The pathwidth of a graph is a measure of how path-like the graph is. The Pathwidth One Vertex Deletion (POVD) problem asks whether, given an undirected graph G and an integer k, one can delete at most k vertices from G so that the remaining graph has pathwidth at most one. This is a natural variation of the classical Feedback vertex Set (FVS) problem, where the deletion of at most k vertices results in a graph of treewidth at most one. In this work, we investigate POVD in the realm of approximation algorithms. We first design a 3-approximation algorithm for POVD running in polynomial time. Then, using this constant factor approximation algorithm, we obtain a randomized parameterized approximation algorithm for POVD running in time O*(( h(beta))(k)), that improves the fastest existing running times for approximation ratios in the range (1.76147, 3). Here the constant hss depends on the approximation factor beta alone and has value 2((3-beta)), which lies in the range (1, 2.3596), when beta is an element of(1.76147, 3). Taking inspiration from two extensively studied problems, namely Connected FVS and Independent FVS, we investigate two variations of the POVD problem from the perspective of parameterized algorithms. These variations are the connected variant, called Connected pathwidth One Vertex Deletion (CPOVD) and the independent variant, called Independent Pathwidth One Vertex Deletion (IPOVD). While in CPOVD the subgraph G[S] induced by the vertices to be deleted needs to be connected, in IPOVD it needs to be independent. Specifically, we show the following results. = CPOVD can be solved in O *(14(k)) time and admits no polynomial kernel unless NP subset of co-NP/poly. = IPOVD can be solved in O *(7(k)) time, and admits a kernel of size O(k(3)).
We study the public-coin randomized communication complexity of the equality function. The communication complexity of this function is known to be low when the error probability is constant and the players have access to many random bits. The complexity grows, however, if the allowed error probability and the amount of randomness are restricted. We show that public-coin randomized protocols for equality and error-correcting codes are essentially the same object. That is, given a protocol for equality, we can construct a code, and vice versa. We substantially extend the protocol-implies-code direction: any protocol computing a function with a large fooling set can be converted into an error-correcting code. As a corollary, we show that among functions with a fooling set of size s, equality on log s bits has the least randomized communication complexity, regardless of the restrictions on the error probability and the amount of randomness. Finally, we use the connection to error-correcting codes to analyze the randomized communication complexity of equality for varying restrictions on the error probability and the amount of randomness. In most cases, we provide tight bounds. We pinpoint the setting in which tight bounds are still unknown.
With the recent advent of exploits like Spectre and Meltdown, the mitigation of side-channel attacks has become an important concern for security researchers. In this paper, we focus on timing-based side channels introduced through conditional branching on secret information within programs. We introduce a language that allows a programmer to write conditionals branching on secrets within its syntax, but has a semantics that keeps execution time constant with respect to an adversary under an observationally equivalent memory. We differ from other approaches that use program analysis methods, opting instead to modify the operational semantics to enforce the necessary properties. We formalize the semantics for our language with timing leak mitigations in Rocq (previously, Coq) and prove that these semantics satisfy the property of timing-sensitive non-interference. Since our system describes a mitigation approach for timing leaks in a general high-level imperative language, we believe that our semantics can be used as a basis for compiler construction for other high-level imperative languages that seek to be safe from timing side channels.
We present a faster algorithm for finding a minimum dijoin, a smallest set of edges whose contraction makes a directed graph strongly connected. This problem has been studied since the 1960s [Seshu and Reed 1961] and is dual to finding a maximum sized family of disjoint dicuts [Lucchesi and Younger 1978]. Given a directed graph G with n vertices and m edges whose minimum dijoin has size d, our algorithm outputs both a minimum dijoin and a maximum sized family of disjoint dicuts in O(TC center dot d) time, where TC = min( mn, n(omega)) is the time to compute the transitive closure. This improves upon the state of the art of [Gabow 1993], which requires O(TC center dot min(m(1/2), n(2/3))) time when d = o(min(m(1/2), n(2/3))). Our result extends to finding a minimum weighted dijoin. We achieve this by observing that Frank ' s algorithm [Frank 1981] can be sped up when warm-started with a 2-approximation solution, which we observed can be computed in near-linear time.
The paper considers the problem of file sharing among peers who are connected to a common core network through links of differing upload and download capacities, as is the case in networks provisioned according to the hose model. The file is assumed to be divided into equal-sized chunks, and a peer can start sending a "chunk" of the file to another peer only after it has received the entire chunk. The objective is to share a chunk, initially residing on one of the peers, with all other peers in the least time possible. Peers can simultaneously send/receive parts of a chunk to/from multiple peers, subject to the upload and download capacity constraints. We only consider the problem of broadcasting one chunk to all peers. We consider two different models - in the migratory model, a peer can receive the chunk from multiple peers, while in the non-migratory model, any peer can receive the chunk only from one peer. For the migratory model, introduced in this paper, we show a novel integer program and use the optimum solution to the LP-relaxation to give a schedule with makespan e(1/e)OPT + P where P is the time required by the slowest peer to download the chunk. Minimising makespan in the non-migratory model is known to be NP-hard. We give a solution with makespan 18OPT + P and this is the first approximation algorithm for heterogeneous and asymmetric upload/download capacities. We also consider 2 special cases. For uniform download capacities, we obtain a solution with makespan 2OPT extending a result due to Liu [12]. For uniform upload capacities, we give the first approximation algorithm, producing makespan at most 2OPT+ 2P. 2012 ACM Subject Classification Theory of computation -> Design and analysis of algorithms
