Mathematical Foundations of Computer Science

In this survey paper, we present known results and open questions on a proper subclass of the class of regular languages. This class, denoted by W, is especially robust: it is closed under union, intersection, product, shuffle, left and right quotients, inverse of morphisms, length preserving morphisms and commutative closure. It can be defined as the largest positive variety of languages not containing the language (ab) * . It admits a nontrivial algebraic characterization in terms of finite ordered monoids, which implies that W is decidable: given a regular language, one can effectively decide whether or not it belongs to W. We propose as a challenge to find a constructive description and a logical characterization of W. ⋆ The authors acknowledge support from the AutoMathA programme of the European Science Foundation.

We give an attribute-based encryption system for Turing Machines that is provably secure assuming only the existence of identity-based encryption (IBE) for large identity spaces. Currently, IBE is known to be realizable from most mainstream number theoretic assumptions that imply public key cryptography including factoring, the search Diffie-Hellman assumption, and the Learning with Errors assumption. Our core construction provides security against an attacker that makes a single key query for a machine T before declaring a challenge string w * that is associated with the challenge ciphertext. We build our construction by leveraging a Garbled RAM construction of Gentry, Halevi, Raykova and Wichs [GHRW14a]; however, to prove security we need to introduce a new notion of security called iterated simulation security. We then show how to transform our core construction into one that is secure for an a-priori bounded number q = q(λ) of key queries that can occur either before or after the challenge ciphertext. We do this by first showing how one can use a special type of non-committing encryption to transform a system that is secure only if a single key is chosen before the challenge ciphertext is declared into one where the single key can be requested either before or after the challenge ciphertext. We give a simple construction of this non-committing encryption from public key encryption in the Random Oracle Model. Next, one can apply standard combinatorial techniques to lift from single-key adaptive security to q-key adaptive security.

We construct a broadcast and trace scheme (also known as trace and revoke or broadcast, trace and revoke) with N users, where the ciphertext size can be made as low as O(N ε ), for any arbitrarily small constant ε > 0. This improves on the prior best construction of broadcast and trace under standard assumptions by Boneh and Waters (CCS '06), which had ciphertext size O(N 1/2 ). While that construction relied on bilinear maps, ours uses a combination of the learning with errors (LWE) assumption and bilinear maps. Recall that, in both broadcast encryption and traitor-tracing schemes, there is a collection of N users, each of which gets a different secret key ski. In broadcast encryption, it is possible to create ciphertexts targeted to a subset S ⊆ [N ] of the users such that only those users can decrypt it correctly. In a traitor tracing scheme, if a subset of users gets together and creates a decoder box D that is capable of decrypting ciphertexts, then it is possible to trace at least one of the users responsible for creating D. A broadcast and trace scheme intertwines the two properties, in a way that results in more than just their union. In particular, it ensures that if a decoder D is able to decrypt ciphertexts targeted toward a set S of users, then it should be possible to trace one of the users in the set S responsible for creating D, even if other users outside of S also participated. As of recently, we have essentially optimal broadcast encryption (Boneh, Gentry, Waters CRYPTO '05) under bilinear maps and traitor tracing (Goyal, Koppula, Waters STOC '18) under LWE, where the ciphertext size is at most polylogarithmic in N . The main contribution of our paper is to carefully combine LWE and bilinear-map based components, and get them to interact with each other, to achieve broadcast and trace.

In this paper we introduce the notion of lockable obfuscation. In a lockable obfuscation scheme there exists an obfuscation algorithm Obf that takes as input a security parameter λ, a program P , a message msg and "lock value" α and outputs an obfuscated program P . One can evaluate the obfuscated program P on any input x where the output of evaluation is the message msg if P (x) = α and otherwise receives a rejecting symbol ⊥. We proceed to provide a construction of lockable obfuscation and prove it secure under the Learning with Errors (LWE) assumption. Notably, our proof only requires LWE with polynomial hardness and does not require complexity leveraging. We follow this by describing multiple applications of lockable obfuscation. First, we show how to transform any attribute-based encryption (ABE) scheme into one in which the attributes used to encrypt the message are hidden from any user that is not authorized to decrypt the message. (Such a system is also know as predicate encryption with one-sided security.) The only previous construction due to Gorbunov, Vaikuntanathan and Wee is based off of a specific ABE scheme of Boneh et al. By enabling the transformation of any ABE scheme we can inherent different forms and features of the underlying scheme such as: multi-authority, adaptive security from polynomial hardness, regular language policies, etc. We also show applications of lockable obfuscation to separation and uninstantiability results. We first show how to create new separation results in circular encryption that were previously based on indistinguishability obfuscation. This results in new separation results from learning with error including a public key bit encryption scheme that it IND-CPA secure and not circular secure. The tool of lockable obfuscation allows these constructions to be almost immediately realized by translation from previous indistinguishability obfuscation based constructions. In a similar vein we provide random oracle uninstantiability results of the Fujisaki-Okamoto transformation (and related transformations) from the lockable obfuscation combined with fully homomorphic encryption. Again, we take advantage that previous work used indistinguishability obfuscation that obfuscated programs in a form that could easily be translated to lockable obfuscation.

Theoretical study of optimization problems in wireless communication often deals with tasks that concern a single point. For example, the power control problem requires computing a power assignment guaranteeing that each transmitting station s i is successfully received at a single receiver point r i . This paper aims at addressing communication applications that require handling two-dimensional tasks (e.g., guaranteeing successful transmission in entire regions rather than at specific points). The natural approach to two-dimensional optimization tasks is to discretize the optimization domain, e.g., by sampling points within the domain. The straightforward implementation of the discretization approach, however, might incur high time and memory requirements, and moreover, it cannot guarantee exact solutions. The alternative proposed and explored in this paper is based on establishing the minimum principle 1 for the signal to interference and noise ratio (SINR) function with free space path loss (i.e., when the signal decays in proportion to the square of the distance between the transmitter and receiver). Essentially, the minimum principle allows us to reduce the dimension of the optimization domain without losing anything in the accuracy or quality of the solution. More specifically, when the two-dimensional optimization domain is bounded and free from any interfering station, the minimum principle implies that it is sufficient to optimize the SINR function over the boundary of the domain, as the "hardest" points to be satisfied reside on the boundary and not in the interior. We then utilize the minimum principle as the basis for an improved discretization technique for solving two-dimensional problems in the SINR model. This approach is shown to be useful for handling optimization problems over two dimensions (e.g., power control, energy minimization); in providing tight bounds on the number of null cells in the reception map; and in approximating geometric and topological properties of the wireless reception map (e.g., maximum inscribed sphere). 1 A function satisfies the minimum principle if its minimum in any compact domain is attained at the domain's boundary.

Linear optimization is many times algorithmically simpler than non-linear convex optimization. Linear optimization over matroid polytopes, matching polytopes and path polytopes are example of problems for which we have efficient combinatorial algorithms, but whose non-linear convex counterpart is harder and admit significantly less efficient algorithms. This motivates the computational model of online decision making and optimization using a linear optimization oracle. In this computational model we give the first efficient decision making algorithm with optimal regret guarantees, answering an open question of [1], [2], in case the decision set is a polytope. We also give an extension of the algorithm for the partial information setting, i.e. the "bandit" model. Our method is based on a novel variant of the conditional gradient method, or Frank-Wolfe algorithm, that reduces the task of minimizing a smooth convex function over a domain to that of minimizing a linear objective. Whereas previous variants of this method give rise to approximation algorithms, we give such algorithm that converges exponentially faster and thus runs in polynomial-time for a large class of convex optimization problems over polyhedral sets, a result of independent interest.

In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two meta-theorems on kernelzation. The first theorem says that all problems expressible in Counting Monadic Second Order Logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.

This is a collection of open problems presented at the Lorentz Workshop “Enumeration Algorithms using Structure” which took place at the Lorentz Center of the University of Leiden (The Netherlands), August 24 - 30, 2015. The workshop brought together researchers interested in various aspects of enumeration algorithms; in particular classical output-sensitive enumeration including output-polynomial and polynomial delay algorithms and the inputsensitive enumeration mainly dealing with exact exponential time enumeration algorithms and combinatorial bounds.

The first theorem of Cantor's 1891 paper introduced the diagonal method by a specific example; namely, as an argument that any list of the set of real numbers is necessarily incomplete and concluding the set of real numbers is not denumerable. The present paper argues that the diagonal method is not a valid form of argument (i.e., not a valid proof schema). Any of the three independent refutations presented here suffice to prove that Cantor's Diagonal Argument (CDA) cannot logically lead to the conclusion(s) alleged for it. The first of the three refutations is applicable to any alleged proof by diagonal method, including its application to non-numeric sets. "And as of 'no-go theorems'[:] … One always has to take the assumptions into consideration, just as the small print in a contract."-Gerard t'Hooft (1999 Nobelist in Physics) [1] I.

This paper considers completions of tope graphs of COMs (complexes of oriented matroids) to ample partial cubes of the same VC-dimension. We show that these exist for OMs (oriented matroids) and CUOMs (complexes of uniform oriented matroids). This implies that tope graphs of OMs and CUOMs satisfy the sample compression conjecture -one of the central open questions of learning theory. We conjecture that the tope graph of every COM can be completed to an ample partial cube without increasing the VC-dimension. OMs, COMs, and AMPs (viewed as isometric subgraphs of hypercubes) give rise to set families for which the VC-dimension has a particular significance: the VC-dimension of an AMP is the largest dimension of a cube of its cube complex, the VC-dimension of an OM is its rank, and the VC-dimension of a COM is the largest VC-dimension of its faces. Littlestone and Warmuth introduced the sample compression technique for deriving generalization bounds in machine learning. Floyd and Warmuth [19] asked whether any set family of VC-dimension d has a sample compression scheme of size O(d). This question, known as the sample compression conjecture, remains one of the oldest open problems in machine learning. It was shown in [28] that labeled compression schemes of size O(2 d ) exist. Moran and Warmuth [27] designed labeled compression schemes of size d for ample set systems. Chalopin et al. [9] designed (stronger) unlabeled compression schemes of size d for maximum families and characterized such schemes for ample set systems. In view of the above, it was noticed in [30] and [27] that the sample compression conjecture would be solved if any set family of VC-dimension d can be completed to an ample (or maximum) set system of VC-dimension O(d) or covered by O(2 d ) ample set systems of VC-dimension O(d). This opens a perspective that apart from its application to sample compression, is interesting in its own right: ample completions of structured set families. This is extending a given set system to an ample system by adding sets. A natural problem is ample completions of set families defined by partial cubes (i.e., isometric subgraphs of hypercubes). In [14], we prove that any partial cube of VC-dimension 2 admits an ample completion of VC-dimension 2. Moreover, we give a set family of VC-dimension 2 which has no ample completion of the same VC-dimension. In the present paper, we give an example of a partial cube of VC-dimension 3 which cannot be completed to an ample set system of VC-dimension 3. Hence, in higher dimension we cannot complete all partial cubes without increasing the VC-dimension. In the light of the above perspective, one may ask if there exists a constant c such that every partial cube of VC-dimension d admits an ample completion of VC-dimension ≤ cd? Even stronger, we wonder if partial cubes of VC-dimension d admit an ample completion of VC-dimension d + c. Note that no such additive constant c exists for general set families . In [14], we perform the ample completion of a partial cube of VC-dimension 2 in two steps. First, we show that they can be completed to tope graphs of COMs and next we complete the resulting graphs into ample ones, in both cases, without increasing the VC-dimension. In the present article, we are interested in the completion of this intermediate class of partial cubes. For COMs, we are inclined to believe that the following stronger result holds: Conjecture 1. The tope graph of any COM of VC-dimension d has an ample completion of VC-dimension d. COMs of rank 2 have the nice property that their faces are uniform OMs. This is not longer true in higher dimensions: COMs whose faces are uniform OMs constitute a proper subclass of COMs (that we call CUOMs). In this paper, we prove that Conjecture 1 holds for all tope graphs of OMs and CUOMs. This proves that set families arising from topes of OMs and CUOMs satisfy the sample compression conjecture. In Fig. , we present an example of the tope graph of a CUOM of VC-dimension 3, which we further use as a running example. 2.1. VC-dimension. Let S be a family of subsets of an m-element set U . A subset X of U is shattered by S if for all Y ⊆ X there exists S ∈ S such that S ∩ X = Y . The Vapnik-Chervonenkis dimension [35] (the VC-dimension for short) VC-dim(S) of S is the cardinality of the largest subset of U shattered by S. Any set family S ⊆ 2 U can be viewed as a subset of vertices of the m-dimensional hypercube Q m = Q(U ). Denote by G(S) the subgraph of Q m

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex Q - 3 (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -a property naturally generalizing the notion of median graphs.

25 pages, 5 figuresInternational audienceWe investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986). Furthermore we point out relations to tope graphs of COMs of low rank and r...

A common subproblem of DNF approximate counting and derandomizing RL is the discrepancy problem for combinatorial rectangles. We explicitly construct a poly(n)-size sample space that approximates the volume of any combinatorial rectangle in [n] n to within o(1) error (improving on the constructions of [EGLNV92]). The construction extends the techniques of [LLSZ95] for the analogous hitting set problem, most notably via discrepancy preserving reductions.

In [KUW1] we have proposed the setting of independence systems to study the relation between the computational complexity of search and decision problems. The universal problem that captures this relation, which we termed the $S$-search problem, is "Given an oracle for the input system, find a maximal independent subset in it." Many interesting and important search problems can be described by an special class of independence systems, called matroids.

The Boolean Decision tree model is perhaps the simplest model that computes Boolean functions; it charges only for reading an input variable. WE study the power of randomness (vs. both determinism and non-determinism) in this model, and prove separation results between the three complexity measures. These results are obtained via general and efficient methods for computing upper and lower bounds on the probabilistic complexity of evaluating Boolean formulae in which every variable appears exactly once (AND/OR tree with distinct leaves). These bounds are shown to be exactly tight for interesting families of such tree functions. We then apply our results to the complexity of evaluating game trees, which is a central problem in $AI$. These trees are similar to Boolean tree functions, except that input variables (leaves) may take values from a large set (of valuations to game positions) and the AND/OR nodes are replaced by MIN/MAX nodes. Here the cost is the number of positions (leaves) probed by the algorithm. The best-known algorithm for this problem is the alpha-beta pruning method. As a deterministic algorithm, it will in the worst case have to examine all positions. Many papers studied the expected behavior of alpha-beta pruning (or uniform trees) under the unreasonable assumption that position values are drawn independently from some distribution. We analyze a randomized variant of alpha-beta pruning, show that it is a considerably faster than deterministic one in worst case, and prove it optimal for uniform trees.

On a Search Problem Related to Branch-and-Bound Procedures RM Karp,* M. Saks,** A. Wigderson*** 0. INTRODUCTION Branch-and-bound procedures are commonly used in practice for the solution of NP-hard combinatorial ... University of California and MSRI, Berkeley. ...

Various efforts ([?, ?, ?]) have been made in recent years to derandomize probabilistic algorithms using the complexity theoretic assumption that there exists a problem in E = dtime(2 O(n) ), that requires circuits of size s(n), (for some function s). These results are based on the NW-generator [?]. For the strong lower bound s(n) = 2 n , [?], and later [?] get the optimal derandomization, P = BP P . However, for weaker lower bound functions s(n), these constructions fall far short of the natural conjecture for optimal derandomization, namely that bptime(t) ⊆ dtime(2 O(s -1 (t)) ). The gap in these constructions is due to an inherent limitation on efficiency in NW-style pseudo-random generators. In this paper we are able to get derandomization in almost optimal time using any lower bound s(n). We do this by using the NW-generator in a new, more sophisticated way. We view any failure of the generator as a reduction from the given "hard" function to its restrictions on smaller input sizes. Thus, either the original construction works (almost) optimally, or one of the restricted functions is (almost) as hard as the original. Any such restriction can then be plugged into the NW-generator recursively. This process generates many "candidate" generators -all are (almost) optimal, and at least one is guaranteed to be "good". Then, to perform the approximation of the acceptance probability of the given circuit (which is the key to derandomization), we use ideas from [?]: we run a tournament between the "candidate" generators which yields an accurate estimate. Following Trevisan, we explore information theoretic analogs of our new construction. Trevisan [?] (and then [?]) used the NW-generator to construct efficient extractors. However, the inherent limitation of the NW-generator mentioned above makes the extra randomness required by that extractor suboptimal (for certain parameters). Applying our construction, we show how to use a weak random souce with optimal amount of extra randomness, for the (simpler than extraction) task of estimating the probability of any event (which is given by an oracle).

What is the least surface area of a shape that tiles R d under translations by Z d ? Any such shape must have volume 1 and hence surface area at least that of the volume-1 ball, namely Ω( √ d). Our main result is a construction with surface area O( √ d), matching the lower bound up to a constant factor of 2 p 2π/e ≈ 3. The best previous tile known was only slightly better than the cube, having surface area on the order of d. We generalize this to give a construction that tiles R d by translations of any full rank discrete lattice Λ with surface area 2π ‚ ‚ V -1 ‚ ‚ fb , where V is the matrix of basis vectors of Λ, and • fb denotes the Frobenius norm. We show that our bounds are optimal within constant factors for rectangular lattices. Our proof is via a random tessellation process, following recent ideas of Raz [11] in the discrete setting. Our construction gives an almost optimal noise-resistant rounding scheme to round points in R d to rectangular lattice points.

This paper contains two main results. The first is an explicit construction of bipartite graphs which do not contain certain complete bipartite subgraphs and have maximal density, up to a constant factor, under this constraint. This construction represents the first significant progress in three decades on this old problem in extremal graph theory. The construction beats the previously known probabilistic lower bound on density. The proof uses the elements of commutative algebra and algebraic geometry (theory of ideals, integral extensions, valuation rings). The second result concerns monotone span programs. We obtain the first superpolynomial lower bounds for explicit functions in this model. The best previous lower bound was Ω(n 5/2 ) by Beimel, Gál, Paterson (FOCS'95); our analysis exploits a general combinatorial lower bound criterion from that paper. We give two proofs of superpolynomial lower bounds; one based on the construction in the first part, the other based on an analysis of Paley-type bipartite graphs via Weil's character sum estimates. A third result demonstrates the power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae 1 . * Part of this work was done while two of the authors, Gál and Rónyai, were visiting the University of Chicago.

We investigate the feasibility of a variety of cryptographic tasks with imperfect randomness. The kind of imperfect randomness we consider are entropy sources, such as those considered by Santha and Vazirani, Chor and Goldreich, and Zuckerman. We show the following: ¯Certain cryptographic tasks like bit commitment, encryption, secret sharing, zero-knowledge, noninteractive zero-knowledge, and secure two-party computation for any non-trivial function are impossible to realize if parties have access to entropy sources with slightly less-than-perfect entropy, i.e., sources with imperfect randomness. These results are unconditional and do not rely on any unproven assumption. ¯On the other hand, based on stronger variants of standard assumptions, secure signature schemes are possible with imperfect entropy sources. As another positive result, we show (without any unproven assumption) that interactive proofs can be made sound with respect to imperfect entropy sources.

In this work, we study indistinguishability obfuscation and functional encryption for general circuits: Indistinguishability obfuscation requires that given any two equivalent circuits C0 and C1 of similar size, the obfuscations of C0 and C1 should be computationally indistinguishable. In functional encryption, ciphertexts encrypt inputs x and keys are issued for circuits C. Using the key SKC to decrypt a ciphertext CTx = Enc(x), yields the value C(x) but does not reveal anything else about x. Furthermore, no collusion of secret key holders should be able to learn anything more than the union of what they can each learn individually. We give constructions for indistinguishability obfuscation and functional encryption that supports all polynomial-size circuits. We accomplish this goal in three steps: • We describe a candidate construction for indistinguishability obfuscation for NC 1 circuits. The security of this construction is based on a new algebraic hardness assumption. The candidate and assumption use a simplified variant of multilinear maps, which we call Multilinear Jigsaw Puzzles. • We show how to use indistinguishability obfuscation for NC 1 together with Fully Homomorphic Encryption (with decryption in NC 1 ) to achieve indistinguishability obfuscation for all circuits. • Finally, we show how to use indistinguishability obfuscation for circuits, public-key encryption, and non-interactive zero knowledge to achieve functional encryption for all circuits. The functional encryption scheme we construct also enjoys succinct ciphertexts, which enables several other applications.

The concept of forwarding sets is widely adopted in many broadcast protocols for wireless multihop networks to alleviate the broadcast storm problem. In these protocols, after receiving a broadcast message, each node that is requested to relay the message instructs a subset of its 1-hop neighbors, a.k.a. the forwarding set, to further relay it. In this paper, we propose to use the Minimum Local Disk Cover Set (MLDCS) as the forwarding set in heterogeneous multihop wireless networks, where nodes may have different transmission ranges.We show that the minimum local disk cover set of a node in heterogeneous networks is equivalent to its skyline set, and then we propose a divide-and-conquer algorithm with the optimal time complexity to compute the skyline set locally and statelessly. Moreover, unlike other forwarding heuristics, the proposed algorithm requires only 1-hop neighbor information. This helps to reduce the forwarding set formation latency and thus will be more suitable for en...

We introduce an extension of the restricted shuffle operator on binary words considered by Atanasiu and Teh (2016). We then derive properties on Parikh matrix equivalence of words over a binary alphabet based on this extended shuffle operator and a weak-ratio property of words. We also examine the recently introduced concept of core Parikh matrix equivalence of binary words in the context of the restricted shuffle operator.

Parikh matrix mapping or Parikh matrix of a word has been introduced in the literature to count the scattered subwords in the word. Several properties of a Parikh matrix have been extensively investigated. A picture array is a two-dimensional connected digitized rectangular array consisting of a finite number of pixels with each pixel in a cell having a label from a finite alphabet. Here we extend the notion of Parikh matrix of a word to a picture array and associate with it two kinds of Parikh matrices, called row Parikh matrix and column Parikh matrix. Two picture arrays A and B are defined to be M-equivalent if their row Parikh matrices are the same and their column Parikh matrices are the same. This enables to extend the notion of M-ambiguity to a picture array. In the binary and ternary cases, conditions that ensure M-ambiguity are then obtained.

We present a randomized NC solution to the problem of constructing a maximum (cardinality) f -matching. As a corollary, we obtain a randomized NC algorithm for the problem of constructing a graph satisfying a sequence d 1 ; d 2 ;...; d n of equality degree constraints. We provide an optimal NC algorithm for the decision version of the degree sequence problem and an approximation NC algorithm for the construction version of this problem. Our main result is an NC algorithm for constructing if possible a graph satisfying the degree constraints d 1 ; d 2 ;...; d n in case d i q P n j=1 d j =5

The buy-at-bulk network design problem has been extensively studied in the general graph model. In this paper, we consider geometric versions of the problem, where all points in a Euclidean space are candidates for network nodes, and present the first general approach for solving them. It enables us to obtain quasi-polynomial-time approximation schemes for basic variants of the buy-at-bulk geometric network design problem with polynomial total demand. Then, for instances with a single sink and low capacity links, we design fast polynomial-time, low-constant approximation algorithms.

Let k be a positive integer, a subset Q of the set of vertices of a graph G is k-dependent in G if each vertex of Q has no more than k neighbours in Q. We present a parallel algorithm which computes a maximal k-dependent set in a graph on n nodes in time O( log 4 n) on an EREW PRAM with O(n2) processors. In this way, we establish the membership of the problem of constructing a maximal k-dependent set in the class NC. Our algorithm can be easily adapted to compute a maximal k-dependent set in a graph of bounded valence in time O( log * n) using only O(n) EREW PRAM processors. Let f be a positive integer function defined on the set V of vertices of a graph G. A subset F of the set of edges of G is said to be an f-matching if every vertex vɛV is adjacent to at most f(v) edges in-F. We present the first NC algorithm for constructing a maximal f-matching. For a graph on n nodes and m edges the algorithm runs in time O( log 4 n) and uses O(n+m) EREW PRAM processors. For graphs of constant...

We consider the following clustering problems: given an undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP), or the number of edges between clusters is minimized (Min-ECP). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

This paper directly refutes the motivating points of §8: Application of the diagonal process from Alan Turing’s paper On Computable Numbers. After briefly touching upon the uncontested fact that computational machines are necessarily fully enumerable, we will discuss an alternative to Turing’s algorithm for computing direct diagonal across the computable numbers. This alternative not only avoids an infinite recursion, but also any sort of decision paradox. Then, by using techniques described in §3 of how to resolve a halting paradox to correct the interface of decision machine 𝓓, we will mitigate the decision paradox that occurs in Turing’s attempt at computing a direct diagonal, and show that it can actually compute a direct diagonal. Finally, we will analogously fix the decision paradox found in trying to compute an inverse diagonal, but in this case we will demonstrate that the resulting computation is not sufficient to produce a complete inverse diagonal. Opposed to Turing’s several objections, there is no way to utilize a paradox-resistant correction of 𝓓 to compute an inconsistency that would disprove its existence. This undermines the foundation which Turing builds his uncomputability arguments on, and leaves us with an open question on the true nature of computability.

Existing work on "rational cryptographic protocols" treats each party (or coalition of parties) running the protocol as a selfish agent trying to maximize its utility. In this work we propose a fundamentally different approach that is better suited to modeling a protocol under attack from an external entity. Specifically, we consider a two-party game between an protocol designer and an external attacker. The goal of the attacker is to break security properties such as correctness or privacy, possibly by corrupting protocol participants; the goal of the protocol designer is to prevent the attacker from succeeding. We lay the theoretical groundwork for a study of cryptographic protocol design in this setting by providing a methodology for defining the problem within the traditional simulation paradigm. Our framework provides ways of reasoning about important cryptographic concepts (e.g., adaptive corruptions or attacks on communication resources) not handled by previous game-theoretic treatments of cryptography. We also prove composition theorems that-for the first time-provide a sound way to design rational protocols assuming "ideal communication resources" (e.g., broadcast or authenticated channels) and then instantiate these resources using standard cryptographic tools. Finally, we investigate the problem of secure function evaluation in our framework, where the attacker has to pay for each party it corrupts. Our results demonstrate how knowledge of the attacker's incentives can be used to circumvent known impossibility results in this setting.

We revisit the following question: what is the optimal round complexity of verifiable secret sharing (VSS)? We focus here on the case of perfect VSS where the number of corrupted parties t satisfies t < n/3, with n the total number of parties. Work of Gennaro et al. (STOC 2001) and Fitzi et al. (TCC 2006) shows that, assuming a broadcast channel, 3 rounds are necessary and sufficient for efficient VSS. Existing protocols, however, treat the broadcast channel as being available "for free" and do not attempt to minimize its usage. This approach leads to relatively poor round complexity when such protocols are compiled to run over a point-to-point network. We show here a VSS protocol that is simultaneously optimal in terms of both the number of rounds and the number of invocations of broadcast. Our protocol also satisfies a certain "2-level sharing" property that makes it useful for constructing protocols for general secure computation.

Combinations are a foundational element of discrete mathematics with applications across probability, computer science, and number theory. In this paper, we introduce a novel combinatorial model-Patterned Combinations with Positional Restrictions (PCPR)-to extend traditional combination structures by incorporating modular and positional constraints. This paper presents the formal definition of PCPR, provides examples, outlines potential computational algorithms, and proposes new directions for its application in cryptographic systems and optimized data structures. Our analysis demonstrates that PCPR offers a promising new avenue in combinatorial mathematics with broad interdisciplinary potential.

One of the most trade-off aspects in the design of NoCs is the improvement of the network performance, in terms of throughput and latency, while minimizing power consumption. 2D-mesh has become the preferred topology, since it offers low and constant link delay. This paper proposes a Power efficient, Oblivious, Path-diverse, Minimal routing (POPM) for mesh-based Networks-on-Chip. In order to improve the performance of the network, POPM makes routing decisions locally at each hop rather than establishing a fixed and deterministic path between the source and destination nodes. POPM routes each packet separately through a path selected from among all minimal paths. Detailed simulations on a set of synthetic traffic patterns as well as a real application traffic pattern show that POPM competes favorably to existing routing algorithms, including dimensionordered (DOR) routing, North-Last turn model, and PROM routing.

We consider a generalization of the maximum flow problem in which the amounts of flow entering and leaving an arc are linearly related. More precisely, if x(e) units of flow enter an arc e, z(e)7(e) units arrive at the other end. For instance, nodes of the graph can correspond to different currencies, with the multipliers being the exchange rates. We require conservation of flow at every node except a given source node. The goal is to maximize the amount of flow excess at the source. This problem is a special case of linear programming, and therefore can be solved in polynomial time. In this paper we present the first polynomial time combinat1o ial algorithms for this problem. The algorithms are simple and intuitive.

The verification of monadic second-order (MSO) graph properties, equivalently, the model-checking problem for MSO logic over finite binary relational structures, is fixed-parameter tractable (FPT) for the parameter consisting of the formula that expresses the property and the tree-width or the clique-width of the input graph or structure. How to build usable algorithms for this problem ? The proof of the general theorem (an algorithmic meta-theorem, cf. ) is based on the description of the inputs by algebraic terms and the construction of finite automata that accepts the terms describing the satisfying inputs. But these automata are in practice much too large to be constructed . A typical number of states is 2 2 10 and lower-bounds match this number. Can one use automata and overcome this difficulty ? We propose to use fly-automata (FA) . They are automata whose states are described and not listed, and whose transitions are computed on the fly and not tabulated. When running on a term of size 1000, a fly-automaton with 2 2 10 states computes only 1000 transitions if it is deterministic. FA can have infinitely many states. For example, a state can record, among other things, the (unbounded)

We address the concrete problem of implementing huge bottom-up term automata. Such automata arise from the verification of Monadic Second Order propositions on graphs of bounded tree-width or clique-width. This applies to graphs of bounded tree-width because bounded tree-width implies bounded clique-width. An automaton which has so many transitions that they cannot be stored in a transition table is represented be a fly-automaton in which the transition function is represented by a finite set of meta-rules. Fly-automata have been implemented inside the Autowrite 1 software and experiments have been run in the domain of graph model checking 2 .

In [H] it is proven, among other things, that the size of any depth k circuit computing the parity or the majority function is n(2o. 1 (o.311)""-u). In th'is note we generalize th~ptoof given there to yield similar lower bounds•for arbitrary .symmetric functions of {O,l}". This irpprpves results in [FKPS], where it was shown that the non-existence cf polynomi'!1-size; constant-depth circuits for the majority function implies the non-existence of such circuits for other symmetric fUl'lctions. • • . . f d • ,.

What is the minimum number 7 of edges to be added to a given graph G so that in the resulting graph the edge-connectivity between every pair {U, U} of nodes is at least a prescribed value T(U, v ) ? Generalizing earlier results of S. Sridhar and R. Chandrasekaran (when G is the empty graph), of K.P. Eswaran and R.E. Tarjan (when T(U,V) E 2), and of G-R. Cai and Y.-G. G k 2 2) we derive a min-max formula for 7 and describe a polynomial time algorithm to compute 7. The directed counterpart of the problem is also solved for the case when ~( u , w ) E k 2 1. A fundamental tool in the proof is the splitting theorems of W. Mader and L. LovLz. We also rely extensively on techniques concerning submodular functions. The method makes it possible to solve a degree-constrained version of the problem. The minimum-cost augmentation problem can also be solved in polynomial time provided that the edge-costs arise from node-costs while the problem for arbitrary edge-costs was known to be NP-complete even for T(U, v) 2.

The aim of this paper is to study communication in networks where nodes fail in a random dependent way. In order to capture fault dependencies, we introduce the neighborhood fault model, where damaging events, called spots, occur randomly and independently with probability p at nodes of a network, and cause faults in the given node and all of its neighbors. Faults at distance at most 2 become dependent in this model and are positively correlated. We investigate the impact of spot probability on feasibility and time of communication in the fault-free part of the network. We show a network which supports fast communication with high probability, if p ≤ 1/c log n. We also show that communication is not feasible with high probability in most classes of networks, for constant spot probabilities. For smaller spot probabilities, high probability communication is supported even by bounded degree networks. It is shown that the torus supports communication with high probability when p decreases faster than 1/n 1/2 , and does not when p ∈ 1/O(n 1/2 ). Furthermore, a network built of tori is designed, with the same fault-tolerance properties and additionally supporting fast communication. We show, however, that networks of degree bounded by a constant d do not support communication with high probability, if p ∈ 1/O(n 1/d ). While communication in networks with independent faults was widely studied, this is the first analytic paper which investigates network communication for random dependent faults.