies, as corollaries, NP-, DP- and DELTA-2- completeness results. We also prove separation results on subclasses of FPSAT by restricting the number of calls to the NP oracle. For example, TRAVELLING SALESPERSON is complete for O(n) queries, CLIQUE is complete for O(log n) queries and BIN PACKING can be solved in O(log log n) queries. We prove these classes distinct under the assumption that P does not equal NP. Finally, we consider generalizations of OptP to higher levels in the Polynomial-Time Hierarchy. We define the DOUBLE KNAPSACK problem and prove that it is complete for DELTA-3, the first example of a natural complete problem for this class, and the highest level in the Polynomial Hierarchy with a known natural complete problem.

Average-case complexity, which examines the tractability of computational problems on ‘random instances,’ is a major topic in complexity theory with at least two distinct motivations. On one hand, it may provide a more realistic model than worst-case complexity for the problem instances actually encountered in practice. On the other hand, it provides us with methods to generate hard instances, allowing us to harness intractability for useful ends such as cryptography and derandomization. These two motivations are actually supported by a variety of different notions of average-case complexity (surveyed in [17, 13, 6]) and relating these notions is an important direction for research in the area. An even more ambitious goal is to understand the relationship between average-case complexity and worst-case complexity, e.g., whether NP = P implies that NP has problems that are hard on average. In recent years, there has been substantial progress on this front. This special issue aims to present a small sample of papers that are representative of the different types of results that have been obtained:

Boolean networks are a general model of interacting entities, with applications to biological phenomena such as gene regulation. Attractors play a central role, and the schedule of entity updates is a priori unknown. This article presents results on the computational complexity of problems related to the existence of update schedules such that some limit-cycle lengths are possible or not. We first prove that given a Boolean network updated in parallel, knowing whether it has at least one limit-cycle of length k is \({\mathsf {NP}}\)-complete. Adding an existential quantification on the block-sequential update schedule does not change the complexity class of the problem, but the following alternation brings us one level above in the polynomial hierarchy: given a Boolean network, knowing whether there exists a block-sequential update schedule such that it has no limit-cycle of length k is \({\mathsf {NP^{NP}}}\)-complete.

We initiate the study of the computational complexity of the covering radius problem for point lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor /spl gamma/(n) > 1 in random exponential time 2/sup O(n)/, it is in AM for /spl gamma/(n) = 2, in coAM for /spl gamma/(n) = /spl radic/(n log n), and in NP /spl cap/ coNP for /spl gamma/(n) = /spl radic/n. For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor /spl gamma/(n) = log n, but cannot be solved in polynomial time for some /spl gamma/(n) = /spl Omega/(log log n) unless NP can be simulated in deterministic n/sup O(log log log n)/ time. Moreover, we prove that the problem is NP-hard for every constant approximation factor, it is /spl Pi//sub 2/-hard for some constant approximation factor, and it is in AM for approximation factor 2. So, it is unlikely to be /spl Pi//sub 2/-hard for approximation factors larger than 2. This is a natural hardness of approximation result in the polynomial hierarchy. For the shortest independent vectors problem, we give a coAM protocol achieving approximation factor /spl gamma/(n) = /spl radic/(n/log n), solving an open problem of Blomer and Seifert (1999), and prove that the problem is also in coNP for /spl gamma/(n) = /spl radic/n. Both results are obtained by giving a gap-preserving nondeterministic polynomial time reduction to the closest vector problem.

This paper presents the following results on sets that are complete for NP. If there is a problem in NP that requires $2^{n^{\Omega(1)}}$ time at almost all lengths, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. If there is a problem in co-NP that cannot be solved by polynomial-size nondeterministic circuits, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in NP∩co-NP, then there is a Turing complete language for NP that is not many-one complete. Our first two results use worst-case hardness hypotheses whereas earlier work that showed similar results relied on average-case or almost-everywhere hardness assumptions. The use of average-case and worst-case hypotheses in the last result is unique as previous results obtaining the same consequence relied on almost-everywhere hardness results.

Article Interpolation of depth-3 arithmetic circuits with two multiplication gates Author: Amir Shpilka Technion, Haifa, Israel Technion, Haifa, IsraelView Profile Authors Info & Claims STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computingJune 2007Pages 284–293https://doi.org/10.1145/1250790.1250833Published:11 June 2007Publication History 12citation214DownloadsMetricsTotal Citations12Total Downloads214Last 12 Months4Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

\n Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a small circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erds and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called robust sunflowers, to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossmans bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain the most significant progress on the sunflower conjecture since its inception in 1960. In this work, we will show how the recent progresses in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, thus breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes.\n

Alternation is a generalized principle of nondeterminism. The alternating turing machine is used to characterize the polynomial hierarchy. In this paper we show, that a hierarchy can be characterized with alternating pushdown automata, which we expect to be strict in contrast to a hierarchy with alternating finite automata or alternating space bounded automata. We describe a similar oracle hierarchy over the context-free languages, for which we construct complete languages. We show, that each level of the hierarchy with alternating pushdown automata is included in the corresponding level of the oracle hierarchy and that the logarithmic closure over both levels is the corresponding level of the polynomial hierarchy with one alternation less.The principle of the alternation is also transfered to grammars. Hereby we prove, that the hierarchy with alternating context-free grammars is identical with the oracle hierarchy over the context-free languages and that in case of unbounded alternation context-free and context-sensitive grammars have the same power.KeywordsFinite AutomatonUniversal StateSentential FormPolynomial HierarchyExistential VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper has two main focal points. We first consider an important class of machine learning algorithms: large margin classifiers, such as Support Vector Machines. The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constant, margin complexity is equal to the inverse of discrepancy. This establishes a strong tie between seemingly very different notions from two distinct areas.In the same way that matrix rigidity is related to rank, we introduce the notion of rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare. This leads to the question of proving lower bounds on the rigidity of margin complexity. Quite surprisingly, this question turns out to be closely related to basic open problems in communication complexity, e.g., whether PSPACE can be separated from the polynomial hierarchy in communication complexity.Communication is a key ingredient in many types of learning. This explains the relations between the field of learning theory and that of communication complexity [6, l0, 16, 26]. The results of this paper constitute another link in this rich web of relations. These new results have already been applied toward the solution of several open problems in communication complexity [18, 20, 29].

Lyapunov design has never been systematic. In the adaptive control of complex multi-input non-linear systems, physical considerations, such as conservation of energy or entropy increase, represent one of the major tools in building Lyapunov-like functions and providing stability and performance guarantees. In this paper we show that a physically motivated Lyapunov-like function based on the concept of total information can be derived for large classes of non-linear physical systems. We study how this function may be used for designing estimation, adaptation and learning mechanisms for such systems. In the process we revisit familiar notions such as controllability and observability from an information perspective, which in turns allows us to define 'natural' space-time scales at which to observe and control a given complex system. By formulating control problems in algorithmic form, we emphasize the importance of computability and computational complexity for issues of control. Generic control problems are shown to be NP-hard: each additional complication, such as the presence of noise or the absence of complete system identification, moves the control problem further up the polynomial hierarchy of computational complexity. In some cases, requirements of 'optimality' may be unrealistic or irrelevant, since the solution to the problem of finding the optimal algorithm for control is uncomputable.

A wide variety of problems, ranging from planning and scheduling tasks in industry to making autonomous decisions in smart environments, can be modelled as constraint satisfaction problems (CSPs). While this general mathematical framework facilitates the reuse of algorithms for many different types of problems, the fact that CSPs are NP-complete gives them an exponential worst-case computational complexity. This complexity imposes severe restrictions on the size of the problem, as virtually all applications require problems to be solved in a timely fashion. However, despite being NP-complete problems, CSPs can often be solved efficiently in practice, making this worst-case complexity largely uninformative. The structure of a problem and of the search space have been used to identify classes of tractable problems and design more efficient solving techniques with better theoretical guarantees. Unfortunately, the worst-case bounds of these algorithms are hardly more informative when it comes to their practical efficiency, and exploiting this structure is challenging because efficient algorithms do not exist for many related tasks. This dissertation studies the structure of problems and their search space and how these influence the search times of algorithms. It provides additional insights into the computational complexity and practical efficiency of search algorithms, which can, for instance, be used to select the best algorithm for a given problem and aid in the design of more efficient solving techniques. This work also contributes towards improving the solving efficiency more directly by proposing improved ways to exploit the structure of a problem and the search space algorithmically.