On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the F -Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k , F -Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and | X | ≤ k . Here, G/X is the graph obtained from G by contracting edges in X . We obtain the following results for the F - Contraction problem: • Clique Contraction is known to be FPT . However, unless NP⊆ coNP/ poly , it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme ( PSAKS ). That is, it admits a (1 + ε)-approximate kernel with O ( k f(ε)) vertices for every ε > 0. • Split Contraction is known to be W[1]-Hard . We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)- FPT -approximation algorithm for Split Contraction ). Furthermore, we show that, assuming Gap-ETH , there is no (5/4-δ)- FPT -approximation algorithm for Split Contraction . Here, ε, δ > 0 are fixed constants. • Chordal Contraction is known to be W[2]-Hard . We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1] , there is no F(k) - FPT -approximation algorithm for Chordal Contraction . Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k) - FPT -approximation algorithm for the F -Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and | X | ≤ k , it outputs an edge set Y of size at most h(k) ċ k for which G/Y is in F .

List Approximation for Increasing Kolmogorov Complexity

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. However, is it possible to construct a few strings, no longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that takes as input a string x of length n and returns a list with O(n2) strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than n−loglogn−O(1). We also present an algorithm that obtains a list of quasi-polynomial size in which each element can be produced in polynomial time.

Lower Bounds on OBDD Proofs with Several Orders

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

Double-authentication-preventing signatures in the standard model

A double-authentication preventing signature (DAPS) scheme is a digital signature scheme equipped with a self-enforcement mechanism. Messages consist of an address and a payload component, and a signer is penalized if she signs two messages with the same addresses but different payloads. The penalty is the disclosure of the signer’s signing key. Most of the existing DAPS schemes are proved secure in the random oracle model (ROM), while the efficient ones in the standard model only support address spaces of polynomial size. We present DAPS schemes that are efficient, secure in the standard model under standard assumptions and support large address spaces. Our main construction builds on vector commitments (VC) and double-trapdoor chameleon hash functions (DCH). We also provide a DAPS realization from Groth–Sahai (GS) proofs that builds on a generic construction by Derler et al., which they instantiate in the ROM. The GS-based construction, while less efficient than our main one, shows that a general yet efficient instantiation of DAPS in the standard model is possible. An interesting feature of our main construction is that it can be easily modified to guarantee security even in the most challenging setting where no trusted setup is provided. To the best of our knowledge, ours seems to be the first construction achieving this in the standard model.

Quasi-Popular Matchings, Optimality, and Extended Formulations

Let [Formula: see text] be an instance of the stable marriage problem in which every vertex ranks its neighbors in a strict order of preference. A matching [Formula: see text] in [Formula: see text] is popular if [Formula: see text] does not lose a head-to-head election against any matching. Popular matchings generalize stable matchings. Unfortunately, when there are edge costs, to find or even approximate up to any factor a popular matching of minimum cost is NP-hard. Let [Formula: see text] be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most [Formula: see text] by paying the price of mildly relaxing popularity. Our main positive results are two bicriteria algorithms that find in polynomial time a “quasi-popular” matching of cost at most [Formula: see text]. Moreover, one of the algorithms finds a quasi-popular matching of cost at most that of a min-cost popular fractional matching, which could be much smaller than [Formula: see text]. Key to the other algorithm is a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity.

Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials

Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs in high orders. In particular, this paper discusses the development of a new algorithm and presents a new mathematical model for computing the initial value of the KP parameter. In addition, a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal recurrence algorithm was derived from the existing n direction and x direction recurrence algorithms. The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP coefficients. First, the KP coefficients were computed for one partition after dividing the KP plane into four. To compute the KP coefficients in the other partitions, the symmetry relations were exploited. The performance evaluation of the proposed recurrence algorithm was determined through different comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polynomial size, and computation cost. The obtained results indicate that the proposed algorithm is reliable and computes lesser coefficients when compared to the existing algorithms across wide ranges of parameter values of p and polynomial sizes N. The results also show that the improvement ratio of the computed coefficients ranges from 18.64% to 81.55% in comparison to the existing algorithms. Besides this, the proposed algorithm can generate polynomials of an order ∼8.5 times larger than those generated using state-of-the-art algorithms.

On the Power of Symmetric Linear Programs

We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs, while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991), showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.

Small Circuits and Dual Weak PHP in the Universal Theory of p-time Algorithms

We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending bits to bits violates the dual weak pigeonhole principle: Every string equals the value of the function for some . The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size .