On Transformations of Interactive Proofs that Preserve the Prover's Complexity

Goldwasser and Sipser [GS89] proved that every interactive proof system can be transformed into a public-coin one (a.k.a., an Arthur-Merlin game). Their transformation has the drawback that the computational complexity of the prover's strategy is not preserved. We show that this is inherent, by proving that the same must be true of any transformation which only uses the original prover and verifier strategies as "black boxes". Our negative result holds even if the original proof system is restricted to be honest-verifier perfect zero knowledge and the transformation can also use the simulator as a black box. We also examine a similar deficiency in a transformation of Fürer et al. [FGM+89] from interactive proofs to ones with perfect completeness. We argue that the increase in prover complexity incurred by their transformation is necessary, given that their construction is a black-box transformation which works regardless of the verifier's computational complexity.Engineering and Applied Science

Probabilistic Proof Systems

Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems -- interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs -- stressing the essential role of randomness in each of them. This exposition is an expanded version of a survey written for the proceedings of the International Congress of Mathematicians (ICM94) held in Zurich in 1994. It is hope that this exposition may be accessible to a broad audience of computer scientists and mathematians

On Efficiently Solvable Cases of Quantum k-SAT

The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k >= 3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. Whereas k-SAT has been well-studied for special tractable cases, as well as from a parameterized complexity perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open. Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two clauses. (2) We give a parameterized algorithm for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry; we hope these prove useful elsewhere

The (Im)possibility of Simple Search-To-Decision Reductions for Approximation Problems

We study the question of when an approximate search optimization problem is harder than the associated decision problem. Specifically, we study a natural and quite general model of black-box search-to-decision reductions, which we call branch-and-bound reductions (in analogy with branch-and-bound algorithms). In this model, an algorithm attempts to minimize (or maximize) a function f: D ? ?_{? 0} by making oracle queries to h_f : ? ? ?_{? 0} satisfying min_{x ? S} f(x) ? h_f(S) ? ? ? min_{x ? S} f(x) (*) for some ? ? 1 and any subset S in some allowed class of subsets ? of the domain D. (When the goal is to maximize f, h_f instead yields an approximation to the maximal value of f over S.) We show tight upper and lower bounds on the number of queries q needed to find even a ?\u27-approximate minimizer (or maximizer) for quite large ?\u27 in a number of interesting settings, as follows. - For arbitrary functions f : {0,1}? ? ?_{? 0}, where ? contains all subsets of the domain, we show that no branch-and-bound reduction can achieve ?\u27 ? ?^{n/log q}, while a simple greedy approach achieves essentially ?^{n/log q}. - For a large class of MAX-CSPs, where ? := {S_w} contains each set of assignments to the variables induced by a partial assignment w, we show that no branch-and-bound reduction can do significantly better than essentially a random guess, even when the oracle h_f guarantees an approximation factor of ? ? 1+?{log(q)/n}. - For the Traveling Salesperson Problem (TSP), where ? := {S_p} contains each set of tours extending a path p, we show that no branch-and-bound reduction can achieve ?\u27 ? (?-1) n/log q. We also prove a nearly matching upper bound in our model. These results show an oracle model in which approximate search and decision are strongly separated. (In particular, our result for TSP can be viewed as a negative answer to a question posed by Bellare and Goldwasser (SIAM J. Comput. 1994), though only in an oracle model.) We also note two alternative interpretations of our results. First, if we view h_f as a data structure, then our results unconditionally rule out black-box search-to-decision reductions for certain data structure problems. Second, if we view h_f as an efficiently computable heuristic, then our results show that any reasonably efficient branch-and-bound algorithm requires more guarantees from its heuristic than simply Eq. (*). Behind our results is a "useless oracle lemma," which allows us to argue that under certain conditions the oracle h_f is "useless," and which might be of independent interest. See also the full version [Alexander Golovnev et al., 2022]

The Journey from NP to TFNP Hardness

The class TFNP is the search analog of NP with the additional guarantee that any instance has a solution. TFNP has attracted extensive attention due to its natural syntactic subclasses that capture the computational complexity of important search problems from algorithmic game theory, combinatorial optimization and computational topology. Thus, one of the main research objectives in the context of TFNP is to search for efficient algorithms for its subclasses, and at the same time proving hardness results where efficient algorithms cannot exist. Currently, no problem in TFNP is known to be hard under assumptions such as NP hardness, the existence of one-way functions, or even public-key cryptography. The only known hardness results are based on less general assumptions such as the existence of collision-resistant hash functions, one-way permutations less established cryptographic primitives (e.g. program obfuscation or functional encryption). Several works explained this status by showing various barriers to proving hardness of TFNP. In particular, it has been shown that hardness of TFNP hardness cannot be based on worst-case NP hardness, unless NP=coNP. Therefore, we ask the following question: What is the weakest assumption sufficient for showing hardness in TFNP? In this work, we answer this question and show that hard-on-average TFNP problems can be based on the weak assumption that there exists a hard-on-average language in NP. In particular, this includes the assumption of the existence of one-way functions. In terms of techniques, we show an interesting interplay between problems in TFNP, derandomization techniques, and zero-knowledge proofs

Many-one reductions and the category of multivalued functions

Multi-valued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees). A more general understanding is possible, if the category-theoretic properties of multi-valued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multi-valued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch). Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multi-valued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multi-valued functions.Comment: an earlier version was titled "Many-one reductions between search problems". in Mathematical Structures in Computer Science, 201