Two-Way Automata Versus Logarithmic Space

We strengthen a previously known connection between the size complexity of two-way finite automata ( ) and the space complexity of Turing machines (tms). Specifically, we prove that

every s-state has a poly(s)-state that agrees with it on all inputs of length ≤s if and only if NL?L/poly, and

every s-state has a poly(s)-state that agrees with it on all inputs of length ≤2 ^s if and only if NLL?LL/polylog.

Here, and are the deterministic and nondeterministic , NL and L/poly are the standard classes of languages recognizable in logarithmic space by nondeterministic tms and by deterministic tms with access to polynomially long advice, and NLL and LL/polylog are the corresponding complexity classes for space O(loglogn) and advice length poly(logn). Our arguments strengthen and extend an old theorem by Berman and Lingas and can be used to obtain variants of the above statements for other modes of computation or other combinations of bounds for the input length, the space usage, and the length of advice.     

Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 42(1):51–63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction:

• Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution.
• Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs.
• If T is a given tree, deciding whether a circle graph G has a dominating set inducing a graph isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by t=|V(T)|. We prove that the FPT algorithm runs in subexponential time, namely $2^{\mathcal{O}(t \cdot\frac{\log\log t}{\log t})} \cdot n^{\mathcal{O}(1)}$ , where n=|V(G)|.

     

On Uniformity and Circuit Lower Bounds

We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts:

1. Lower bounds against medium-uniform circuits. Informally, a circuit class is “medium uniform” if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium-uniform circuit classes, including: ? For all k, P is not contained in P-uniform SIZE(n ^k ). That is, for all k, there is a language \({L_k \in {\textsf P}}\) that does not have O(n ^k )-size circuits constructible in polynomial time. This improves Kannan’s lower bound from 1982 that NP is not in P-uniform SIZE(n ^k ) for any fixed k. ? For all k, NP is not in \({{\textsf P}^{\textsf NP}_{||}-{\textsf {uniform SIZE}}(n^k)}\) .This also improves Kannan’s theorem, but in a different way: the uniformity condition on the circuits is stronger than that on the language itself. ? For all k, LOGSPACE does not have LOGSPACE-uniform branching programs of size n ^k .
2. Eliminating non-uniformity and (non-uniform) circuit lower bounds. We complement these results by showing how to convert any potential simulation of LOGTIME-uniform NC ¹ in ACC ⁰/poly or TC ⁰/poly into a medium-uniform simulation using small advice. This lemma can be used to simplify the proof that faster SAT algorithms imply NEXP circuit lower bounds and leads to the following new connection: ? Consider the following task: given a TC ⁰ circuit C of n ^O(1) size, output yes when C is unsatisfiable, and output no when C has at least 2 ^n-2 satisfying assignments. (Behavior on other inputs can be arbitrary.) Clearly, this problem can be solved efficiently using randomness. If this problem can be solved deterministically in 2 ^n-ω(log n) time, then \({{\textsf{NEXP}} \not \subset {\textsf{TC}}^0/{\rm poly}}\) .

Another application is to derandomize randomized TC ⁰ simulations of NC ¹ on almost all inputs: ?Suppose \({{\textsf{NC}}^1 \subseteq {\textsf{BPTC}}^0}\) . Then, for every ε > 0 and every language L in NC ¹, there is a LOGTIME?uniform TC ⁰ circuit family of polynomial size recognizing a language L′ such that L and L′ differ on at most \({2^{n^{\epsilon}}}\) inputs of length n, for all n.     

Log-Space Algorithms for Paths and Matchings in k-Trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 (Jakoby and Tantau in Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007). In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect matching (decision version), and if so, finding a perfect matching (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs, as shown in Datta et al. (Theory Comput. Syst. 47:737–757, 2010). Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of the divide-and-conquer approach in log-space, along with some ideas from Jakoby and Tantau (Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007) and Limaye et al. (Theory Comput. Syst. 46(3):499–522, 2010).     

DNF sparsification and a faster deterministic counting algorithm

Given a DNF formula f on n variables, the two natural size measures are the number of terms or size s(f) and the maximum width of a term w(f). It is folklore that small DNF formulas can be made narrow: if a formula has m terms, it can be ${\epsilon}$ -approximated by a formula with width ${{\rm log}(m/{\epsilon})}$ . We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be ${\epsilon}$ -approximated by a width w DNF with at most ${(w\, {\rm log}(1/{\epsilon}))^{O(w)}}$ terms. We combine our sparsification result with the work of Luby & Velickovic (1991, Algorithmica 16(4/5):415–433, 1996) to give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF. Given a formula on n variables with poly(n) terms, we give a deterministic ${n^{\tilde{O}({\rm log}\, {\rm log} (n))}}$ time algorithm that computes an additive ${\epsilon}$ approximation to the fraction of satisfying assignments of f for ${\epsilon = 1/{\rm poly}({\rm log}\, n)}$ . The previous best result due to Luby and Velickovic from nearly two decades ago had a run time of ${n^{{\rm exp}(O(\sqrt{{\rm log}\, {\rm log} n}))}}$ (Luby & Velickovic 1991, in Algorithmica 16(4/5):415–433, 1996).     

The Parallel Complexity of Graph Canonization Under Abelian Group Action

We study the problem of computing canonical forms for graphs and hypergraphs under Abelian group action and show tight complexity bounds. Our approach is algebraic. We transform the problem of computing canonical forms for graphs to the problem of computing canonical forms for associated algebraic structures, and we develop parallel algorithms for these associated problems.

1. In our first result we show that the problem of computing canonical labelings for hypergraphs of color class size 2 is logspace Turing equivalent to solving a system of linear equations over the field $\mathbb {F} _{2}$ . This implies a deterministic NC ² algorithm for the problem.
2. Similarly, we show that the problem of canonical labeling graphs and hypergraphs under arbitrary Abelian permutation group action is fairly well characterized by the problem of computing the integer determinant. In particular, this yields deterministic NC ³ and randomized NC ² algorithms for the problem.

     

Resource Trade-offs in Syntactically Multilinear Arithmetic Circuits

The class of polynomials computable by polynomial size log-depth arithmetic circuits (VNC ¹) is known to be computable by constant width polynomial degree circuits (VsSC ⁰), but whether the converse containment holds is an open problem. As a partial answer to this question, we give a construction which shows that syntactically multilinear circuits of constant width and polynomial degree can be depth-reduced, which in our notation shows that sm-VsSC ⁰ ${\subseteq}$ ? sm-VNC ¹. We further strengthen this inclusion, by giving a separate construction that provides a width-efficient simulation for constant width syntactically multilinear circuits by constant width syntactically multilinear algebraic branching programs; in our notation, sm-VsSC ⁰ ${\subseteq}$ ? sm-VBWBP. We then focus on polynomial size syntactically multilinear circuits and study relationships between classes of functions obtained by imposing various resource (width, depth, degree) restrictions on these circuits. Along the way, we also observe a characterization of the class NC ¹ in terms of a restricted class of planar branching programs of polynomial size. Finally, in contrast to the general case, we report closure and stability of coefficient functions for the syntactically multilinear classes studied in this paper.     

Small Space Analogues of Valiant’s Classes and the Limitations of Skew Formulas

In the uniform circuit model of computation, the width of a boolean circuit exactly characterizes the “space” complexity of the computed function. Looking for a similar relationship in Valiant’s algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width. We introduce the class VL as an algebraic variant of deterministic log-space L; VL is a subclass of VP. Further, to define algebraic variants of non-deterministic space-bounded classes, we introduce the notion of “read-once” certificates for arithmetic circuits. We show that polynomial-size algebraic branching programs (an algebraic analog of NL) can be expressed as read-once exponential sums over polynomials in ${{\sf VL}, {\it i.e.}\quad{\sf VBP} \in \Sigma^R \cdot {\sf VL}}$ . Thus, read-once exponential sums can be viewed as a reasonable way of capturing space-bounded non-determinism. We also show that Σ ^R ·VBP = VBP, i.e. VBPs are stable under read-once exponential sums. Though the best upper bound we have for Σ ^R ·VL itself is VNP, we can obtain better upper bounds for width-bounded multiplicatively disjoint (md-) circuits. Without the width restriction, md- arithmetic circuits are known to capture all of VP. We show that read-once exponential sums over md- constant-width arithmetic circuits are within VP and that read-once exponential sums over md- polylog-width arithmetic circuits are within VQP. We also show that exponential sums of a skew formula cannot represent the determinant polynomial.     

Amplifying circuit lower bounds against polynomial time, with applications

We give a self-reduction for the Circuit Evaluation problem (CircEval) and prove the following consequences.

1. Amplifying size–depth lower bounds. If CircEval has Boolean circuits of n ^k size and n ^1?δ depth for some k and δ, then for every ${\epsilon > 0}$ , there is a δ′ > 0 such that CircEval has circuits of ${n^{1 + \epsilon}}$ size and ${n^{1- \delta^{\prime}}}$ depth. Moreover, the resulting circuits require only ${\tilde{O}(n^{\epsilon})}$ bits of non-uniformity to construct. As a consequence, strong enough depth lower bounds for Circuit Evaluation imply a full separation of P and NC (even with a weak size lower bound).
2. Lower bounds for quantified Boolean formulas. Let c, d > 1 and e < 1 satisfy c < (1 ? e + d )/d. Either the problem of recognizing valid quantified Boolean formulas (QBF) is not solvable in TIME[n ^c ], or the Circuit Evaluation problem cannot be solved with circuits of n ^d size and n ^e depth. This implies unconditional polynomial-time uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n ^c -time uniform NC circuits, for all c < 2.

     

A Competitive Analysis for Balanced Transactional Memory Workloads

We consider transactional memory contention management in the context of balanced workloads, where if a transaction is writing, the number of write operations it performs is a constant fraction of its total reads and writes. We explore the theoretical performance boundaries of contention management in balanced workloads from the worst-case perspective by presenting and analyzing two new polynomial time contention management algorithms. We analyze the performance of a contention management algorithm by comparison with an optimal offline contention management algorithm to provide a competitive ratio. The first algorithm Clairvoyant is $O(\sqrt{s})$ -competitive, where s is the number of shared resources. This algorithm depends on explicitly knowing the conflict graph at each time step of execution. The second algorithm Non-Clairvoyant is $O(\sqrt{s} \cdot \log n)$ -competitive, with high probability, which is only a O(log?n) factor worse, but does not require knowledge of the conflict graph, where n is the number of transactions. Both of these algorithms are greedy. We also prove that the performance of Clairvoyant is close to optimal, since there is no polynomial time contention management algorithm for the balanced transaction scheduling problem that is better than $O((\sqrt{s})^{1-\varepsilon})$ -competitive for any constant ε>0, unless NP?ZPP. To our knowledge, these results are significant improvements over the best previously known O(s) competitive ratio bound.     

Degree Constrained Node-Connectivity Problems

We consider Degree Constrained Survivable Network problems. For the directed Degree Constrained k -Edge-Outconnected Subgraph problem, we slightly improve the best known approximation ratio, by a simple proof. Our main contribution is giving a framework to handle node-connectivity degree constrained problems with the iterative rounding method. In particular, for the degree constrained versions of the Element-Connectivity Survivable Network problem on undirected graphs, and of the k -Outconnected Subgraph problem on both directed and undirected graphs, our algorithm computes a solution J of cost O(logk) times the optimal, with degrees O(2 ^k )?b(v). Similar result are obtained for the k -Connected Subgraph problem. The latter improves on the only degree approximation O(klogn)?b(v) in O(n ^k ) time on undirected graphs by Feder, Motwani, and Zhu.     

Complexity of Rational and Irrational Nash Equilibria

We introduce two new natural decision problems, denoted as ? RATIONAL NASH and ? IRRATIONAL NASH, pertinent to the rationality and irrationality, respectively, of Nash equilibria for (finite) strategic games. These problems ask, given a strategic game, whether or not it admits (i) a rational Nash equilibrium where all probabilities are rational numbers, and (ii) an irrational Nash equilibrium where at least one probability is irrational, respectively. We are interested here in the complexities of ? RATIONAL NASH and ? IRRATIONAL NASH. Towards this end, we study two other decision problems, denoted as NASH-EQUIVALENCE and NASH-REDUCTION, pertinent to some mutual properties of the sets of Nash equilibria of two given strategic games with the same number of players. The problem NASH-EQUIVALENCE asks whether or not the two sets of Nash equilibria coincide; we identify a restriction of its complementary problem that witnesses ? RATIONAL NASH. The problem NASH-REDUCTION asks whether or not there is a so called Nash reduction: a suitable map between corresponding strategy sets of players that yields a Nash equilibrium of the former game from a Nash equilibrium of the latter game; we identify a restriction of NASH-REDUCTION that witnesses ? IRRATIONAL NASH. As our main result, we provide two distinct reductions to simultaneously show that (i) NASH-EQUIVALENCE is co- $\mathcal{NP}$ -hard and ? RATIONAL NASH is $\mathcal{NP}$ -hard, and (ii) NASH-REDUCTION and ? IRRATIONAL NASH are both $\mathcal{NP}$ -hard, respectively. The reductions significantly extend techniques previously employed by Conitzer and Sandholm (Proceedings of the 18th Joint Conference on Artificial Intelligence, pp. 765–771, 2003; Games Econ. Behav. 63(2), 621–641, 2008).     

Detecting Fixed Patterns in Chordal Graphs in Polynomial Time

The Contractibility problem takes as input two graphs G and H, and the task is to decide whether H can be obtained from G by a sequence of edge contractions. The Induced Minor and Induced Topological Minor problems are similar, but the first allows both edge contractions and vertex deletions, whereas the latter allows only vertex deletions and vertex dissolutions. All three problems are NP-complete, even for certain fixed graphs H. We show that these problems can be solved in polynomial time for every fixed H when the input graph G is chordal. Our results can be considered tight, since these problems are known to be W[1]-hard on chordal graphs when parameterized by the size of H. To solve Contractibility and Induced Minor, we define and use a generalization of the classic Disjoint Paths problem, where we require the vertices of each of the k paths to be chosen from a specified set. We prove that this variant is NP-complete even when k=2, but that it is polynomial-time solvable on chordal graphs for every fixed k. Our algorithm for Induced Topological Minor is based on another generalization of Disjoint Paths called Induced Disjoint Paths, where the vertices from different paths may no longer be adjacent. We show that this problem, which is known to be NP-complete when k=2, can be solved in polynomial time on chordal graphs even when k is part of the input. Our results fit into the general framework of graph containment problems, where the aim is to decide whether a graph can be modified into another graph by a sequence of specified graph operations. Allowing combinations of the four well-known operations edge deletion, edge contraction, vertex deletion, and vertex dissolution results in the following ten containment relations: (induced) minor, (induced) topological minor, (induced) subgraph, (induced) spanning subgraph, dissolution, and contraction. Our results, combined with existing results, settle the complexity of each of the ten corresponding containment problems on chordal graphs.     

Approximating Node-Connectivity Augmentation Problems

We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J=(V,E _J ) and connectivity requirements $\{r(u,v): u,v \in V\}$ , find a minimum size set I of new edges (any edge is allowed) such that the graph J∪I contains r(u,v) internally-disjoint uv-paths, for all u,v∈V. In Rooted NCA there is s∈V such that r(u,v)>0 implies u=s or v=s. For large values of k=max? _u,v∈V r(u,v), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit an approximation ratio polylogarithmic in k. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(kln?n) for NCA and O(ln?n) for Rooted NCA. In this paper we give an approximation algorithm with ratios O(kln?² k) for NCA and O(ln?² k) for Rooted NCA. This is the first approximation algorithm with ratio independent of?n, and thus is a constant for any fixed k. Our algorithm is based on the following new structural result which is of independent interest. If $\mathcal{D}$ is a set of node pairs in a graph?J, then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in $\mathcal{D}$ is O(? ²), where ? is the maximum connectivity in J of a pair in $\mathcal{D}$ .     

Prehistoric Graph in Modal Derivations and Self-Referentiality

By terms-allowed-in-formulas capacity, Artemov’s Logic of Proofs LP Artemov includes self-referential formulas of the form t:?(t) that play a crucial role in the realization of modal logic S4 in LP, and in the Brouwer–Heyting–Kolmogorov semantics of intuitionistic logic via LP. In an earlier work appeared in the Proceedings of CSR 2010 the author defined prehistoric loop in a sequent calculus of S4, and verified its necessity to self-referentiality in S4?LP realization. In this extended version we generalize results there to T and K4, two modal logics smaller than S4 that yet call for self-referentiality in their realizations into corresponding justification logics JT and J4.     

ATP and Presentation Service for Mizar Formalizations

This paper describes the Automated Reasoning for Mizar ( $\textsf{Miz}\mathbb{AR}$ ) service, which integrates several automated reasoning, artificial intelligence, and presentation tools with Mizar and its authoring environment. The service provides ATP assistance to Mizar authors in finding and explaining proofs, and offers generation of Mizar problems as challenges to ATP systems. The service is based on a sound translation from the Mizar language to that of first-order ATP systems, and relies on the recent progress in application of ATP systems in large theories containing tens of thousands of available facts. We present the main features of $\textsf{Miz}\mathbb{AR}$ services, followed by an account of initial experiments in finding proofs with the ATP assistance. Our initial experience indicates that the tool offers substantial help in exploring the Mizar library and in preparing new Mizar articles.     

Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded-genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in $\mathcal{O}(n \log n)$ time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS framework from planar graphs to bounded-genus graphs: any problem that is shown to be approximable by the planar PTAS framework of Borradaile et al. (ACM Trans. Algorithms 5(3), 2009) will also be approximable in bounded-genus graphs by our extension.     

Computing Sparse Multiples of Polynomials

We consider the problem of finding a sparse multiple of a polynomial. Given f??F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h??F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=? and t is constant, we give an algorithm which requires polynomial-time in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.     

Efficient Fully-Compressed Sequence Representations

We present a data structure that stores a sequence s[1..n] over alphabet [1..σ] in $n\mathcal{H}_{0}(s) + o(n)(\mathcal {H}_{0}(s){+}1)$ bits, where $\mathcal{H}_{0}(s)$ is the zero-order entropy of s. This structure supports the queries access, rank and select, which are fundamental building blocks for many other compressed data structures, in worst-case time ${\mathcal{O} ( {\lg\lg\sigma} )}$ and average time ${\mathcal{O} ( {\lg\mathcal{H}_{0}(s)} )}$ . The worst-case complexity matches the best previous results, yet these had been achieved with data structures using $n\mathcal{H}_{0}(s)+o(n\lg \sigma)$ bits. On highly compressible sequences the o(nlgσ) bits of the redundancy may be significant compared to the $n\mathcal{H}_{0}(s)$ bits that encode the data. Our representation, instead, compresses the redundancy as well. Moreover, our average-case complexity is unprecedented. Our technique is based on partitioning the alphabet into characters of similar frequency. The subsequence corresponding to each group can then be encoded using fast uncompressed representations without harming the overall compression ratios, even in the redundancy. The result also improves upon the best current compressed representations of several other data structures. For example, we achieve (i) compressed redundancy, retaining the best time complexities, for the smallest existing full-text self-indexes; (ii) compressed permutations π with times for π() and π ^?1() improved to loglogarithmic; and (iii) the first compressed representation of dynamic collections of disjoint sets. We also point out various applications to inverted indexes, suffix arrays, binary relations, and data compressors. Our structure is practical on large alphabets. Our experiments show that, as predicted by theory, it dominates the space/time tradeoff map of all the sequence representations, both in synthetic and application scenarios.