1. Let { a b) For each w wi wn E define wR the everse of w. to be wn ay, with the special case e-e. For example, (abaab-baaba Consider the language( twwR E . ) To tacil tate the detection of the lef rnost tape cell every input is presunned to begin with # This language can be decided by an NTM mplementing the following algorithm. i. Scan the input string from left to right and nondeterministically select ("guess") the rightmost symbol of w. 11. Starting rom the gh most symbol ofw selected 1n step 1 cross off a matching pair of a s or o bs n inside out manner by moving the hea ack an for or example, #abaab aaba woul lead to #abaaxxaaba, #abaxxxxaba. #abxxx xxxba, so on. In sufficient detail, concisely describe a verification TM that is essentially equivalent to this NTM. Describe the nature of a proposed certificate and how it is used 2. In sufficient detail, describe a polynomial-time nondeterministic algorithm to decide each of the following problems: o VERTEX COVER o SUBSET-SUM Your algorithm should explicitly use nondeterministic choices at branching points, not one that generates a proposed certificate nondeterministically. Show that the worst-case runtime of your algorithm is polynomially bounded. You may illustrate the algorithms with help of small examples. 1. Let { a b) For each w wi wn E define wR the everse of w. to be wn ay, with the special case e-e. For example, (abaab-baaba Consider the language( twwR E . ) To tacil tate the detection of the lef rnost tape cell every input is presunned to begin with # This language can be decided by an NTM mplementing the following algorithm. i. Scan the input string from left to right and nondeterministically select ("guess") the rightmost symbol of w. 11. Starting rom the gh most symbol ofw selected 1n step 1 cross off a matching pair of a s or o bs n inside out manner by moving the hea ack an for or example, #abaab aaba woul lead to #abaaxxaaba, #abaxxxxaba. #abxxx xxxba, so on. In sufficient detail, concisely describe a verification TM that is essentially equivalent to this NTM. Describe the nature of a proposed certificate and how it is used 2. In sufficient detail, describe a polynomial-time nondeterministic algorithm to decide each of the following problems: o VERTEX COVER o SUBSET-SUM Your algorithm should explicitly use nondeterministic choices at branching points, not one that generates a proposed certificate nondeterministically. Show that the worst-case runtime of your algorithm is polynomially bounded. You may illustrate the algorithms with help of small examples