1 Defining Network Flow A flow network is a directed graph G = (V,E) in which each edge (u, v) ∈ E has non-negative capacity c(u, v) ≥ 0. We require that if (u, v) ∈ E, then (v, u) / ∈ E. That is, if an edge exists, then the edge between the same vertices going the reverse direction does not exist. Every flow network has a source s and a sink t, and we assume that for every v ∈ V , there is some path s→ · · · → v → · · · → t. Note that this implies that flow networks are connected. Informally, the intuition behind network flow is to think of the edges as pipes and the weights on the edges as the capacity its corresponding pipe per unit of time. The question we are trying to ask is: how many units of water can we send from the source to the sink per unit of time? Formally, a flow in G is a function f : V × V → R that satisfies the following: • Capacity constraint. For all u, v ∈ V , we require 0 ≤ f(u, v) ≤ c(u, v). Our pipe cannot hold more than is allowed as dictated by its capacity. • Flow conservation. For u ∈ V − {s, t}, we require ∑

Network Flows

A program and underlying algorithms are described which quasi-optimally partition a set of logic equations into subsets which represent physical packaging entities. The criterion for partitioning is the signal communication between equations. Several different methods of measuring and evaluating the criterion are built into the program as optional strategies. The effectiveness of these and other strategies are compared, based on accumulated experience with the program. The important characteristics of the program and its data structures which allow very efficient computer execution time are described. The plans and techniques for further development of the program are also presented.

Logic partitioning

We present a simple and fast deterministic algorithm for the minimum k-way cut problem in a capacitated graph, that is, finding a set of edges with minimum total capacity whose removal splits the graph into at least k components. The algorithm packs O(mk3 log n) trees. Each new tree is a minimal spanning tree with respect to the edge utilizations, and the utilization of an edge is the number of times it has been used in previous spanning trees divided by its capacity. We prove that each minimum k-way cut is crossed at most 2k-2 times by one of the trees. We can enumerate all such cuts in ~O(n2k) time, which is hence the running time of our algorithm producing all minimum k-way cuts. The previous fastest deterministic algorithm of Kamidoi et al. [SICOMP'06] took O(n(4+o(1))k) time, so this is a near-quadratic improvement. Moreover, we essentially match the O(n(2-o(1))k) running time of the Monto Carlo (no correctness guarantee) randomized algorithm of Karger and Stein [JACM'96].

Minimum k-way cuts via deterministic greedy tree packing

We consider the minimum $k$-way cut problem for unweighted undirected graphs with a size bound $s$ on the number of cut edges allowed. Thus we seek to remove as few edges as possible so as to split a graph into $k$ components, or report that this requires cutting more than $s$ edges. We show that this problem is fixed-parameter tractable (FPT) with the standard parameterization in terms of the solution size $s$. More precisely, for $s=O(1)$, we present a quadratic time algorithm. Moreover, we present a much easier linear time algorithm for planar graphs and bounded genus graphs. Our tractability result stands in contrast to known W[1] hardness of related problems. Without the size bound, Downey et al.~[2003] proved that the minimum $k$-way cut problem is W[1] hard with parameter $k$, and this is even for simple unweighted graphs. Downey et al.~asked about the status for planar graphs. We get linear time with fixed parameter $k$ for simple planar graphs since the minimum $k$-way cut of a planar graph is of size at most $6k$. More generally, we get FPT with parameter $k$ for any graph class with bounded average degree. A simple reduction shows that vertex cuts are at least as hard as edge cuts, so the minimum $k$-way vertex cut is also W[1] hard with parameter $k$. Marx [2004] proved that finding a minimum $k$-way vertex cut of size $s$ is also W[1] hard with parameter $s$. Marx asked about the FPT status with edge cuts, which we prove tractable here. We are not aware of any other cut problem where the vertex version is W[1] hard but the edge version is FPT, e.g., Marx [2004] proved that the $k$-terminal cut problem is FPT parameterized by the cut size, both for edge and vertex cuts.

The Minimum k-way Cut of Bounded Size is Fixed-Parameter Tractable

Let $G=(V,E)$ be an edge-weighted undirected graph with $n$ vertices and $m$ edges. We present a deterministic algorithm to compute a minimum $k$-way cut of $G$ for a given $k$. Our algorithm is a divide-and-conquer method based on a procedure that reduces an instance of the minimum $k$-way cut problem to $O(n^{2k-5})$ instances of the minimum $(\lfloor (k+\sqrt{k})/2\rfloor+1)$-way cut problem, and can be implemented to run in $O(n^{4k/(1-1.71/\sqrt{k}) -31} )$ time. With a slight modification, the algorithm can find all minimum $k$-way cuts in $O(n^{4k/(1-1.71/\sqrt{k}) -16} )$ time.

A Deterministic Algorithm for Finding All Minimum $k$-Way Cuts

This paper presents algorithms for computing a minimum 3 -way cut and a minimum 4 -way cut of an undirected weighted graph G . Let G=(V, E) be an undirected graph with n vertices, m edges, and positive edge weights. Goldschmidt and Hochbaum presented an algorithm for the minimum k -way cut problem with fixed k , that requires O(n
 4
 ) and O(n
 6
 ) maximum flow computations, respectively, to compute a minimum 3 -way cut and a minimum 4 -way cut of G . In this paper we first show some properties on minimum 3 -way cuts and minimum 4 -way cuts, which indicate a recursive structure of the minimum k -way cut problem when k = 3 and 4 . Then, based on those properties, we give divide-and-conquer algorithms for computing a minimum 3 -way cut and a minimum 4 -way cut of G , which require O(n
 3
 ) and O(n
 4
 ) maximum flow computations, respectively.

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A Divide-and-Conquer Approach to the Minimum k -Way Cut Problem

This paper proposes a new approach that makes it possible for every undergraduate student to perform experiments of developing a pipelined RISC processor within limited time available for the course. The approach consists of 4 steps; at the first step, modeling of pipelined RISC processor is simplified by avoiding structural hazard and by ignoring other hazards, and in the succeeding steps, students learn difficulties of pipelining by themselves. An educational FPGA board ASAver.1 and results of feasibility study are also shown.

ASAver.1 : An FPGA-Based Education Board for Computer Architecture/System Design (Special Section on VLSI Design and CAD Algorithms)

Minimum cost subpartitions in graphs

In recent years, blockchain is utilized practically as a distributed secure digital ledger of some sorts of transactions. Blockchain is regarded as one of the most important next generation infrastructure technologies of the financial industry, as well as artificial intelligence and big data. In 2020, cryptocurrencies based on blockchain, such as Bitcoin, Ethereum, or XRP, have a value of more than $450 billion in the market capitalization. Furthermore, on blockchains such as Ethereum, transactions can also represent automatic executions of programs, which are called smart contracts. Thus, many institutes in various categories show their positive attitude toward processing financial transactions or non-financial contracts on blockchain. Although many researchers have studied for various types of issues on blockchain, there always exist security and privacy concerns for blockchain. In this paper, we point out a new concern for abusing the publicity of blockchain and also show the possibility of suspicions aroused by the concern. Then we propose a selective mechanism for self-protecting against the approach from crimes or computer viruses on blockchain, whether the disclosure of user’s privacy occurs or not. Next, we also propose a concrete implementation of our proposed selective mechanism with two new address types. We aim to incorporate the mechanism in Bitcoin Core, which is the official Bitcoin client software, and using libbitcoin library functions for Bitcoin software development. We show experimental results to estimate overhead costs for processing our proposed address types toward processing the current standard address type in nodes on the peer-to-peer network.

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Yoko Kamidoi

Papers

A Deterministic Algorithm for Finding All Minimum $k$-Way Cuts

A Divide-and-Conquer Approach to the Minimum k -Way Cut Problem

ASAver.1 : An FPGA-Based Education Board for Computer Architecture/System Design (Special Section on VLSI Design and CAD Algorithms)

Minimum cost subpartitions in graphs

A Protocol for Preventing Transaction Commitment Without Recipient’s Authorization on Blockchain and It’s Implementation